Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion

Abstract

First, we revisit basic theory of functional Itô/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness.

1 Introduction

The present work collects several results obtained in the papers [9, 10], focusing on the study of some specific examples and particular cases, for which an ad hoc analysis is developed. This work is an improved version of [8], trying to explain more precisely some details. For example, in [8] a slightly more restrictive definition of strong-viscosity solution was adopted, see Remark 12.

Recently, a new branch of stochastic calculus has appeared, known as functional Itô calculus, which results to be an extension of classical Itô calculus to functionals depending on the entire path of a stochastic process and not only on its current value, see Dupire [17], Cont and Fournié [5–7]. Independently, Di Girolami and Russo, and more recently Fabbri, Di Girolami, and Russo, have introduced a stochastic calculus via regularizations for processes taking values in a separable Banach space B (see [12–16]), including the case $B = C([-T,0])$, which concerns the applications to the path-dependent calculus.

In the first part of the present paper, we follow [9] and revisit functional Itô calculus by means of stochastic calculus via regularization. We recall that Cont and Fournié [5–7] developed functional Itô calculus and derived a functional Itô’s formula using discretization techniques of Föllmer [23] type, instead of regularization techniques, which in our opinion, better fit to the notion of derivative. Let us illustrate another difference with respect to [5]. One of the main issues of functional Itô calculus is the definition of the functional (or pathwise) derivatives, i.e., the horizontal derivative (calling in only the past values of the trajectory) and the vertical derivative (calling in only the present value of the trajectory). In [5], it is essential to consider functionals defined on the space of càdlàg trajectories, since the definition of functional derivatives necessitates of discontinuous paths. Therefore, if a functional is defined only on the space of continuous trajectories (because, e.g., it depends on the paths of a continuous process as Brownian motion), we have to extend it anyway to the space of càdlàg trajectories, even though, in general, there is no unique way to extend it. In contrast to this approach, we introduce a new space larger than the space of continuous trajectories $C([-T,0])$, denoted by $\mathscr {C}([-T,0])$, which allows us to define functional derivatives. $\mathscr {C}([-T,0])$ is the space of bounded trajectories on $[-T,0]$, continuous on $[-T,0[$ and with possibly a jump at 0. We endow $\mathscr {C}([-T,0])$ with a topology such that $C([-T,0])$ is dense in $\mathscr {C}([-T,0])$ with respect to this topology. Therefore, any functional ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$, continuous with respect to the topology of $\mathscr {C}([-T,0])$, admits a unique extension to $\mathscr {C}([-T,0])$, denoted $u:[0,T]\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$. We present some significant functionals for which a continuous extension exists. Then, we develop the functional Itô calculus for $u:[0,T]\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$.

Notice that we use a slightly different notation compared with [5]. In particular, in place of a map ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$, in [5] a family of maps $F=(F_t)_{t\in [0,T]}$, with $F_t:C([0,t])\rightarrow {\mathbb {R}}$, is considered. However, we can always move from one formulation to the other. Indeed, given $F=(F_t)_{t\in [0,T]}$, where each $F_t:C([0,t])\rightarrow {\mathbb {R}}$, we can define ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ as follows:

$$ {\mathscr {U}}(t,\eta ) \ := \ F_t(\eta (\cdot +T)|_{[0,t]}), \qquad (t,\eta )\in [0,T]\times C([-T,0]). $$

Vice-versa, let ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ and define $F=(F_t)_{t\in [0,T]}$ as

$$\begin{aligned} F_t(\tilde{\eta }) \ := \ {\mathscr {U}}(t,\eta ), \qquad (t,\tilde{\eta })\in [0,T]\times C([0,t]), \end{aligned}$$

(1)

where $\eta$ is the element of $C([-T,0])$ obtained from $\tilde{\eta }$ firstly translating $\tilde{\eta }$ on the interval $[-t,0]$, then extending it in a constant way up to $-T$, namely $\eta (x) := \tilde{\eta }(x\,+\,t)1_{[-t,0]}(x)\,+\,\tilde{\eta }(-t)1_{[-T,-t)}(x)$, for any $x\in [-T,0]$. Observe that, in principle, the map ${\mathscr {U}}$ contains more information than F, since in (1) we do not take into account the values of ${\mathscr {U}}$ at $(t,\eta )\in [0,T]\times C([-T,0])$ with $\eta$ not constant on the interval $[-T,-t]$. Despite this, the equivalence between the two notations is guaranteed; indeed, when we consider the composition of ${\mathscr {U}}$ with a stochastic process, this extra information plays no role. Our formulation has two advantages. Firstly, we can work with a single map instead of a family of maps. In addition, the time variable and the path have two distinct roles in our setting, as for the time variable and the space variable in the classical Itô calculus. This, in particular, allows us to define the horizontal derivative independently of the time derivative, so that, the horizontal derivative defined in [5] corresponds to the sum of our horizontal derivative and of the time derivative. We mention that an alternative approach to functional derivatives was introduced in [1].

We end the first part of the paper showing how our functional Itô’s formula is strictly related to the Itô’s formula derived in the framework of Banach space valued stochastic calculus via regularization, for the case of window processes. This new branch of stochastic calculus has been recently conceived and developed in many directions in [12, 14–16]; for more details see [13]. For the particular case of window processes, we also refer to Theorem 6.3 and Sect. 7.2 in [12]. In the present paper, we prove formulae which allow to express functional derivatives in terms of differential operators arising in the Banach space valued stochastic calculus via regularization, with the aim of identifying the building blocks of our functional Itô’s formula with the terms appearing in the Itô’s formula for window processes.

Dupire [17] introduced also the concept of path-dependent partial differential equation, to which the second part of the present paper is devoted. Di Girolami and Russo, in Chap. 9 of [13], considered existence of regular solutions associated with a path dependent heat equation (which is indeed the Kolmogorov equation related to window Brownian motion) with a Fréchet smooth final condition. This was performed in the framework of Banach space valued calculus, for which we refer also to [22]. A flavour of the notion of regular solution in the Banach space framework, appeared in Chap. IV of [30] which introduced the notion of weak infinitesimal generator (in some weak sense) of the window Brownian motion and more general solutions of functional dependent stochastic differential equations. Indeed, the monograph [30] by Mohammed constitutes an excellent early contribution to the theory of this kind of equations.

We focus on semilinear parabolic path-dependent equations associated to the window Brownian motion. For more general equations we refer to [9] (for strict solutions) and to [10] (for strong-viscosity solutions). First, we consider regular solution, which we call strict solutions, in the framework of functional Itô calculus. We prove a uniqueness result for this kind of solution, showing that, if a strict solution exists, then it can be expressed through the unique solution to a certain backward stochastic differential equation (BSDE). Then, we prove an existence result for strict solutions.

However, this notion of solution turns out to be unsuitable to deal with all significant examples. As a matter of fact, if we consider the path-dependent PDE arising in the hedging problem of lookback contingent claims, we can not expect too much regularity of the solution (this example is studied in detail in Sect. 3.2). Therefore, we are led to consider a weaker notion of solution. In particular, we are interested in a viscosity-type solution, namely a solution which is not required to be differentiable.

The issue of providing a suitable definition of viscosity solutions for path-dependent PDEs has attracted a great interest, see Peng [33] and Tang and Zhang [42], Ekren et al. [18–20], Ren et al. [34]. In particular, the definition of viscosity solution provided by [18–20, 34] is characterized by the fact that the classical minimum/maximum property, which appears in the standard definition of viscosity solution, is replaced with an optimal stopping problem under nonlinear expectation [21]. Then, probability plays an essential role in this latter definition, which can not be considered as a purely analytic object as the classical definition of viscosity solution is; it is, more properly, a probabilistic version of the classical definition of viscosity solution. We also emphasize that a similar notion of solution, called stochastic weak solution, has been introduced in the recent paper [29] in the context of variational inequalities for the Snell envelope associated to a non-Markovian continuous process X. Those authors also revisit functional Itô calculus, making use of stopping times. This approach seems very promising. Instead, our aim is to provide a definition of viscosity type solution, which has the peculiarity to be a purely analytic object; this will be called a strong-viscosity solution to distinguish it from the classical notion of viscosity solution. A strong-viscosity solution to a path-dependent partial differential equation is defined, in a few words, as the pointwise limit of strict solutions to perturbed equations. We notice that the definition of strong-viscosity solution is similar in spirit to the vanishing viscosity method, which represents one of the primitive ideas leading to the conception of the modern definition of viscosity solution. Moreover, it has also some similarities with the definition of good solution, which turned out to be equivalent to the definition of $L^p$-viscosity solution for certain fully nonlinear partial differential equations, see, e.g., [3, 11, 27, 28]. Finally, our definition is likewise inspired by the notion of strong solution (which justifies the first word in the name of our solution), as defined for example in [2, 24, 25], even though strong solutions are required to be more regular (this regularity is usually required to prove uniqueness of strong solutions, which for example in [24, 25] is based on a Fukushima-Dirichlet decomposition). Instead, our definition of strong-viscosity solution to the path-dependent semilinear Kolmogorov equation is not required to be continuous, as in the spirit of viscosity solutions. The term viscosity in the name of our solution is also justified by the fact that in the finite dimensional case we have an equivalence result between the notion of strong-viscosity solution and that of viscosity solution, see Theorem 3.7 in [8]. We prove a uniqueness theorem for strong-viscosity solutions using the theory of backward stochastic differential equations and we provide an existence result. We refer to [10] for more general results (when the path-dependent equation is not the path-dependent heat equation) and also for the application of strong-viscosity solutions to standard semilinear parabolic PDEs.

The paper is organized as follows. In Sect. 2 we develop functional Itô calculus via regularization following [9]: after a brief introduction on finite dimensional stochastic calculus via regularization in Sect. 2.1, we introduce and study the space $\mathscr {C}([-T,0])$ in Sect. 2.2; then, we define the pathwise derivatives and we prove the functional Itô’s formula in Sect. 2.3; in Sect. 2.4, instead, we discuss the relation between functional Itô calculus via regularization and Banach space valued stochastic calculus for window processes. In Sect. 3, on the other hand, we study path-dependent PDEs following [10]. More precisely, in Sect. 3.1 we discuss strict solutions; in Sect. 3.2 we present a significant hedging example to motivate the introduction of a weaker notion of solution; finally, in Sect. 3.3 we provide the definition of strong-viscosity solution.

2 Functional Itô Calculus: A Regularization Approach

2.1 Background: Finite Dimensional Calculus via Regularization

The theory of stochastic calculus via regularization has been developed in several papers, starting from [37, 38]. We recall below only the results used in the present paper, and we refer to [40] for a survey on the subject. We emphasize that integrands are allowed to be anticipating. Moreover, the integration theory and calculus appear to be close to a pure pathwise approach even though there is still a probability space behind.

Fix a probability space $(\varOmega ,{\mathscr {F}},{\mathbb {P}})$ and $T\in ]0,\infty [$. Let ${\mathbb {F}}=({\mathscr {F}}_t)_{t\in [0,T]}$ denote a filtration satisfying the usual conditions. Let $X=(X_t)_{t\in [0,T]}$ (resp. $Y=(Y_t)_{t\in [0,T]}$) be a real continuous (resp. ${\mathbb {P}}$-a.s. integrable) process. Every real continuous process $X = (X_t)_{t\in [0,T]}$ is naturally extended to all $t\in {\mathbb {R}}$ setting $X_t = X_0$, $t\le 0$, and $X_t=X_T$, $t\ge T$. We also define a $C([-T,0])$-valued process ${\mathbb {X}}=({\mathbb {X}}_t)_{t\in {\mathbb {R}}}$, called the window process associated with X, defined by

$$\begin{aligned} {\mathbb {X}}_t := \{X_{t+x},\,x\in [-T,0]\}, \qquad t\in {\mathbb {R}}. \end{aligned}$$

(2)

This corresponds to the so-called segment process which appears for instance in [43].

Definition 1

Suppose that, for every $t\in [0,T]$, the limit

$$\begin{aligned} \int _{0}^{t}Y_sd^-X_s \ := \ \lim _{\varepsilon \rightarrow 0^+}\int _{0}^{t} Y_s\frac{X_{s+\varepsilon }-X_s}{\varepsilon }ds, \end{aligned}$$

(3)

exists in probability. If the obtained random function admits a continuous modification, that process is denoted by $\int _0^\cdot Yd^-X$ and called forward integral of Y with respect to X .

Definition 2

A family of processes $(H_t^{(\varepsilon )})_{t\in [0,T]}$ is said to converge to $(H_t)_{t\in [0,T]}$ in the ucp sense, if $\sup _{0\le t\le T}|H_t^{(\varepsilon )}-H_t|$ goes to 0 in probability, as $\varepsilon \rightarrow 0^+$.

Proposition 1

Suppose that the limit (3) exists in the ucp sense. Then, the forward integral $\int _0^\cdot Yd^-X$ of Y with respect to X exists.

Let us introduce the concept of covariation, which is a crucial notion in stochastic calculus via regularization. Let us suppose that X, Y are continuous processes.

Definition 3

The covariation of X and Y is defined by

$$ \left[ X,Y\right] _{t} \ = \ \left[ Y,X\right] _{t} \ = \ \lim _{\varepsilon \rightarrow 0^{+}} \frac{1}{\varepsilon } \int _{0}^{t} (X_{s+\varepsilon }-X_{s})(Y_{s+\varepsilon }-Y_{s})ds, \qquad t \in [0,T], $$

if the limit exists in probability for every $t \in [0,T]$, provided that the limiting random function admits a continuous version (this is the case if the limit holds in the ucp sense). If $X=Y,$ X is said to be a finite quadratic variation process and we set $[X]:=[X,X]$.

The forward integral and the covariation generalize the classical Itô integral and covariation for semimartingales. In particular, we have the following result, for a proof we refer to, e.g., [40].

Proposition 2

The following properties hold.

• (i) Let $S^1,S^2$ be continuous ${\mathbb {F}}$-semimartingales. Then, $[S^1,S^2]$ is the classical bracket $[S^1,S^2]=\langle M^1,M^2\rangle$, where $M^1$ (resp. $M^2$ $)$ is the local martingale part of $S^1$ (resp. $S^2$ $)$.

• (ii) Let V be a continuous bounded variation process and Y be a càdlàg process (or vice-versa$)$ $;$ then $[V] =[Y,V]= 0$. Moreover $\int _0^\cdot Y d^-V=\int _0^\cdot Y dV$, is the Lebesgue-Stieltjes integral.

• (iii) If W is a Brownian motion and Y is an ${\mathbb {F}}$-progressively measurable process such that $\int _0^T Y^2_s ds < \infty$, ${\mathbb {P}}$-a.s., then $\int _0^\cdot Yd^- W$ exists and equals the Itô integral $\int _0^\cdot YdW$.

We could have defined the forward integral using limits of non-anticipating Riemann sums. Another reason to use the regularization procedure is due to the fact that it extends the Itô integral, as Proposition 2(iii) shows. If the integrand had uncountable jumps (as Y being the indicator function of the rational number in [0, 1]) then, the Itô integral $\int _0^\cdot YdW$ would be zero $Y = 0$ a.e. The limit of Riemann sums $\sum _i Y_{t_i} (W_{t_{i+1}} - W_{t_{i}})$ would heavily depend on the discretization grid.

We end this crash introduction to finite dimensional stochastic calculus via regularization presenting one of its cornerstones: Itô’s formula. It is a well-known result in the theory of semimartingales, but it also extends to the framework of finite quadratic variation processes. For a proof we refer to Theorem 2.1 of [39].

Theorem 1

Let $F:[0,T]\times {\mathbb {R}}\longrightarrow {\mathbb {R}}$ be of class $C^{1,2}\left( [0,T]\times {\mathbb {R}}\right)$ and $X=(X_t)_{t\in [0,T]}$ be a real continuous finite quadratic variation process. Then, the following Itô’s formula holds, ${\mathbb {P}}$-a.s.,

$$\begin{aligned} F(t,X_{t})&= \ F(0,X_{0}) + \int _{0}^{t}\partial _t F(s,X_s)ds + \int _{0}^{t} \partial _{x} F(s,X_s)d^-X_s \nonumber \\&\quad \ + \frac{1}{2}\int _{0}^{t} \partial ^2_{x\, x} F(s,X_s)d[X]_s, \qquad \qquad \quad 0\le t\le T. \end{aligned}$$

(4)

2.1.1 The Deterministic Calculus via Regularization

A useful particular case of finite dimensional stochastic calculus via regularization arises when $\varOmega$ is a singleton, i.e., when the calculus becomes deterministic. In addition, in this deterministic framework we will make use of the definite integral on an interval [a, b], where $a<b$ are two real numbers. Typically, we will consider $a=-T$ or $a=-t$ and $b=0$.

We start with two conventions. By default, every bounded variation function $f:[a,b]\rightarrow {\mathbb {R}}$ will be considered as càdlàg. Moreover, given a function $f:[a,b]\rightarrow {\mathbb {R}}$, we will consider the following two extensions of f to the entire real line:

$$ f_J(x) := {\left\{ \begin{array}{ll} 0, &{} x>b, \\ f(x), &{} x\in [a,b], \\ f(a), &{} x<a, \end{array}\right. } \qquad \qquad f_{\overline{J}}(x) := {\left\{ \begin{array}{ll} f(b), &{} x>b, \\ f(x), &{} x\in [a,b], \\ 0, &{} x<a, \end{array}\right. } $$

where $J:=\,]a,b]$ and $\overline{J}=[a,b[$.

Definition 4

Let $f,g:[a,b]\rightarrow {\mathbb {R}}$ be càdlàg functions.

(i) Suppose that the following limit

$$\begin{aligned} \int _{[a,b]}g(s)d^-f(s) \ := \ \lim _{\varepsilon \rightarrow 0^+}\int _{\mathbb {R}}g_J(s)\frac{f_{\overline{J}}(s+\varepsilon )-f_{\overline{J}}(s)}{\varepsilon }ds, \end{aligned}$$

(5)

exists and it is finite. Then, the obtained quantity is denoted by $\int _{[a,b]} gd^-f$ and called (deterministic, definite) forward integral of g with respect to f (on [a, b]).

(ii) Suppose that the following limit

$$\begin{aligned} \int _{[a,b]}g(s)d^+f(s) \ := \ \lim _{\varepsilon \rightarrow 0^+}\int _{\mathbb {R}}g_J(s)\frac{f_{\overline{J}}(s)-f_{\overline{J}}(s-\varepsilon )}{\varepsilon }ds, \end{aligned}$$

(6)

exists and it is finite. Then, the obtained quantity is denoted by $\int _{[a,b]} gd^+f$ and called (deterministic, definite) backward integral of g with respect to f (on [a, b]).

The notation concerning this integral is justified since when the integrator f has bounded variation the previous integrals are Lebesgue-Stieltjes integrals on [a, b].

Proposition 3

Suppose $f:[a,b]\rightarrow {\mathbb {R}}$ with bounded variation and $g:[a,b]\rightarrow {\mathbb {R}}$ càdlàg. Then, we have

$$\begin{aligned} \int _{[a,b]} g(s) d^-f(s)&= \ \int _{[a,b]} g(s^-) df(s) \ := \ g(a)f(a) + \int _{]a,b]} g(s^-) df(s), \end{aligned}$$

(7)

$$\begin{aligned} \int _{[a,b]} g(s) d^+f(s)&= \ \int _{[a,b]} g(s) df(s) \ := \ g(a)f(a) + \int _{]a,b]} g(s) df(s). \end{aligned}$$

(8)

Proof

Identity (7). We have

$$\begin{aligned} \int _{\mathbb {R}}g_J(s) \frac{f_{\overline{J}}(s+\varepsilon )-f_{\overline{J}}(s)}{\varepsilon } ds&= \ \frac{1}{\varepsilon } g(a) \int _{a-\varepsilon }^a f(s+\varepsilon ) ds \nonumber \\&\quad \ + \int _a^b g(s) \frac{f((s+\varepsilon )\wedge b) - f(s)}{\varepsilon } ds. \end{aligned}$$

(9)

The second integral on the right-hand side of (9) gives, by Fubini’s theorem,

$$\begin{aligned} \int _a^b g(s) \bigg (\frac{1}{\varepsilon }\int _{]s,(s+\varepsilon )\wedge b]} df(y)\bigg ) ds&= \ \int _{]a,b]} \bigg (\frac{1}{\varepsilon }\int _{[a\vee (y-\varepsilon ),y]}g(s)ds\bigg ) df(y) \\&\overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ \int _{]a,b]} g(y^-) df(y). \end{aligned}$$

The first integral on the right-hand side of (9) goes to g(a)f(a) as $\varepsilon \rightarrow 0^+$, so the result follows.

Identity (8). We have

$$\begin{aligned} \int _{\mathbb {R}}g_J(s) \frac{f_{\overline{J}}(s)-f_{\overline{J}}(s-\varepsilon )}{\varepsilon } ds&= \ \int _{a+\varepsilon }^b g(s) \frac{f(s) - f(s-\varepsilon )}{\varepsilon } ds \nonumber \\&\quad \ + \frac{1}{\varepsilon } \int _a^{a+\varepsilon } g(s)f(s) ds. \end{aligned}$$

(10)

The second integral on the right-hand side of (10) goes to g(a)f(a) as $\varepsilon \rightarrow 0^+$. The first one equals

$$\begin{aligned} \int _{a+\varepsilon }^b g(s) \bigg (\frac{1}{\varepsilon }\int _{]s-\varepsilon ,s]}df(y)\bigg ) ds \ = \ \int _{]a,b]} \bigg (\frac{1}{\varepsilon }\int _{]y,(y+\varepsilon )\wedge b]}g(s)ds\bigg ) df(y) \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ \int _{]a,b]} g(y) df(y), \end{aligned}$$

from which the claim follows. $\Box$

Let us now introduce the deterministic covariation.

Definition 5

Let $f,g:[a,b]\rightarrow {\mathbb {R}}$ be continuous functions and suppose that $0\in [a,b]$. The (deterministic) covariation of f and g (on [a, b]) is defined by

$$\begin{aligned} \left[ f,g\right] (x) \ = \ \left[ g,f\right] (x) \ = \ \lim _{\varepsilon \rightarrow 0^{+}} \frac{1}{\varepsilon } \int _{0}^x (f(s+\varepsilon )-f(s))(g(s+\varepsilon )-g(s))ds, \qquad x\in [a,b], \end{aligned}$$

if the limit exists and it is finite for every $x\in [a,b]$. If $f=g$, we set $[f]:=[f,f]$ and it is called quadratic variation of f (on [a, b]).

We notice that in Definition 5 the quadratic variation [f] is continuous on [a, b], since f is a continuous function.

Remark 1

Notice that if f is a fixed Brownian path and $g(s)=\varphi (s,f(s))$, with $\varphi \in C^1([a,b]\times {\mathbb {R}})$. Then $\int _{[a,b]} g(s) d^- f(s)$ exists for almost all (with respect to the Wiener measure on C([a, b])) Brownian paths f. This latter result can be shown using Theorem 2.1 in [26] (which implies that the deterministic bracket exists, for almost all Brownian paths f, and $[f](s)=s$) and then applying Itô’s formula in Theorem 1 above, with ${\mathbb {P}}$ given by the Dirac delta at a Brownian path f. $\Box$

We conclude this subsection with an integration by parts formula for the deterministic forward and backward integrals.

Proposition 4

Let $f:[a,b]\rightarrow {\mathbb {R}}$ be a càdlàg function and $g:[a,b]\rightarrow {\mathbb {R}}$ be a bounded variation function. Then, the following integration by parts formulae hold:

$$\begin{aligned} \int _{[a,b]} g(s) d^-f(s) \ = \ g(b) f(b) - \int _{]a,b]} f(s) dg(s), \end{aligned}$$

(11)

$$\begin{aligned} \int _{[a,b]} g(s) d^+f(s) \ = \ g(b) f(b^-) - \int _{]a,b]} f(s^-) dg(s). \end{aligned}$$

(12)

Proof

Identity (11). The left-hand side of (11) is the limit, when $\varepsilon \rightarrow 0^+$, of

$$\begin{aligned} \frac{1}{\varepsilon }\int _a^{b-\varepsilon } g(s) f(s+\varepsilon ) ds - \frac{1}{\varepsilon }\int _a^b g(s) f(s) ds + \frac{1}{\varepsilon }\int _{b-\varepsilon }^b g(s) f(b) ds + \frac{1}{\varepsilon } \int _{a-\varepsilon }^a g(a) f(s+\varepsilon ) ds. \end{aligned}$$

This gives

$$\begin{aligned}&\frac{1}{\varepsilon }\int _{a+\varepsilon }^b g(s-\varepsilon )f(s)ds - \frac{1}{\varepsilon }\int _a^b g(s)f(s) ds + \frac{1}{\varepsilon }\int _{b-\varepsilon }^b g(s) f(b) ds + \frac{1}{\varepsilon }\int _{a-\varepsilon }^a g(a)f(s+\varepsilon ) ds \\&= \ - \int _{a+\varepsilon }^b \frac{g(s) - g(s-\varepsilon )}{\varepsilon }f(s)ds - \frac{1}{\varepsilon }\int _a^{a+\varepsilon } g(s)f(s) ds + \frac{1}{\varepsilon }\int _{b-\varepsilon }^b g(s) f(b) ds \\&\quad \ + \frac{1}{\varepsilon }\int _{a-\varepsilon }^a g(a)f(s+\varepsilon ) ds. \end{aligned}$$

We see that

$$\begin{aligned} \frac{1}{\varepsilon }\int _{b-\varepsilon }^b g(s) f(b) ds&\overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ g(b^-) f(b), \\ \frac{1}{\varepsilon }\int _{a-\varepsilon }^a g(a)f(s+\varepsilon ) ds - \frac{1}{\varepsilon }\int _a^{a+\varepsilon } g(s)f(s) ds&\overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0. \end{aligned}$$

Moreover, we have

$$\begin{aligned} - \int _{a+\varepsilon }^b \frac{g(s) - g(s-\varepsilon )}{\varepsilon }f(s)ds&= \ - \int _{a+\varepsilon }^b ds f(s) \frac{1}{\varepsilon } \int _{]s-\varepsilon ,s]} dg(y) \\&= \ - \int _{]a,b]} dg(y) \frac{1}{\varepsilon } \int _{y\vee (a+\varepsilon )}^{b\wedge (y+\varepsilon )} f(s) ds \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ - \int _{]a,b[} dg(y) f(y). \end{aligned}$$

In conclusion, we find

$$\begin{aligned} \int _{[a,b]} g(s) d^-f(s)&= \ - \int _{]a,b]} dg(y) f(y) + (g(b) - g(b^-)) f(b) + g(b^-) f(b) \\&= \ - \int _{]a,b]} dg(y) f(y) + g(b) f(b). \end{aligned}$$

Identity (12). The left-hand side of (12) is given by the limit, as $\varepsilon \rightarrow 0^+$, of

$$\begin{aligned} \frac{1}{\varepsilon }\int _a^b g(s)f(s) ds - \frac{1}{\varepsilon }\int _{a+\varepsilon }^b g(s)f(s-\varepsilon ) ds \ = \ \frac{1}{\varepsilon }\int _a^b g(s)f(s) ds - \frac{1}{\varepsilon }\int _a^{b-\varepsilon } g(s+\varepsilon )f(s) ds&\\ = \ - \int _a^{b-\varepsilon } f(s)\frac{g(s+\varepsilon )-g(s)}{\varepsilon } ds + \frac{1}{\varepsilon }\int _{b-\varepsilon }^b g(s)f(s) ds&\end{aligned}$$

The second integral on the right-hand side goes to $g(b^-)f(b^-)$ as $\varepsilon \rightarrow 0^+$. The first integral expression equals

$$\begin{aligned} -\int _{\mathbb {R}}f_J(s) \frac{g_{\overline{J}}(s+\varepsilon ) - g_{\overline{J}}(s)}{\varepsilon } ds + \frac{1}{\varepsilon } f(a) \int _{a-\varepsilon }^a g(s+\varepsilon ) ds + \int _{b-\varepsilon }^b f(s)\frac{g(b)-g(s)}{\varepsilon } ds \\ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ - \int _{]a,b]} f(s^-)dg(s) - f(a)g(a) + f(a)g(a) + (g(b)-g(b^-))f(b^-), \end{aligned}$$

taking into account identity (7). This gives us the result. $\Box$

2.2 The Spaces $\mathscr {C}([-T,0])$ and $\mathscr {C}([-T,0[)$

Let $C([-T,0])$ denote the set of real continuous functions on $[-T,0]$, endowed with supremum norm $\Vert \eta \Vert _\infty = \sup _{x\in [-T,0]}|\eta (x)|$, for any $\eta \in C([-T,0])$.

Remark 2

We shall develop functional Itô calculus via regularization firstly for time-independent functionals ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$, since we aim at emphasizing that in our framework the time variable and the path play two distinct roles, as emphasized in the introduction. This, also, allows us to focus only on the definition of horizontal and vertical derivatives. Clearly, everything can be extended in an obvious way to the time-dependent case ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$, as we shall illustrate later. $\Box$

Consider a map ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$. Our aim is to derive a functional Itô’s formula for ${\mathscr {U}}$. To do this, we are led to define the functional (i.e., horizontal and vertical) derivatives for ${\mathscr {U}}$ in the spirit of [5, 17]. Since the definition of functional derivatives necessitates of discontinuous paths, in [5] the idea is to consider functionals defined on the space of càdlàg trajectories ${\mathbb {D}}([-T,0])$. However, we can not, in general, extend in a unique way a functional ${\mathscr {U}}$ defined on $C([-T,0])$ to ${\mathbb {D}}([-T,0])$. Our idea, instead, is to consider a larger space than $C([-T,0])$, denoted by $\mathscr {C}([-T,0])$, which is the space of bounded trajectories on $[-T,0]$, continuous on $[-T,0[$ and with possibly a jump at 0. We endow $\mathscr {C}([-T,0])$ with a (inductive) topology such that $C([-T,0])$ is dense in $\mathscr {C}([-T,0])$ with respect to this topology. Therefore, if ${\mathscr {U}}$ is continuous with respect to the topology of $\mathscr {C}([-T,0])$, then it admits a unique continuous extension $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$.

Definition 6

We denote by $\mathscr {C}([-T,0])$ the set of bounded functions $\eta :[-T,0]\rightarrow {\mathbb {R}}$ such that $\eta$ is continuous on $[-T,0[$, equipped with the topology we now describe.

Convergence We endow $\mathscr {C}([-T,0])$ with a topology inducing the following convergence: $(\eta _n)_n$ converges to $\eta$ in $\mathscr {C}([-T,0])$ as n tends to infinity if the following holds.

• (i) $\Vert \eta _n\Vert _\infty \le C$, for any $n\in \mathbb {N}$, for some positive constant C independent of $n$ $;$

• (ii) $\sup _{x\in K}|\eta _n(x)-\eta (x)|\rightarrow 0$ as n tends to infinity, for any compact set $K\subset [-T,0[$ $;$

• (iii) $\eta _n(0)\rightarrow \eta (0)$ as n tends to infinity.

Topology For each compact $K\subset [-T,0[$ define the seminorm $p_K$ on $\mathscr {C}([-T,0])$ by

$$ p_K(\eta ) \ = \ \sup _{x\in K}|\eta (x)| + |\eta (0)|, \qquad \forall \,\eta \in \mathscr {C}([-T,0]). $$

Let $M>0$ and $\mathscr {C}_M([-T,0])$ be the set of functions in $\mathscr {C}([-T,0])$ which are bounded by M. Still denote $p_K$ the restriction of $p_K$ to $\mathscr {C}_M([-T,0])$ and consider the topology on $\mathscr {C}_M([-T,0])$ induced by the collection of seminorms $(p_K)_K$. Then, we endow $\mathscr {C}([-T,0])$ with the smallest topology (inductive topology) turning all the inclusions $i_M:\mathscr {C}_M([-T,0])\rightarrow \mathscr {C}([-T,0])$ into continuous maps.

Remark 3

(i) Notice that $C([-T,0])$ is dense in $\mathscr {C}([-T,0])$, when endowed with the topology of $\mathscr {C}([-T,0])$. As a matter of fact, let $\eta \in \mathscr {C}([-T,0])$ and define, for any $n\in \mathbb {N}\backslash \{0\}$,

$$ \varphi _n(x)= {\left\{ \begin{array}{ll} \eta (x), \qquad &{}-T\le x\le -1/n, \\ n(\eta (0)-\eta (-1/n))x + \eta (0), &{}-1/n<x\le 0. \end{array}\right. } $$

Then, we see that $\varphi _n\in C([-T,0])$ and $\varphi _n\rightarrow \eta$ in $\mathscr {C}([-T,0])$.

Now, for any $a\in {\mathbb {R}}$ define

$$\begin{aligned} C_a([-T,0])&:= \ \{\eta \in C([-T,0]):\eta (0)=a\}, \\ \mathscr {C}_a([-T,0])&:= \ \{\eta \in \mathscr {C}([-T,0]):\eta (0)=a\}. \end{aligned}$$

Then, $C_a([-T,0])$ is dense in $\mathscr {C}_a([-T,0])$ with respect to the topology of $\mathscr {C}([-T,0])$.

(ii) We provide two examples of functionals ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$, continuous with respect to the topology of $\mathscr {C}([-T,0])$, and necessarily with respect to the topology of $C([-T,0])$ (the proof is straightforward and not reported):

• (a) ${\mathscr {U}}(\eta ) = g(\eta (t_1),\ldots ,\eta (t_n))$, for all $\eta \in C([-T,0])$, with $-T\le t_1<\cdots <t_n\le 0$ and $g:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ continuous.

• (b) ${\mathscr {U}}(\eta ) = \int _{[-T,0]}\varphi (x)d^-\eta (x)$, for all $\eta \in C([-T,0])$, with $\varphi :[0,T]\rightarrow {\mathbb {R}}$ a càdlàg bounded variation function. Concerning this example, keep in mind that, using the integration by parts formula, ${\mathscr {U}}(\eta )$ admits the representation (11).

(iii) Consider the functional ${\mathscr {U}}(\eta ) = \sup _{x\in [-T,0]}\eta (x)$, for all $\eta \in C([-T,0])$. It is obviously continuous, but it is not continuous with respect to the topology of $\mathscr {C}([-T,0])$. As a matter of fact, for any $n\in \mathbb {N}$ consider $\eta _n\in C([-T,0])$ given by

$$ \eta _n(x) \ = \ {\left\{ \begin{array}{ll} 0, \qquad \qquad \qquad &{}-T\le x\le -\frac{T}{2^n}, \\ \frac{2^{n+1}}{T}x+2, &{}-\frac{T}{2^n}<x\le -\frac{T}{2^{n+1}}, \\ -\frac{2^{n+1}}{T}x, &{}-\frac{T}{2^{n+1}}<x\le 0. \end{array}\right. } $$

Then, ${\mathscr {U}}(\eta _n)=\sup _{x\in [-T,0]}\eta _n(x)=1$, for any n. However, $\eta _n$ converges to the zero function in $\mathscr {C}([-T,0])$, as n tends to infinity. This example will play an important role in Sect. 3 to justify a weaker notion of solution to the path-dependent semilinear Kolmogorov equation. $\Box$

To define the functional derivatives, we shall need to separate the “past” from the “present” of $\eta \in \mathscr {C}([-T,0])$. Indeed, roughly speaking, the horizontal derivative calls in the past values of $\eta$, namely $\{\eta (x):x\in [-T,0[\}$, while the vertical derivative calls in the present value of $\eta$, namely $\eta (0)$. To this end, it is useful to introduce the space $\mathscr {C}([-T,0[)$.

Definition 7

We denote by $\mathscr {C}([-T,0[)$ the set of bounded continuous functions $\gamma :[-T,0[$ $\rightarrow {\mathbb {R}}$, equipped with the topology we now describe.

Convergence We endow $\mathscr {C}([-T,0[)$ with a topology inducing the following convergence: $(\gamma _n)_n$ converges to $\gamma$ in $\mathscr {C}([-T,0[)$ as n tends to infinity if:

• (i) $\sup _{x\in [-T,0[}|\gamma _n(x)| \le C$, for any $n\in \mathbb {N}$, for some positive constant C independent of $n$ $;$

• (ii) $\sup _{x\in K}|\gamma _n(x)-\gamma (x)|\rightarrow 0$ as n tends to infinity, for any compact set $K\subset [-T,0[$.

Topology For each compact $K\subset [-T,0[$ define the seminorm $q_K$ on $\mathscr {C}([-T,0[)$ by

$$ q_K(\gamma ) \ = \ \sup _{x\in K}|\gamma (x)|, \qquad \forall \,\gamma \in \mathscr {C}([-T,0[). $$

Let $M>0$ and $\mathscr {C}_M([-T,0[)$ be the set of functions in $\mathscr {C}([-T,0[)$ which are bounded by M. Still denote $q_K$ the restriction of $q_K$ to $\mathscr {C}_M([-T,0[)$ and consider the topology on $\mathscr {C}_M([-T,0[)$ induced by the collection of seminorms $(q_K)_K$. Then, we endow $\mathscr {C}([-T,0[)$ with the smallest topology (inductive topology) turning all the inclusions $i_M:\mathscr {C}_M([-T,0[)\rightarrow \mathscr {C}([-T,0[)$ into continuous maps.

Remark 4

(i) Notice that $\mathscr {C}([-T,0])$ is isomorphic to $\mathscr {C}([-T,0[)\times {\mathbb {R}}$. As a matter of fact, it is enough to consider the map

$$\begin{aligned} J :{\mathscr {C}}([-T,0])&\rightarrow \mathscr {C}([-T,0[)\times {\mathbb {R}}\\ \eta&\mapsto (\eta _{|[-T,0[},\eta (0)). \end{aligned}$$

Observe that $J^{-1}:\mathscr {C}([-T,0[)\times {\mathbb {R}}\rightarrow \mathscr {C}([-T,0])$ is given by $J^{-1}(\gamma ,a) = \gamma 1_{[-T,0[} + a1_{\{0\}}$.

(ii) $\mathscr {C}([-T,0])$ is a space which contains $C([-T,0])$ as a dense subset and it has the property of separating “past” from “present”. Another space having the same property is $L^2([-T,0]; d \mu )$ where $\mu$ is the sum of the Dirac measure at zero and Lebesgue measure. Similarly as for item (i), that space is isomorphic to $L^2([-T,0]) \times {\mathbb {R}}$, which is a very popular space appearing in the analysis of functional dependent (as delay) equations, starting from [4]. $\Box$

For every $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$, we can now exploit the space $\mathscr {C}([-T,0[)$ to define a map $\tilde{u}:\mathscr {C}([-T,0[)\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ where “past” and “present” are separated.

Definition 8

Let $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ and define $\tilde{u}:\mathscr {C}([-T,0[)\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ as

$$\begin{aligned} \tilde{u}(\gamma ,a) \ := \ u(\gamma 1_{[-T,0[} + a1_{\{0\}}), \qquad \forall \,(\gamma ,a)\in \mathscr {C}([-T,0[)\times {\mathbb {R}}. \end{aligned}$$

(13)

In particular, we have $u(\eta ) = \tilde{u}(\eta _{|[-T,0[},\eta (0))$, for all $\eta \in \mathscr {C}([-T,0])$.

We conclude this subsection with a characterization of the dual spaces of $\mathscr {C}([-T,0])$ and $\mathscr {C}([-T,0[)$, which has an independent interest. Firstly, we need to introduce the set ${\mathscr {M}}([-T,0])$ of finite signed Borel measures on $[-T,0]$. We also denote ${\mathscr {M}}_0([-T,0])\subset {\mathscr {M}}([-T,0])$ the set of measures $\mu$ such that $\mu (\{0\})=0$.

Proposition 5

Let $\varLambda \in \mathscr {C}([-T,0])^*$, the dual space of $\mathscr {C}([-T,0])$. Then, there exists a unique $\mu \in {\mathscr {M}}([-T,0])$ such that

$$ \varLambda \eta \ = \ \int _{[-T,0]} \eta (x) \mu (dx), \qquad \forall \,\eta \in \mathscr {C}([-T,0]). $$

Proof

Let $\varLambda \in \mathscr {C}([-T,0])^*$ and define

$$ \tilde{\varLambda }\varphi \ := \ \varLambda {\varphi }, \qquad \forall \,\varphi \in C([-T,0]). $$

Notice that $\tilde{\varLambda }:C([-T,0])\rightarrow {\mathbb {R}}$ is a continuous functional on the Banach space $C([-T,0])$ endowed with the supremum norm $\Vert \cdot \Vert _\infty$. Therefore $\tilde{\varLambda }\in C([-T,0])^*$ and it follows from Riesz representation theorem (see, e.g., Theorem 6.19 in [36]) that there exists a unique $\mu \in {\mathscr {M}}([-T,0])$ such that

$$\begin{aligned} \tilde{\varLambda }\varphi \ = \ \int _{[-T,0]} \varphi (x)\mu (dx), \qquad \forall \,\varphi \in C([-T,0]). \end{aligned}$$

Obviously $\tilde{\varLambda }$ is also continuous with respect to the topology of $\mathscr {C}([-T,0])$. Since $C([-T,0])$ is dense in $\mathscr {C}([-T,0])$ with respect to the topology of $\mathscr {C}([-T,0])$, we deduce that there exists a unique continuous extension of $\tilde{\varLambda }$ to $\mathscr {C}([-T,0])$, which is clearly given by

$$ \varLambda \eta \ = \ \int _{[-T,0]} \eta (x)\mu (dx), \qquad \forall \,\eta \in \mathscr {C}([-T,0]). $$

$\Box$

Proposition 6

Let $\varLambda \in \mathscr {C}([-T,0[)^*$, the dual space of $\mathscr {C}([-T,0[)$. Then, there exists a unique $\mu \in {\mathscr {M}}_0([-T,0])$ such that

$$ \varLambda {\gamma } \ = \ \int _{[-T,0[} \gamma (x) \mu (dx), \qquad \forall \,\gamma \in \mathscr {C}([-T,0[). $$

Proof

Let $\varLambda \in \mathscr {C}([-T,0[)^*$ and define

$$\begin{aligned} \tilde{\varLambda }\eta \ := \ \varLambda (\eta _{|[-T,0[}), \qquad \forall \,\eta \in \mathscr {C}([-T,0]). \end{aligned}$$

(14)

Notice that $\tilde{\varLambda }:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ is a continuous functional on $\mathscr {C}([-T,0])$. It follows from Proposition 5 that there exists a unique $\mu \in {\mathscr {M}}([-T,0])$ such that

$$\begin{aligned} \tilde{\varLambda }\eta \ = \ \int _{[-T,0]} \eta (x)\mu (dx) \ = \ \int _{[-T,0[} \eta (x)\mu (dx) + \eta (0)\mu (\{0\}), \qquad \forall \,\eta \in \mathscr {C}([-T,0]). \end{aligned}$$

(15)

Let $\eta _1,\eta _2\in \mathscr {C}([-T,0])$ be such that $\eta _1 1_{[-T,0[}=\eta _2 1_{[-T,0[}$. Then, we see from (14) that $\tilde{\varLambda }\eta _1=\tilde{\varLambda }\eta _2$, which in turn implies from (15) that $\mu (\{0\})=0$. In conclusion, $\mu \in {\mathscr {M}}_0([-T,0])$ and $\varLambda$ is given by

$$ \varLambda {\gamma } \ = \ \int _{[-T,0[} \gamma (x)\mu (dx), \qquad \forall \,\gamma \in \mathscr {C}([-T,0[). $$

2.3 Functional Derivatives and Functional Itô’s Formula

In the present section we shall prove one of the main result of this section, namely the functional Itô’s formula for ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$ and, more generally, for ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$. We begin introducing the functional derivatives, firstly for a functional $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$, and then for ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$.

Definition 9

Consider $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ and $\eta \in \mathscr {C}([-T,0])$.

(i) We say that u admits the horizontal derivative at $\eta$ if the following limit exists and it is finite:

$$\begin{aligned} D^H u(\eta ) \ := \ \lim _{\varepsilon \rightarrow 0^+} \frac{u(\eta (\cdot )1_{[-T,0[}+\eta (0)1_{\{0\}}) - u(\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}})}{\varepsilon }. \end{aligned}$$

(16)

(i)’ Let $\tilde{u}$ be as in (13), then we say that $\tilde{u}$ admits the horizontal derivative at $(\gamma ,a)\in \mathscr {C}([-T,0[)\times {\mathbb {R}}$ if the following limit exists and it is finite:

$$\begin{aligned} D^H\tilde{u}(\gamma ,a) \ := \ \lim _{\varepsilon \rightarrow 0^+} \frac{\tilde{u}(\gamma (\cdot ),a) - \tilde{u}(\gamma (\cdot -\varepsilon ),a)}{\varepsilon }. \end{aligned}$$

(17)

Notice that if $D^H u(\eta )$ exists then $D^H\tilde{u}(\eta _{|[-T,0[},\eta (0))$ exists and they are equal; viceversa, if $D^H\tilde{u}(\gamma ,a)$ exists then $D^H u(\gamma 1_{[-T,0[} + a 1_{\{0\}})$ exists and they are equal.

(ii) We say that u admits the first-order vertical derivative at $\eta$ if the first-order partial derivative $\partial _a\tilde{u}(\eta _{|[-T,0[},\eta (0))$ at $(\eta _{|[-T,0[},\eta (0))$ of $\tilde{u}$ with respect to its second argument exists and we set

$$ D^V u(\eta ) \ := \ \partial _a \tilde{u}(\eta _{|[-T,0[},\eta (0)). $$

(iii) We say that u admits the second-order vertical derivative at $\eta$ if the second-order partial derivative at $(\eta _{|[-T,0[},\eta (0))$ of $\tilde{u}$ with respect to its second argument, denoted by $\partial _{aa}^2\tilde{u}(\eta _{|[-T,0[},\eta (0))$, exists and we set

$$ D^{VV} u(\eta ) \ := \ \partial _{aa}^2 \tilde{u}(\eta _{|[-T,0[},\eta (0)). $$

Definition 10

We say that $u:{\mathscr {C}}([-T,0])\rightarrow {\mathbb {R}}$ is of class $\mathscr {C}^{1,2}({\text {past}}\times {\text {present}})$ if

• (i) u is continuous; 

• (ii) $D^H u$ exists everywhere on $\mathscr {C}([-T,0])$ and for every $\gamma \in \mathscr {C}([-T,0[)$ the map

  $$ (\varepsilon ,a)\longmapsto D^H\tilde{u}(\gamma (\cdot -\varepsilon ),a), \qquad (\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}$$

  is continuous on $[0,\infty [\times {\mathbb {R}}$ $;$

• (iii) $D^V u$ and $D^{VV}u$ exist everywhere on $\mathscr {C}([-T,0])$ and are continuous.

Remark 5

Notice that in Definition 10 we still obtain the same class of functions $\mathscr {C}^{1,2}({\text {past}}\times {\text {present}})$ if we substitute point (ii) with

• (ii’) $D^H u$ exists everywhere on $\mathscr {C}([-T,0])$ and for every $\gamma \in \mathscr {C}([-T,0[)$ there exists $\delta (\gamma )>0$ such that the map

  $$\begin{aligned} (\varepsilon ,a)\longmapsto D^H\tilde{u}(\gamma (\cdot -\varepsilon ),a), \qquad (\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}\end{aligned}$$

  (18)

  is continuous on $[0,\delta (\gamma )[\times {\mathbb {R}}$.

In particular, if (ii’) holds then we can always take $\delta (\gamma ) = \infty$ for any $\gamma \in \mathscr {C}([-T,0[)$, which implies (ii). To prove this last statement, let us proceed by contradiction assuming that

$$ \delta ^*(\gamma ) \ = \ \sup \big \{\delta (\gamma )>0:\text {the map (17) is continuous on }[0,\delta (\gamma )[\times {\mathbb {R}}\big \} \ < \ \infty . $$

Notice that $\delta ^*(\gamma )$ is in fact a max, therefore the map (18) is continuous on $[0,\delta ^*(\gamma )[\times {\mathbb {R}}$. Now, define $\bar{\gamma }(\cdot ) := \gamma (\cdot -\delta ^*(\gamma ))$. Then, by condition (ii’) there exists $\delta (\bar{\gamma })>0$ such that the map

$$ (\varepsilon ,a)\longmapsto D^H\tilde{u}(\bar{\gamma }(\cdot -\varepsilon ),a) = D^H\tilde{u}(\gamma (\cdot -\varepsilon -\delta ^*(\gamma )),a) $$

is continuous on $[0,\delta (\bar{\gamma })[\times {\mathbb {R}}$. This shows that the map (18) is continuous on $[0,\delta ^*(\gamma )+\delta (\bar{\gamma })[\times {\mathbb {R}}$, a contradiction with the definition of $\delta ^*(\gamma )$. $\Box$

We can now provide the definition of functional derivatives for a map ${\mathscr {U}}:$ $C([-T,0])$ $\rightarrow$ ${\mathbb {R}}$.

Definition 11

Let ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$ and $\eta \in C([-T,0])$. Suppose that there exists a unique extension $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ of ${\mathscr {U}}$ (e.g., if ${\mathscr {U}}$ is continuous with respect to the topology of $\mathscr {C}([-T,0])$ $)$. Then we define the following concepts.

(i) The horizontal derivative of ${\mathscr {U}}$ at $\eta$ as:

$$ D^H {\mathscr {U}}(\eta ) \ := \ D^H u(\eta ). $$

(ii) The first-order vertical derivative of ${\mathscr {U}}$ at $\eta$ as:

$$ D^V {\mathscr {U}}(\eta ) \ := \ D^V u(\eta ). $$

(iii) The second-order vertical derivative of ${\mathscr {U}}$ at $\eta$ as:

$$ D^{VV} {\mathscr {U}}(\eta ) \ := \ D^{VV} u(\eta ). $$

Definition 12

We say that ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$ is $C^{1,2}({\text {past}}\times {\text {present}})$ if ${\mathscr {U}}$ admits a (necessarily unique) extension $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ of class $\mathscr {C}^{1,2}({\text {past}}\times {\text {present}})$.

Theorem 2

Let ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$ be of class $C^{1,2}({\text {past}}\times {\text {present}})$ and $X=(X_t)_{t\in [0,T]}$ be a real continuous finite quadratic variation process. Then, the following functional Itô’s formula holds, ${\mathbb {P}}$-a.s.,

$$\begin{aligned} {\mathscr {U}}({\mathbb {X}}_t) \ = \ {\mathscr {U}}({\mathbb {X}}_0) + \int _0^t D^H {\mathscr {U}}({\mathbb {X}}_s)ds + \int _0^t D^V {\mathscr {U}}({\mathbb {X}}_s) d^- X_s + \frac{1}{2}\int _0^t D^{VV}{\mathscr {U}}({\mathbb {X}}_s)d[X]_s, \end{aligned}$$

(19)

for all $0 \le t \le T$, where the window process ${\mathbb {X}}$ was defined in (2).

Proof

Fix $t\in [0,T]$ and consider the quantity

$$\begin{aligned} I_0(\varepsilon ,t) \ = \ \int _0^t \frac{{\mathscr {U}}({\mathbb {X}}_{s+\varepsilon }) - {\mathscr {U}}({\mathbb {X}}_s)}{\varepsilon } ds \ = \ \frac{1}{\varepsilon } \int _t^{t+\varepsilon } {\mathscr {U}}({\mathbb {X}}_s) ds - \frac{1}{\varepsilon } \int _0^\varepsilon {\mathscr {U}}({\mathbb {X}}_s) ds, \qquad \varepsilon >0. \end{aligned}$$

Since the process $({\mathscr {U}}({\mathbb {X}}_s))_{s\ge 0}$ is continuous, $I_0(\varepsilon ,t)$ converges ucp to ${\mathscr {U}}({\mathbb {X}}_t) - {\mathscr {U}}({\mathbb {X}}_0)$, namely $\sup _{0 \le t \le T}|I_0(\varepsilon ,t)-({\mathscr {U}}({\mathbb {X}}_t) - {\mathscr {U}}({\mathbb {X}}_0))|$ converges to zero in probability when $\varepsilon \rightarrow 0^+$. On the other hand, we can write $I_0(\varepsilon ,t)$ in terms of the function $\tilde{u}$, defined in (13), as follows

$$ I_0(\varepsilon ,t) \ = \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s+\varepsilon |[-T,0[},X_{s+\varepsilon }) - \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s)}{\varepsilon } ds. $$

Now we split $I_0(\varepsilon ,t)$ into the sum of two terms

$$\begin{aligned} I_1(\varepsilon ,t)&= \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s+\varepsilon |[-T,0[},X_{s+\varepsilon }) - \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_{s+\varepsilon })}{\varepsilon } ds, \end{aligned}$$

(20)

$$\begin{aligned} I_2(\varepsilon ,t)&= \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s|[-T,0[},X_{s+\varepsilon }) - \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s)}{\varepsilon } ds. \end{aligned}$$

(21)

We begin proving that

$$\begin{aligned} I_1(\varepsilon ,t) \underset{\varepsilon \rightarrow 0^+}{\overset{\text {ucp}}{\longrightarrow }} \int _0^t D^H {\mathscr {U}}({\mathbb {X}}_s) ds. \end{aligned}$$

(22)

Firstly, fix $\gamma \in \mathscr {C}([-T,0[)$ and define

$$ \phi (\varepsilon ,a) \ := \ \tilde{u}(\gamma (\cdot -\varepsilon ),a), \qquad (\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}. $$

Then, denoting by $\partial _\varepsilon ^+ \phi$ the right partial derivative of $\phi$ with respect to $\varepsilon$ and using formula (17), we find

$$\begin{aligned} \partial _\varepsilon ^+ \phi (\varepsilon ,a)&= \ \lim _{r\rightarrow 0^+} \frac{\phi (\varepsilon +r,a) - \phi (\varepsilon ,a)}{r} \\&= \ -\lim _{r\rightarrow 0^+} \frac{\tilde{u}(\gamma (\cdot -\varepsilon ),a) - \tilde{u}(\gamma (\cdot -\varepsilon -r),a)}{r} \\&= \ -D^H \tilde{u}(\gamma (\cdot -\varepsilon ),a), \qquad \forall \,(\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}. \end{aligned}$$

Since $u\in \mathscr {C}^{1,2}({\text {past}}\times {\text {present}})$, we see from Definition 10(ii), that $\partial _\varepsilon ^+ \phi$ is continuous on $[0,\infty [\times {\mathbb {R}}$. It follows from a standard differential calculus result (see for example Corollary 1.2, Chap. 2, in [32]) that $\phi$ is continuously differentiable on $[0,\infty [\times {\mathbb {R}}$ with respect to its first argument. Then, for every $(\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}$, from the fundamental theorem of calculus, we have

$$ \phi (\varepsilon ,a) - \phi (0,a) \ = \ \int _0^\varepsilon \partial _\varepsilon \phi (r,a) dr, $$

which in terms of $\tilde{u}$ reads

$$\begin{aligned} \tilde{u}(\gamma (\cdot ),a) - \tilde{u}(\gamma (\cdot -\varepsilon ),a) \ = \ \int _0^\varepsilon D^H \tilde{u}(\gamma (\cdot -r),a) dr. \end{aligned}$$

(23)

Now, we rewrite, by means of a shift in time, the term $I_1(\varepsilon ,t)$ in (20) as follows:

$$\begin{aligned} I_1(\varepsilon ,t)&= \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) - \tilde{u}({\mathbb {X}}_{s-\varepsilon |[-T,0[},X_s)}{\varepsilon } ds \nonumber \\&\quad + \int _t^{t+\varepsilon } \frac{\tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) - \tilde{u}({\mathbb {X}}_{s-\varepsilon |[-T,0[},X_s)}{\varepsilon } ds \nonumber \\&\quad - \int _0^{\varepsilon } \frac{\tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) - \tilde{u}({\mathbb {X}}_{s-\varepsilon |[-T,0[},X_s)}{\varepsilon } ds. \end{aligned}$$

(24)

Plugging (23) into (24), setting $\gamma = {\mathbb {X}}_s, a = X_s$, we obtain

$$\begin{aligned} I_1(\varepsilon ,t)&= \ \int _0^t \frac{1}{\varepsilon } \bigg (\int _0^\varepsilon D^H \tilde{u}({\mathbb {X}}_{s-r|[-T,0[},X_s)dr\bigg )ds \nonumber \\&\quad + \int _t^{t+\varepsilon } \frac{1}{\varepsilon } \bigg (\int _0^\varepsilon D^H \tilde{u}({\mathbb {X}}_{s-r|[-T,0[},X_s)dr\bigg )ds \nonumber \\&\quad - \int _0^\varepsilon \frac{1}{\varepsilon } \bigg (\int _0^\varepsilon D^H \tilde{u}({\mathbb {X}}_{s-r|[-T,0[},X_s)dr\bigg )ds. \end{aligned}$$

(25)

Observe that

$$ \int _0^t \frac{1}{\varepsilon } \bigg (\int _0^\varepsilon D^H \tilde{u}({\mathbb {X}}_{s-r|[-T,0[},X_s)dr\bigg )ds \underset{\varepsilon \rightarrow 0^+}{\overset{\text {ucp}}{\longrightarrow }} \int _0^t D^H u({\mathbb {X}}_s) ds. $$

Similarly, we see that the other two terms in (25) converge ucp to zero. As a consequence, we get (22).

Regarding $I_2(\varepsilon ,t)$ in (21), it can be written, by means of the following standard Taylor’s expansion for a function $f\in C^2({\mathbb {R}})$:

$$\begin{aligned} f(b)&= \ f(a)+f'(a)(b-a)+\frac{1}{2}f''(a)(b-a)^2 \\&\quad + \int _0^1(1-\alpha )\big (f''(a+\alpha (b-a))-f''(a)\big )(b-a)^2d\alpha , \end{aligned}$$

as the sum of the following three terms:

$$\begin{aligned} I_{21}(\varepsilon ,t)&= \ \int _0^t \partial _a \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) \frac{X_{s+\varepsilon }-X_s}{\varepsilon } ds \\ I_{22}(\varepsilon ,t)&= \ \frac{1}{2}\int _0^t \partial _{aa}^2 \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) \frac{(X_{s+\varepsilon }-X_s)^2}{\varepsilon } ds \\ I_{23}(\varepsilon ,t)&= \ \int _0^t \bigg (\int _0^1(1-\alpha ) \big ( \partial _{aa}^2 \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s + \alpha (X_{s+\varepsilon }-X_s)) \\&\quad \;\, - \partial _{aa}^2 \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) \big ) \frac{(X_{s+\varepsilon }-X_s)^2}{\varepsilon } d\alpha \bigg ) ds. \end{aligned}$$

By similar arguments as in Proposition 1.2 of [39], we have

$$ I_{22}(\varepsilon ,t) \underset{\varepsilon \rightarrow 0^+}{\overset{\text {ucp}}{\longrightarrow }} \frac{1}{2}\int _0^t\partial _{aa}^2 \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) d[X]_s = \frac{1}{2}\int _0^t D^{VV} u({\mathbb {X}}_s) d[X]_s. $$

Regarding $I_{23}(\varepsilon ,t)$, for every $\omega \in \varOmega$, define $\psi _\omega :[0,T]\times [0,1]\times [0,1]\rightarrow {\mathbb {R}}$ as

$$ \psi _\omega (s,\alpha ,\varepsilon ) \ := \ (1-\alpha ) \partial _{aa}^2 \tilde{u}\big ({\mathbb {X}}_{s|[-T,0[}(\omega ),X_s(\omega ) + \alpha (X_{s+\varepsilon }(\omega )-X_s(\omega ))\big ), $$

for all $(s,\alpha ,\varepsilon )\in [0,T]\times [0,1]\times [0,1]$. Notice that $\psi _\omega$ is uniformly continuous. Denote $\rho _{\psi _\omega }$ its continuity modulus, then

$$ \sup _{t\in [0,T]} |I_{23}(\varepsilon ,t)| \ \le \ \int _0^T\rho _{\psi _\omega }(\varepsilon ) \frac{(X_{s+\varepsilon }-X_s)^2}{\varepsilon }ds. $$

Since X has finite quadratic variation, we deduce that $I_{23}(\varepsilon ,t)\rightarrow 0$ ucp as $\varepsilon \rightarrow 0^+$. Finally, because of $I_0(\varepsilon ,t)$, $I_1(\varepsilon ,t)$, $I_{22}(\varepsilon ,t)$, and $I_{23}(\varepsilon ,t)$ converge ucp, it follows that the forward integral exists:

$$ I_{21}(\varepsilon ,t) \underset{\varepsilon \rightarrow 0^+}{\overset{\text {ucp}}{\longrightarrow }} \int _0^t \partial _a \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_s) d^- X_s = \int _0^t D^V u({\mathbb {X}}_s) d^- X_s, $$

from which the claim follows.

Remark 6

Notice that, under the hypotheses of Theorem 2, the forward integral $\int _0^t D^V{\mathscr {U}}({\mathbb {X}}_s)d^-X_s$ exists as a ucp limit, which is generally not required. $\Box$

Remark 7

The definition of horizontal derivative. Notice that our definition of horizontal derivative differs from that introduced in [17], since it is based on a limit on the left, while the definition proposed in [17] would conduct to the formula

$$\begin{aligned} D^{H,+} u(\eta ) \ := \ \lim _{\varepsilon \rightarrow 0^+} \frac{\tilde{u}(\eta (\cdot +\varepsilon ) 1_{[-T,0[},\eta (0)) - \tilde{u}(\eta (\cdot ) 1_{[-T,0[},\eta (0))}{\varepsilon }. \end{aligned}$$

(26)

To give an insight into the difference between (16) and (26), let us consider a real continuous finite quadratic variation process X with associated window process ${\mathbb {X}}$. Then, in the definition (26) of $D^{H,+}u({\mathbb {X}}_t)$ we consider the increment $\tilde{u}({\mathbb {X}}_{t|[-T,0[}(\cdot +\varepsilon ),X_t) - \tilde{u}({\mathbb {X}}_{t|[-T,0[},X_t)$, comparing the present value of $u({\mathbb {X}}_t)=\tilde{u}({\mathbb {X}}_{t|[-T,0[},X_t)$ with an hypothetical future value $\tilde{u}({\mathbb {X}}_{t|[-T,0[}(\cdot +\varepsilon ),X_t)$, obtained assuming a constant time evolution for X. On the other hand, in our definition (16) we consider the increment $\tilde{u}({\mathbb {X}}_{t|[-T,0[},X_t) - \tilde{u}({\mathbb {X}}_{t-\varepsilon |[-T,0[},X_t)$, where only the present and past values of X are taken into account, and where we also extend in a constant way the trajectory of X before time 0. In particular, unlike (26), since we do not call in the future in our formula (16), we do not have to specify a future time evolution for X, but only a past evolution before time 0. This difference between (16) and (26) is crucial for the proof of the functional Itô’s formula. In particular, the adoption of (26) as definition for the horizontal derivative would require an additional regularity condition on u in order to prove an Itô formula for the process $t\mapsto u({\mathbb {X}}_t)$. Indeed, as it can be seen from the proof of Theorem 2, to prove Itô’s formula we are led to consider the term

$$ I_1(\varepsilon ,t) \ = \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s+\varepsilon |[-T,0[},X_{s+\varepsilon }) - \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_{s+\varepsilon })}{\varepsilon } ds. $$

When adopting definition (26) it is convenient to write $I_1(\varepsilon ,t)$ as the sum of the two integrals

$$\begin{aligned} I_{11}(\varepsilon ,t)&= \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s+\varepsilon |[-T,0[},X_{s+\varepsilon }) - \tilde{u}({\mathbb {X}}_{s|[-T,0[}(\cdot +\varepsilon ),X_{s+\varepsilon })}{\varepsilon } ds, \\ I_{12}(\varepsilon ,t)&= \ \int _0^t \frac{\tilde{u}({\mathbb {X}}_{s|[-T,0[}(\cdot +\varepsilon ),X_{s+\varepsilon }) - \tilde{u}({\mathbb {X}}_{s|[-T,0[},X_{s+\varepsilon })}{\varepsilon } ds. \end{aligned}$$

It can be shown quite easily that, under suitable regularity conditions on u (more precisely, if u is continuous, $D^{H,+}u$ exists everywhere on $\mathscr {C}([-T,0])$, and for every $\gamma \in \mathscr {C}([-T,0[)$ the map $(\varepsilon ,a)\longmapsto D^{H,+}\tilde{u}(\gamma (\cdot +\varepsilon ),a)$ is continuous on $[0,\infty )\times {\mathbb {R}}[$, we have

$$ I_{12}(\varepsilon ,t) \underset{\varepsilon \rightarrow 0^+}{\overset{\text {ucp}}{\longrightarrow }} \int _0^t D^{H,+} u({\mathbb {X}}_s) ds. $$

To conclude the proof of Itô’s formula along the same lines as in Theorem 2, we should prove

$$\begin{aligned} I_{11}(\varepsilon ,t) \underset{\varepsilon \rightarrow 0^+}{\overset{\text {ucp}}{\longrightarrow }} 0. \end{aligned}$$

(27)

In order to guarantee (27), we need to impose some additional regularity condition on $\tilde{u}$, and hence on u. As an example, (27) is satisfied if we assume the following condition on $\tilde{u}$: there exists a constant $C>0$ such that, for every $\varepsilon >0$,

$$ |\tilde{u}(\gamma _1,a) - \tilde{u}(\gamma _2,a)| \ \le \ C \varepsilon \sup _{x\in [-\varepsilon ,0[}|\gamma _1(x)-\gamma _2(x)|, $$

for all $\gamma _1,\gamma _2\in \mathscr {C}([-T,0[)$ and $a\in {\mathbb {R}}$, with $\gamma _1(x) = \gamma _2(x)$ for any $x\in [-T,-\varepsilon ]$. This last condition is verified if, for example, $\tilde{u}$ is uniformly Lipschitz continuous with respect to the $L^1([-T,0])$-norm on $\mathscr {C}([-T,0[)$, namely: there exists a constant $C>0$ such that

$$ |\tilde{u}(\gamma _1,a) - \tilde{u}(\gamma _2,a)| \ \le \ C \int _{[-T,0[} |\gamma _1(x) - \gamma _2(x)| dx, $$

for all $\gamma _1,\gamma _2\in \mathscr {C}([-T,0[)$ and $a\in {\mathbb {R}}$. $\Box$

We conclude this subsection providing the functional Itô’s formula for a map ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ depending also on the time variable. Firstly, we notice that for a map ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ (resp. $u:[0,T]\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$) the functional derivatives $D^H{\mathscr {U}}$, $D^V{\mathscr {U}}$, and $D^{VV}{\mathscr {U}}$ (resp. $D^Hu$, $D^Vu$, and $D^{VV}u$) are defined in an obvious way as in Definition 11 (resp. Definition 9). Moreover, given $u:[0,T]\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ we can define, as in Definition 8, a map $\tilde{u}:[0,T]\times \mathscr {C}([-T,0[)\times {\mathbb {R}}\rightarrow {\mathbb {R}}$. Then, we can give the following definitions.

Definition 13

Let I be [0, T[ or [0, T]. We say that $u:I\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ is of class $\mathscr {C}^{1,2}((I\times {\text {past}})\times {\text {present}})$ if the properties below hold.

• (i) u is continuous; 

• (ii) $\partial _tu$ exists everywhere on $I\times \mathscr {C}([-T,0])$ and is continuous; 

• (iii) $D^H u$ exists everywhere on $I\times \mathscr {C}([-T,0])$ and for every $\gamma \in \mathscr {C}([-T,0[)$ the map

  $$ (t,\varepsilon ,a)\longmapsto D^H\tilde{u}(t,\gamma (\cdot -\varepsilon ),a), \qquad (t,\varepsilon ,a)\in I\times [0,\infty [\times {\mathbb {R}}$$

  is continuous on $I\times [0,\infty [\times {\mathbb {R}}$ $;$

• (iv) $D^V u$ and $D^{VV}u$ exist everywhere on $I\times \mathscr {C}([-T,0])$ and are continuous.

Definition 14

Let I be [0, T[ or [0, T]. We say that ${\mathscr {U}}:I\times C([-T,0])\rightarrow {\mathbb {R}}$ is $C^{1,2}((I\times {\text {past}})\times {\text {present}})$ if ${\mathscr {U}}$ admits a (necessarily unique) extension $u:I\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ of class $\mathscr {C}^{1,2}((I\times {\text {past}})\times {\text {present}})$.

We can now state the functional Itô’s formula, whose proof is not reported, since it can be done along the same lines as Theorem 2.

Theorem 3

Let ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ be of class $C^{1,2}(([0,T]\times {\text {past}})\times {\text {present}})$ and $X=(X_t)_{t\in [0,T]}$ be a real continuous finite quadratic variation process. Then, the following functional Itô’s formula holds, ${\mathbb {P}}$-a.s.,

$$\begin{aligned} {\mathscr {U}}(t,{\mathbb {X}}_t)&= \ {\mathscr {U}}(0,{\mathbb {X}}_0) + \int _0^t \big (\partial _t{\mathscr {U}}(s,{\mathbb {X}}_s) + D^H {\mathscr {U}}(s,{\mathbb {X}}_s)\big )ds + \int _0^t D^V {\mathscr {U}}(s,{\mathbb {X}}_s) d^- X_s \nonumber \\&\quad \ + \frac{1}{2}\int _0^t D^{VV}{\mathscr {U}}(s,{\mathbb {X}}_s)d[X]_s, \end{aligned}$$

(28)

for all $0 \le t \le T$.

Remark 8

Notice that, as a particular case, choosing ${\mathscr {U}}(t,\eta )=F(t,\eta (0))$, for any $(t,\eta )\in [0,T]\times C([-T,0])$, with $F\in C^{1,2}([0,T]\times {\mathbb {R}})$, we retrieve the classical Itô formula for finite quadratic variation processes, i.e. (4). More precisely, in this case ${\mathscr {U}}$ admits as unique continuous extension the map $u:[0,T]\times \mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$ given by $u(t,\eta )=F(t,\eta (0))$, for all $(t,\eta )\in [0,T]\times \mathscr {C}([-T,0])$. Moreover, we see that $D^H{\mathscr {U}}\equiv 0$, while $D^V{\mathscr {U}}=\partial _x F$ and $D^{VV}{\mathscr {U}}=\partial _{xx}^2 F$, where $\partial _x F$ (resp. $\partial _{xx}^2F$) denotes the first-order (resp. second-order) partial derivative of F with respect to its second argument. $\Box$

2.4 Comparison with Banach Space Valued Calculus via Regularization

In the present subsection our aim is to make a link between functional Itô calculus, as derived in this paper, and Banach space valued stochastic calculus via regularization for window processes, which has been conceived in [13], see also [12, 14–16] for more recent developments. More precisely, our purpose is to identify the building blocks of our functional Itô’s formula (19) with the terms appearing in the Itô formula derived in Theorem 6.3 and Sect. 7.2 in [12]. While it is expected that the vertical derivative $D^V{\mathscr {U}}$ can be identified with the term $D_{dx}^{\delta _0}{\mathscr {U}}$ of the Fréchet derivative, it is more difficult to guess to which terms the horizontal derivative $D^H{\mathscr {U}}$ corresponds. To clarify this latter point, in this subsection we derive two formulae which express $D^H{\mathscr {U}}$ in terms of Fréchet derivatives of ${\mathscr {U}}$.

Let us introduce some useful notations. We denote by $BV([-T,0])$ the set of càdlàg bounded variation functions on $[-T,0]$, which is a Banach space when equipped with the norm

$$ \Vert \eta \Vert _{BV([-T,0])} \ := \ |\eta (0)| + \Vert \eta \Vert _{\text {Var}([-T,0])}, \qquad \eta \in BV([-T,0]), $$

where $\Vert \eta \Vert _{\text {Var}([-T,0])}=|d\eta |([-T,0])$ and $|d\eta |$ is the total variation measure associated to the measure $d\eta \in {\mathscr {M}}([-T,0])$ generated by $\eta$: $d\eta (]-T,-t])=\eta (-t)-\eta (-T)$, $t\in [-T,0]$. We recall from Sect. 2.1 that we extend $\eta \in BV([-T,0])$ to all $x\in {\mathbb {R}}$ setting $\eta (x) = 0$, $x<-T$, and $\eta (x)=\eta (0)$, $x\ge 0$. Let us now introduce some useful facts about tensor products of Banach spaces.

Definition 15

Let $(E,\Vert \cdot \Vert _E)$ and $(F,\Vert \cdot \Vert _F)$ be two Banach spaces.

(i) We shall denote by $E\otimes F$ the algebraic tensor product of E and F, defined as the set of elements of the form $v = \sum _{i=1}^n e_i\otimes f_i$, for some positive integer n, where $e\in E$ and $f\in F$. The map $\otimes :E\times F\rightarrow E\otimes F$ is bilinear.

(ii) We endow $E\otimes F$ with the projective norm $\pi$:

$$ \pi (v) \ := \ \inf \bigg \{\sum _{i=1}^n \Vert e_i\Vert _E\Vert f_i\Vert _F \ :\ v = \sum _{i=1}^n e_i\otimes f_i\bigg \}, \qquad \forall \,v\in E\otimes F. $$

(iii) We denote by $E\hat{\otimes }_\pi F$ the Banach space obtained as the completion of $E\otimes F$ for the norm $\pi$. We shall refer to $E\hat{\otimes }_\pi F$ as the tensor product of the Banach spaces E and F .

(iv) If E and F are Hilbert spaces, we denote $E\hat{\otimes }_h F$ the Hilbert tensor product, which is still a Hilbert space obtained as the completion of $E\otimes F$ for the scalar product $\langle e'\otimes f',e''\otimes f''\rangle := \langle e',e''\rangle _E\langle f',f''\rangle _F$, for any $e',e''\in E$ and $f',f''\in F$.

(v) The symbols $E\hat{\otimes }_\pi ^2$ and $e\otimes ^2$ denote, respectively, the Banach space $E\hat{\otimes }_\pi E$ and the element $e\otimes e$ of the algebraic tensor product $E\otimes E$.

Remark 9

(i) The projective norm $\pi$ belongs to the class of the so-called reasonable crossnorms $\alpha$ on $E\otimes F$, verifying $\alpha (e\otimes f)=\Vert e\Vert _E\Vert f\Vert _F$.

(ii) We notice, proceeding for example as in [16] (see, in particular, formula (2.1) in [16]; for more information on this subject we refer to [41]), that the dual $(E\hat{\otimes }_\pi F)^*$ of $E\hat{\otimes }_\pi F$ is isomorphic to the space of continuous bilinear forms ${\mathscr {B}}i(E,F)$, equipped with the norm $\Vert \cdot \Vert _{E,F}$ defined as

$$ \Vert \varPhi \Vert _{E,F} \ := \ \sup _{\begin{array}{c} e\in E, f\in F \\ \Vert e\Vert _E,\Vert f\Vert _F \le 1 \end{array}} |\varPhi (e,f)|, \qquad \forall \,\varPhi \in {\mathscr {B}}i(E,F). $$

Moreover, there exists a canonical isomorphism between ${\mathscr {B}}i(E,F)$ and $L(E,F^*)$, the space of bounded linear operators from E into $F^*$. Hence, we have the following chain of canonical identifications: $(E\hat{\otimes }_\pi F)^* \cong {\mathscr {B}}i(E,F) \cong L(E;F^*)$. $\Box$

Definition 16

Let E be a Banach space. We say that ${\mathscr {U}}:E\rightarrow {\mathbb {R}}$ is of class $C^2(E)$ if

• (i) $D {\mathscr {U}}$, the first Fréchet derivative of ${\mathscr {U}}$, belongs to $C(E; E^*)$ and

• (ii) $D^2 {\mathscr {U}}$, the second Fréchet derivative of ${\mathscr {U}}$, belongs to $C(E; L(E;E^*))$.

Remark 10

Take $E = C([-T,0])$ in Definition 16.

(i) First Fréchet derivative $D{\mathscr {U}}$. We have

$$ D{\mathscr {U}}:C([-T,0]) \ \longrightarrow \ (C([-T,0]))^* \cong {\mathscr {M}}([-T,0]). $$

For every $\eta \in C([-T,0])$, we shall denote $D_{dx}{\mathscr {U}}(\eta )$ the unique measure in ${\mathscr {M}}([-T,0])$ such that

$$ D{\mathscr {U}}(\eta )\varphi \ = \ \int _{[-T,0]} \varphi (x) D_{dx}{\mathscr {U}}(\eta ), \qquad \forall \,\varphi \in C([-T,0]). $$

Notice that ${\mathscr {M}}([-T,0])$ can be represented as the direct sum: ${\mathscr {M}}([-T,0]) = {\mathscr {M}}_0([-T,0])\oplus {\mathscr {D}}_0$, where we recall that ${\mathscr {M}}_0([-T,0])$ is the subset of ${\mathscr {M}}([-T,0])$ of measures $\mu$ such that $\mu (\{0\})=0$, instead ${\mathscr {D}}_0$ (which is a shorthand for ${\mathscr {D}}_0([-T,0])$) denotes the one-dimensional space of measures which are multiples of the Dirac measure $\delta _0$. For every $\eta \in C([-T,0])$ we denote by $(D_{dx}^\perp {\mathscr {U}}(\eta ),D_{dx}^{\delta _0}{\mathscr {U}}(\eta ))$ the unique pair in ${\mathscr {M}}_0([-T,0])\oplus {\mathscr {D}}_0$ such that

$$ D_{dx}{\mathscr {U}}(\eta ) \ = \ D_{dx}^\perp {\mathscr {U}}(\eta ) + D_{dx}^{\delta _0}{\mathscr {U}}(\eta ). $$

(ii) Second Fréchet derivative $D^2{\mathscr {U}}$. We have

$$\begin{aligned} D^2{\mathscr {U}}:C([-T,0]) \ \longrightarrow \ L(C([-T,0]);(C([-T,0]))^*)&\cong {\mathscr {B}}i(C([-T,0]),C([-T,0])) \\&\cong (C([-T,0])\hat{\otimes }_\pi C([-T,0]))^*, \end{aligned}$$

where we used the identifications of Remark 9(ii). Let $\eta \in C([-T,0])$; a typical situation arises when there exists $D_{dx\,dy} {\mathscr {U}}(\eta )\in {\mathscr {M}}([-T,0]^2)$ such that $D^2{\mathscr {U}}(\eta )\in L(C([-T,0]);$ $(C([-T,0]))^*)$ admits the representation

$$ D^2{\mathscr {U}}(\eta )(\varphi ,\psi ) \ = \ \int _{[-T,0]^2} \varphi (x)\psi (y) D_{dx\,dy}{\mathscr {U}}(\eta ), \qquad \forall \,\varphi ,\psi \in C([-T,0]). $$

Moreover, $D_{dx\,dy}{\mathscr {U}}(\eta )$ is uniquely determined. $\Box$

The definition below was given in [13].

Definition 17

Let E be a Banach space. A Banach subspace $(\chi ,\Vert \cdot \Vert _\chi )$ continuously injected into $(E\hat{\otimes }_\pi ^2)^*$, i.e., $\Vert \cdot \Vert _\chi \ge \Vert \cdot \Vert _{(E\hat{\otimes }_\pi ^2)^*}$, will be called a Chi-subspace (of $(E\hat{\otimes }_\pi ^2)^*$ $)$.

Remark 11

Take $E=C([-T,0])$ in Definition 17. As indicated in [13], a typical example of Chi-subspace of $C([-T,0])\hat{\otimes }_\pi ^2$ is ${\mathscr {M}}([-T,0]^2)$ equipped with the usual total variation norm, denoted by $\Vert \cdot \Vert _{\text {Var}}$. Another important Chi-subspace of $C([-T,0])\hat{\otimes }_\pi ^2$ is the following, which is also a Chi-subspace of ${\mathscr {M}}([-T,0]^2)$:

$$\begin{aligned} \chi _0&:= \ \big \{\mu \in {\mathscr {M}}([-T,0]^2):\mu (dx,dy) = g_1(x,y)dxdy + \lambda _1\delta _0(dx)\otimes \delta _0(dy) \\&\quad \ + g_2(x)dx\otimes \lambda _2\delta _0(dy) + \lambda _3\delta _0(dx)\otimes g_3(y)dy + g_4(x)\delta _y(dx)\otimes dy, \\&\quad \ g_1\in L^2([-T,0]^2),\,g_2,g_3\in L^2([-T,0]),\,g_4\in L^\infty ([-T,0]),\,\lambda _1,\lambda _2,\lambda _3\in {\mathbb {R}}\big \}. \end{aligned}$$

Using the notations of Example 3.4 and Remark 3.5 in [16], to which we refer for more details on this subject, we notice that $\chi _0$ is indeed given by the direct sum $\chi _0 = L^2([-T,0]^2) \oplus \big (L^2([-T,0])\hat{\otimes }_h {\mathscr {D}}_0\big ) \oplus \big ({\mathscr {D}}_0 \hat{\otimes }_h L^2([-T,0])\big ) \oplus {\mathscr {D}}_{0,0}([-T,0]^2) \oplus Diag([-T,0]^2)$. In the sequel, we shall refer to the term $g_4(x)\delta _y(dx)\otimes dy$ as the diagonal component and to $g_4(x)$ as the diagonal element of $\mu$. $\Box$

We can now state our first representation result for $D^H{\mathscr {U}}$.

Proposition 7

Let ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$ be continuously Fréchet differentiable. Suppose the following.

• (i) For any $\eta \in C([-T,0])$ there exists $D_\cdot ^{\text {ac}}{\mathscr {U}}(\eta )\in BV([-T,0])$ such that

  $$ D_{dx}^\perp {\mathscr {U}}(\eta ) \ = \ D_x^{\text {ac}}{\mathscr {U}}(\eta )dx. $$

• (ii) There exist continuous extensions (necessarily unique)

  $$ u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}, \qquad \qquad D_\cdot ^{\text {ac}}u:\mathscr {C}([-T,0])\rightarrow BV([-T,0]) $$

  of ${\mathscr {U}}$ and $D_\cdot ^{\text {ac}}{\mathscr {U}}$, respectively.

Then, for any $\eta \in C([-T,0])$,

$$\begin{aligned} D^H {\mathscr {U}}(\eta ) \ = \ \int _{[-T,0]} D_x^{\text {ac}} {\mathscr {U}}(\eta ) d^+ \eta (x), \end{aligned}$$

(29)

where we recall that the previous deterministic integral has been defined in Sect. 2.1.1. In particular, the horizontal derivative $D^H {\mathscr {U}}(\eta )$ and the backward integral in (29) exist.

Proof

Let $\eta \in C([-T,0])$, then starting from the left-hand side of (29), using the definition of $D^H{\mathscr {U}}(\eta )$, we are led to consider the following increment for the function u:

$$\begin{aligned} \frac{u(\eta ) - u(\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}})}{\varepsilon }. \end{aligned}$$

(30)

We shall expand (30) using a Taylor’s formula. Firstly, notice that, since ${\mathscr {U}}$ is $C^1$ Fréchet on $C([-T,0])$, for every $\eta _1\in C([-T,0])$, with $\eta _1(0)=\eta (0)$, from the fundamental theorem of calculus we have

$$ {\mathscr {U}}(\eta ) - {\mathscr {U}}(\eta _1) \ = \ \int _0^1 \bigg ( \int _{-T}^0 D_x^{\text {ac}}{\mathscr {U}}(\eta + \lambda (\eta _1-\eta ))(\eta (x)-\eta _1(x)) dx\bigg ) d\lambda . $$

Recalling from Remark 3 the density of $C_{\eta (0)}([-T,0])$ in $\mathscr {C}_{\eta (0)}([-T,0])$ with respect to the topology of $\mathscr {C}([-T,0])$, we deduce the following Taylor’s formula for u:

$$\begin{aligned} u(\eta ) - u(\eta _1) \ = \ \int _0^1 \bigg ( \int _{-T}^0 D_x^{\text {ac}}u(\eta + \lambda (\eta _1-\eta ))(\eta (x)-\eta _1(x)) dx\bigg ) d\lambda , \end{aligned}$$

(31)

for all $\eta _1\in \mathscr {C}_{\eta (0)}([-T,0])$. As a matter of fact, for any $\delta \in ]0,T/2]$ let (similarly to Remark 3(i))

$$ \eta _{1,\delta }(x) \ := \ {\left\{ \begin{array}{ll} \eta _1(x), \qquad &{}-T\le x\le -\delta , \\ \frac{1}{\delta }(\eta _1(0)-\eta _1(-\delta ))x + \eta _1(0), &{}-\delta <x\le 0 \end{array}\right. } $$

and $\eta _{1,0}:=\eta _1$. Then $\eta _{1,\delta }\in C([-T,0])$, for any $\delta \in ]0,T/2]$, and $\eta _{1,\delta }\rightarrow \eta _1$ in $\mathscr {C}([-T,0])$, as $\delta \rightarrow 0^+$. Now, define $f:[-T,0]\times [0,1]\times [0,T/2]\rightarrow {\mathbb {R}}$ as follows

$$ f(x,\lambda ,\delta ) \ := \ D_x^{\text {ac}}u(\eta + \lambda (\eta _{1,\delta }-\eta ))(\eta (x)-\eta _{1,\delta }(x)), $$

for all $(x,\lambda ,\delta )\in [-T,0]\times [0,1]\times [0,T/2]$. Now $(\lambda , \delta ) \mapsto \eta + \lambda (\eta _{1,\delta }-\eta )$, is continuous. Taking into account that $D_\cdot ^{\text {ac}}u:\mathscr {C}([-T,0])\rightarrow BV([-T,0])$ is continuous, hence bounded on compact sets, it follows that f is bounded. Then, it follows from Lebesgue dominated convergence theorem that

$$\begin{aligned}&\int _0^1 \bigg ( \int _{-T}^0 D_x^{\text {ac}}{\mathscr {U}}(\eta + \lambda (\eta _{1,\delta }-\eta ))(\eta (x)-\eta _{1,\delta }(x)) dx\bigg ) d\lambda \\&= \ \int _0^1 \bigg ( \int _{-T}^0 f(x,\lambda ,\delta ) dx\bigg ) d\lambda \ \overset{\delta \rightarrow 0^+}{\longrightarrow } \ \int _0^1 \bigg ( \int _{-T}^0 f(x,\lambda ,0) dx\bigg ) d\lambda \\&= \ \int _0^1 \bigg ( \int _{-T}^0 D_x^{\text {ac}}u(\eta + \lambda (\eta _1-\eta ))(\eta (x)-\eta _1(x)) dx\bigg ) d\lambda , \end{aligned}$$

from which we deduce (31), since ${\mathscr {U}}(\eta _{1,\delta })\rightarrow u(\eta _1)$ as $\delta \rightarrow 0^+$. Taking $\eta _1(\cdot )=\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}}$, we obtain

$$\begin{aligned}&\frac{u(\eta ) - u(\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}})}{\varepsilon } \\&= \int _0^1 \bigg ( \int _{-T}^0 D_x^{\text {ac}} u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) \frac{\eta (x)-\eta (x-\varepsilon )}{\varepsilon } dx \bigg ) d\lambda \\&= \ I_1(\eta ,\varepsilon ) + I_2(\eta ,\varepsilon )+ I_3(\eta ,\varepsilon ), \end{aligned}$$

where

$$\begin{aligned} I_1(\eta ,\varepsilon )&:= \ \int _0^1 \bigg (\int _{-T}^0 \eta (x)\frac{1}{\varepsilon } \Big ( D_x^{\text {ac}} u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) \\&\quad \ - D_{x+\varepsilon }^{\text {ac}} u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) \Big ) dx \bigg ) d\lambda , \\ I_2(\eta ,\varepsilon )&:= \ \frac{1}{\varepsilon }\int _0^1 \bigg ( \int _{-\varepsilon }^0 \eta (x) D_{x+\varepsilon }^{\text {ac}} u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) dx \bigg ) d\lambda , \\ I_3(\eta ,\varepsilon )&:= \ - \frac{1}{\varepsilon } \int _0^1 \bigg ( \int _{-T-\varepsilon }^{-T} \eta (x) D_{x+\varepsilon }^{\text {ac}} u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) dx \bigg ) d\lambda . \end{aligned}$$

Notice that, since $\eta (x)=0$ for $x<-T$, we see that $I_3(\eta ,\varepsilon )=0$. Moreover $D_x^{\text {ac}} u(\cdot )=D_0^{\text {ac}} u(\cdot )$, for $x\ge 0$, and $\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot )) 1_{[-T,0[}\rightarrow \eta$ in $\mathscr {C}([-T,0])$ as $\varepsilon \rightarrow 0^+$. Since $D_x^{\text {ac}}u$ is continuous from $\mathscr {C}([-T,0])$ into $BV([-T,0])$, we have $D_0^{\text {ac}}u(\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot )) 1_{[-T,0[})\rightarrow D_0^{\text {ac}}u(\eta )$ as $\varepsilon \rightarrow 0^+$. Then

$$\begin{aligned}&\frac{1}{\varepsilon }\int _{-\varepsilon }^0 \eta (x)D_{x+\varepsilon }^{\text {ac}} u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) dx \nonumber \\&= \ \frac{1}{\varepsilon }\int _{-\varepsilon }^0 \eta (x) dx\,D_0^{\text {ac}}u\big (\eta + \lambda \big (\eta (\cdot -\varepsilon )-\eta (\cdot )\big ) 1_{[-T,0[}\big ) \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ \eta (0)D_0^{\text {ac}}u(\eta ). \end{aligned}$$

So $I_2(\eta ,\varepsilon ) \rightarrow \eta (0) D^\mathrm{ac}_0 u(\eta )$. Finally, concerning $I_1(\eta ,\varepsilon )$, from Fubini’s theorem we obtain (denoting $\eta _{\varepsilon ,\lambda } := \eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot )) 1_{[-T,0[}$)

$$\begin{aligned} I_1(\eta ,\varepsilon )&= \ \int _0^1 \bigg (\int _{-T}^0 \eta (x)\frac{1}{\varepsilon } \Big ( D_x^{\text {ac}} u(\eta _{\varepsilon ,\lambda }) - D_{x+\varepsilon }^{\text {ac}} u(\eta _{\varepsilon ,\lambda }) \Big ) dx \bigg ) d\lambda \\&= \ -\int _0^1 \bigg (\int _{-T}^0 \eta (x)\frac{1}{\varepsilon } \bigg ( \int _{]x,x+\varepsilon ]} D_{dy}^{\text {ac}} u(\eta _{\varepsilon ,\lambda })\bigg ) dx \bigg ) d\lambda \\&= \ -\int _0^1 \bigg (\int _{]-T,\varepsilon ]} \frac{1}{\varepsilon } \bigg ( \int _{(-T)\vee (y-\varepsilon )}^{0\wedge y} \eta (x) dx\bigg ) D_{dy}^{\text {ac}} u(\eta _{\varepsilon ,\lambda }) \bigg ) d\lambda \\&= \ I_{11}(\eta ,\varepsilon ) + I_{12}(\eta ,\varepsilon ), \end{aligned}$$

where

$$\begin{aligned} I_{11}(\eta ,\varepsilon )&:= \ -\int _0^1 \bigg (\int _{]-T,\varepsilon ]} \frac{1}{\varepsilon } \bigg ( \int _{(-T)\vee (y-\varepsilon )}^{0\wedge y} \eta (x) dx\bigg ) \Big (D_{dy}^{\text {ac}} u(\eta _{\varepsilon ,\lambda }) - D_{dy}^{\text {ac}} u(\eta )\Big ) \bigg ) d\lambda , \\ I_{12}(\eta ,\varepsilon )&:= \ -\int _0^1 \bigg (\int _{]-T,\varepsilon ]} \frac{1}{\varepsilon } \bigg ( \int _{(-T)\vee (y-\varepsilon )}^{0\wedge y} \eta (x) dx\bigg ) D_{dy}^{\text {ac}} u(\eta ) \bigg ) d\lambda \\&= \ -\bigg (\int _{]-T,\varepsilon ]} \frac{1}{\varepsilon } \bigg ( \int _{(-T)\vee (y-\varepsilon )}^{0\wedge y} \eta (x) dx\bigg ) D_{dy}^{\text {ac}} u(\eta ). \end{aligned}$$

Recalling that $D_x^{\text {ac}} u(\cdot )=D_0^{\text {ac}} u(\cdot )$, for $x\ge 0$, we see that in $I_{11}(\eta ,\varepsilon )$ and $I_{12}(\eta ,\varepsilon )$ the integrals on $]{-}T,\varepsilon ]$ are equal to the same integrals on $]{-}T,0]$, i.e.,

$$\begin{aligned} I_{11}(\eta ,\varepsilon )&= \ -\int _0^1 \bigg (\int _{]-T,0]} \frac{1}{\varepsilon } \bigg ( \int _{(-T)\vee (y-\varepsilon )}^{0\wedge y} \eta (x) dx\bigg ) \Big (D_{dy}^{\text {ac}} u(\eta _{\varepsilon ,\lambda }) - D_{dy}^{\text {ac}} u(\eta )\Big ) \bigg ) d\lambda \\&= \ -\int _0^1 \bigg (\int _{]-T,0]} \frac{1}{\varepsilon } \bigg ( \int _{y-\varepsilon }^y \eta (x) dx\bigg ) \Big (D_{dy}^{\text {ac}} u(\eta _{\varepsilon ,\lambda }) - D_{dy}^{\text {ac}} u(\eta )\Big ) \bigg ) d\lambda , \\ I_{12}(\eta ,\varepsilon )&= \ -\int _{]-T,0]} \frac{1}{\varepsilon } \bigg ( \int _{(-T)\vee (y-\varepsilon )}^{0\wedge y} \eta (x) dx\bigg ) D_{dy}^{\text {ac}} u(\eta ) \\&= \ -\int _{]-T,0]} \frac{1}{\varepsilon } \bigg ( \int _{y-\varepsilon }^y \eta (x) dx\bigg ) D_{dy}^{\text {ac}} u(\eta ). \end{aligned}$$

Now, observe that

$$ |I_{11}(\eta ,\varepsilon )| \ \le \ \Vert \eta \Vert _\infty \Vert D_\cdot ^\text {ac}u(\eta _{\varepsilon ,\lambda })-D_\cdot ^\text {ac}u(\eta )\Vert _{\text {Var}([-T,0])} \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0. $$

Moreover, since $\eta$ is continuous at $y\in ]{-}T,0]$, we deduce that $\int _{y-\varepsilon }^y \eta (x) dx/\varepsilon \rightarrow \eta (y)$ as $\varepsilon \rightarrow 0^+$. Therefore, by Lebesgue’s dominated convergence theorem, we get

$$ I_{12}(\eta ,\varepsilon ) \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ -\int _{]-T,0]} \eta (y) D_{dy}^{\text {ac}} u(\eta ). $$

So $I_1(\eta ,\varepsilon ) \rightarrow -\int _{]-T,0]} \eta (y) D_{dy}^{\text {ac}} u(\eta ).$ In conclusion, we have

$$ D^H{\mathscr {U}}(\eta ) \ = \ \eta (0)D_0^{\text {ac}}u(\eta ) - \int _{]-T,0]} \eta (y) D_{dy}^{\text {ac}} u(\eta ). $$

Notice that we can suppose, without loss of generality, $D_{0^-}^{\text {ac}}{\mathscr {U}}(\eta )=D_0^{\text {ac}}{\mathscr {U}}(\eta )$. Then, the above identity gives (29) using the integration by parts formula (12). $\Box$

For our second representation result of $D^H{\mathscr {U}}$ we need the following generalization of the deterministic backward integral when the integrand is a measure.

Definition 18

Let $a < b$ be two reals. Let $f:[a,b]\rightarrow {\mathbb {R}}$ be a càdlàg function (resp. càdlàg function with f(a) = 0) and $g\in {\mathscr {M}}([-T,0])$. Suppose that the following limit

$$\begin{aligned} \int _{[a,b]}g(ds)d^+f(s)&:= \ \lim _{\varepsilon \rightarrow 0^+}\int _{[a,b]} g(ds) \frac{f_{\overline{J}}(s)-f_{\overline{J}}(s-\varepsilon )}{\varepsilon } \end{aligned}$$

(32)

$$\begin{aligned} \Big (\text {resp. }\int _{[a,b]}g(ds)d^-f(s)&:= \ \lim _{\varepsilon \rightarrow 0^+}\int _{[a,b]} g(ds) \frac{f_{\overline{J}}(s+\varepsilon )-f_{\overline{J}}(s)}{\varepsilon } \Big ) \end{aligned}$$

(33)

exists and it is finite. Then, the obtained quantity is denoted by $\int _{[a,b]} gd^+f$ $($ $\int _{[a,b]} gd^-f$ $)$ and called (deterministic, definite) backward (resp. forward) integral of g with respect to f (on [ a ,  b ]).

Proposition 8

If g is absolutely continuous with density being càdlàg (still denoted with $g$ $)$ then Definition 18 is compatible with the one in Definition 4.

Proof

Suppose that $g(ds)=g(s)ds$ with g càdlàg.

Identity (32). The right-hand side of (6) gives

$$ \int _a^b g(s)\frac{f(s)-f(s-\varepsilon )}{\varepsilon } ds, $$

which is also the right-hand side of (32) in that case.

Identity (33). The right-hand side of (5) gives, since $f(a)=0$,

$$ \frac{1}{\varepsilon }g(a)\int _{a-\varepsilon }^a f(s+\varepsilon ) ds + \int _a^b g(s)\frac{f_{\overline{J}}(s+\varepsilon )-f_{\overline{J}}(s)}{\varepsilon } ds $$

The first integral goes to zero. The second one equals the right-hand side of (33). $\Box$

Proposition 9

Let $\eta \in C([-T,0])$ be such that the quadratic variation on $[-T,0]$ exists. Let ${\mathscr {U}}:C([-T,0])\rightarrow {\mathbb {R}}$ be twice continuously Fréchet differentiable such that

$$\begin{aligned} D^2{\mathscr {U}}:C([-T,0]) \ \longrightarrow \ \chi _0\subset (C([-T,0])\hat{\otimes }_\pi C([-T,0]))^*\text { continuously with respect to }\chi _0. \end{aligned}$$

Let us also suppose the following.

• (i) $D_x^{2,Diag}{\mathscr {U}}(\eta )$, the diagonal element of the second-order derivative at $\eta$, has a set of discontinuity which has null measure with respect to $[\eta ]$ (in particular, if it is countable).

• (ii) There exist continuous extensions (necessarily unique$)$ $:$

  $$ u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}, \qquad \qquad D_{dx\,dy}^2u:\mathscr {C}([-T,0])\rightarrow \chi _0 $$

  of ${\mathscr {U}}$ and $D_{dx\,dy}^2{\mathscr {U}}$, respectively.

• (iii) The horizontal derivative $D^H{\mathscr {U}}(\eta )$ exists at $\eta \in C([-T,0])$.

Then

$$\begin{aligned} D^H {\mathscr {U}}(\eta ) \ = \ \int _{[-T,0]} D_{dx}^\perp {\mathscr {U}}(\eta ) d^+ \eta (x) - \frac{1}{2}\int _{[-T,0]} D_x^{2,Diag}{\mathscr {U}}(\eta ) d[\eta ](x). \end{aligned}$$

(34)

In particular, the backward integral in (34) exists.

Proof

Let $\eta \in C([-T,0])$. Using the definition of $D^H{\mathscr {U}}(\eta )$ we are led to consider the following increment for the function u:

$$\begin{aligned} \frac{u(\eta ) - u(\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}})}{\varepsilon }, \end{aligned}$$

(35)

with $\varepsilon >0$. Our aim is to expand (35) using a Taylor’s formula. To this end, since ${\mathscr {U}}$ is $C^2$ Fréchet, we begin noting that for every $\eta _1\in C([-T,0])$ the following standard Taylor’s expansion holds:

$$\begin{aligned} {\mathscr {U}}(\eta _1)&= \ {\mathscr {U}}(\eta ) + \int _{[-T,0]} D_{dx}{\mathscr {U}}(\eta ) \big (\eta _1(x) - \eta (x)\big ) \\&\quad \ + \frac{1}{2}\int _{[-T,0]^2} D_{dx\,dy}^2{\mathscr {U}}(\eta ) \big (\eta _1(x)-\eta (x)\big ) \big (\eta _1(y)-\eta (y)\big ) \nonumber \\&\quad \ + \int _0^1 (1-\lambda )\bigg ( \int _{[-T,0]^2} \Big ( D_{dx\,dy}^2{\mathscr {U}}(\eta + \lambda (\eta _1-\eta )) \nonumber \\&\quad \ - D_{dx\,dy}^2{\mathscr {U}}(\eta ) \Big ) \big (\eta _1(x)-\eta (x)\big ) \big (\eta _1(y)-\eta (y)\big ) \bigg ) d\lambda . \nonumber \end{aligned}$$

Now, using the density of $C_{\eta (0)}([-T,0])$ into $\mathscr {C}_{\eta (0)}([-T,0])$ with respect to the topology of $\mathscr {C}([-T,0])$ and proceeding as in the proof of Proposition 7, we deduce the following Taylor’s formula for u:

$$\begin{aligned}&\frac{u(\eta ) - u(\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}})}{\varepsilon } \\&= \ \int _{[-T,0]} D_{dx}^\perp {\mathscr {U}}(\eta ) \frac{\eta (x) - \eta (x-\varepsilon )}{\varepsilon } \nonumber \\&\quad \ - \frac{1}{2}\int _{[-T,0]^2} D_{dx\,dy}^2 {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } 1_{[-T,0[\times [-T,0[}(x,y) \nonumber \\&\quad \ - \int _0^1 (1-\lambda )\bigg ( \int _{[-T,0]^2} \Big ( D_{dx\,dy}^2 u(\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}) \nonumber \\&\quad \ - D_{dx\,dy}^2 {\mathscr {U}}(\eta ) \Big ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } 1_{[-T,0[\times [-T,0[}(x,y) \bigg ) d\lambda . \nonumber \end{aligned}$$

(36)

Recalling the definition of $\chi _0$ given in Remark 11, we notice that (due to the presence of the indicator function $1_{[-T,0[\times [-T,0[}$)

$$\begin{aligned}&\int _{[-T,0]^2} D_{dx\,dy}^2 {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } 1_{[-T,0[\times [-T,0[}(x,y) \\&= \ \int _{[-T,0]^2} D_{x\,y}^{2,L^2} {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } dxdy \\&\quad \ + \int _{[-T,0]} D_x^{2,Diag} {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx, \end{aligned}$$

where, by hypothesis, the maps $\eta \in \mathscr {C}([-T,0])\mapsto D_{x\,y}^{2,L^2} u(\eta )\in L^2([-T,0]^2)$ and $\eta \in \mathscr {C}([-T,0])\mapsto D_{x}^{2,Diag}u(\eta )\in L^\infty ([-T,0])$ are continuous. In particular, (36) becomes

$$\begin{aligned} \frac{u(\eta ) - u(\eta (\cdot -\varepsilon )1_{[-T,0[}+\eta (0)1_{\{0\}})}{\varepsilon } \ = \ I_1(\varepsilon ) + I_2(\varepsilon ) + I_3(\varepsilon ) + I_4(\varepsilon ) + I_5(\varepsilon ), \end{aligned}$$

(37)

where

$$\begin{aligned} I_1(\varepsilon )&:= \ \int _{[-T,0]} D_{dx}^\perp {\mathscr {U}}(\eta ) \frac{\eta (x) - \eta (x-\varepsilon )}{\varepsilon }, \\ I_2(\varepsilon )&:= \ - \frac{1}{2}\int _{[-T,0]^2} D_{x\,y}^{2,L^2} {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } dxdy, \\ I_3(\varepsilon )&:= \ - \frac{1}{2}\int _{[-T,0]} D_x^{2,Diag} {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx, \\ I_4(\varepsilon )&:= \ - \int _0^1 (1-\lambda )\bigg ( \int _{[-T,0]^2} \Big ( D_{x\,y}^{2,L^2} u(\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}) \\&\quad \ \ \,- D_{x\,y}^{2,L^2} {\mathscr {U}}(\eta ) \Big ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } dx\,dy \bigg ) d\lambda , \\ I_5(\varepsilon )&:= \ - \int _0^1 (1-\lambda )\bigg ( \int _{[-T,0]} \Big ( D_x^{2,Diag} u(\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}) \\&\quad \ \ \,- D_x^{2,Diag} {\mathscr {U}}(\eta ) \Big ) \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \bigg ) d\lambda . \end{aligned}$$

Firstly, we shall prove that

$$\begin{aligned} I_2(\varepsilon ) \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0. \end{aligned}$$

(38)

To this end, for every $\varepsilon >0$, we define the operator $T_\varepsilon :L^2([-T,0]^2)\rightarrow {\mathbb {R}}$ as follows:

$$\begin{aligned} T_\varepsilon \, g \ = \ \int _{[-T,0]^2} g(x,y) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } dxdy, \qquad \forall \, g\in L^2([-T,0]^2). \end{aligned}$$

Then $T_\varepsilon \in L^2([-T,0])^*$. Indeed, from Cauchy-Schwarz inequality,

$$\begin{aligned} |T_\varepsilon \,g|&\le \ \Vert g\Vert _{L^2([-T,0]^2)} \sqrt{\int _{[-T,0]^2}\frac{(\eta (x)-\eta (x-\varepsilon ))^2 (\eta (y)-\eta (y-\varepsilon ))^2}{\varepsilon ^2}dxdy} \\&= \ \Vert g\Vert _{L^2([-T,0]^2)} \int _{[-T,0]} \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \end{aligned}$$

and the latter quantity is bounded with respect to $\varepsilon$ since the quadratic variation of $\eta$ on $[-T,0]$ exists. In particular, we have proved that for every $g\in L^2([-T,0]^2)$ there exists a constant $M_g\ge 0$ such that

$$ \sup _{0 < \varepsilon < 1} |T_\varepsilon \,g| \ \le \ M_g. $$

It follows from Banach-Steinhaus theorem that there exists a constant $M\ge 0$ such that

$$\begin{aligned} \sup _{0 < \varepsilon < 1} \Vert T_\varepsilon \Vert _{L^2([-T,0])^*} \ \le \ M. \end{aligned}$$

(39)

Now, let us consider the set ${\mathscr {S}}:= \{g\in L^2([-T,0]^2):g(x,y) = e(x)f(y),\,\text {with}\,e,f\in C^1([-T,0])\}$, which is dense in $L^2([-T,0]^2)$. Let us show that

$$\begin{aligned} T_\varepsilon \,g \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0, \qquad \forall \,g\in {\mathscr {S}}. \end{aligned}$$

(40)

Fix $g\in {\mathscr {S}}$, with $g(x,y)=e(x)f(y)$ for any $(x,y)\in [-T,0]$, then

$$\begin{aligned} T_\varepsilon \,g \ = \ \frac{1}{\varepsilon } \int _{[-T,0]} e(x) \big (\eta (x)\,-\,\eta (x-\varepsilon )\big ) dx \int _{[-T,0]} f(y) \big (\eta (y)\,-\,\eta (y-\varepsilon )\big ) dy. \end{aligned}$$

(41)

We have

$$\begin{aligned}&\bigg |\int _{[-T,0]} e(x) \big (\eta (x) - \eta (x-\varepsilon )\big ) dx\bigg | \ = \ \bigg |\int _{[-T,0]} \big (e(x) - e(x+\varepsilon )\big ) \eta (x) dx \\&- \int _{[-T-\varepsilon ,-T]}e(x+\varepsilon )\eta (x) dx + \int _{[-\varepsilon ,0]} e(x+\varepsilon )\eta (x) dx\bigg | \\&\le \ \varepsilon \bigg (\int _{[-T,0]}|\dot{e}(x)|dx + 2\Vert e\Vert _\infty \bigg )\Vert \eta \Vert _\infty . \end{aligned}$$

Similarly,

$$ \bigg |\int _{[-T,0]} f(y) \big (\eta (y) - \eta (y-\varepsilon )\big ) dy\bigg | \ \le \ \varepsilon \bigg (\int _{[-T,0]}|\dot{f}(y)|dy + 2\Vert f\Vert _\infty \bigg )\Vert \eta \Vert _\infty . $$

Therefore, from (41) we find

$$ |T_\varepsilon \,g| \ \le \ \varepsilon \bigg (\int _{[-T,0]}|\dot{e}(x)|dx + 2\Vert e\Vert _\infty \bigg ) \bigg (\int _{[-T,0]}|\dot{f}(y)|dy + 2\Vert f\Vert _\infty \bigg ) \Vert \eta \Vert _\infty ^2, $$

which converges to zero as $\varepsilon$ goes to zero and therefore (40) is established. This in turn implies that

$$\begin{aligned} T_\varepsilon \,g \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0, \qquad \forall \,g\in L^2([-T,0]^2). \end{aligned}$$

(42)

Indeed, fix $g\in L^2([-T,0]^2)$ and let $(g_n)_n\subset {\mathscr {S}}$ be such that $g_n\rightarrow g$ in $L^2([-T,0]^2)$. Then

$$ |T_\varepsilon \,g| \ \le \ |T_\varepsilon (g-g_n)| + |T_\varepsilon \,g_n| \ \le \ \Vert T_\varepsilon \Vert _{L^2([-T,0]^2)^*} \Vert g-g_n\Vert _{L^2([-T,0]^2)} + |T_\varepsilon \,g_n|. $$

From (39) it follows that

$$ |T_\varepsilon \,g| \ \le \ M \Vert g-g_n\Vert _{L^2([-T,0]^2)} + |T_\varepsilon \,g_n|, $$

which implies $\limsup _{\varepsilon \rightarrow 0^+}|T_\varepsilon \,g| \le M \Vert g-g_n\Vert _{L^2([-T,0]^2)}$. Sending n to infinity, we deduce (42) and finally (38).

Let us now consider the term $I_3(\varepsilon )$ in (37). Since the quadratic variation $[\eta ]$ exists, it follows from Portmanteau’s theorem and hypothesis (i) that

$$\begin{aligned} I_3(\varepsilon ) \ = \ \int _{[-T,0]} D_x^{2,Diag} {\mathscr {U}}(\eta ) \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \ \underset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ \int _{[-T,0]} D_x^{2,Diag}{\mathscr {U}}(\eta ) d[\eta ](x). \end{aligned}$$

Regarding the term $I_4(\varepsilon )$ in (37), let $\phi _\eta :[0,1]^2\rightarrow L^2([-T,0]^2)$ be given by

$$ \phi _\eta (\varepsilon ,\lambda )(\cdot ,\cdot ) \ = \ D_{\cdot \,\cdot }^{2,L^2} u\big (\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}\big ). $$

By hypothesis, $\phi _\eta$ is a continuous map, and hence it is uniformly continuous, since $[0,1]^2$ is a compact set. Let $\rho _{\phi _\eta }$ denote the continuity modulus of $\phi _\eta$, then

$$\begin{aligned}&\big \Vert D_{\cdot \,\cdot }^{2,L^2} u\big (\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}\big ) - D_{\cdot \,\cdot }^{2,L^2} {\mathscr {U}}(\eta )\big \Vert _{L^2([-T,0]^2)} \\&= \ \Vert \phi _\eta (\varepsilon ,\lambda ) - \phi _\eta (0,\lambda )\Vert _{L^2([-T,0]^2)} \ \le \ \rho _{\phi _\eta }(\varepsilon ). \end{aligned}$$

This implies, by Cauchy-Schwarz inequality,

$$\begin{aligned}&\bigg |\int _0^1 (1-\lambda )\bigg ( \int _{[-T,0]^2} \Big ( D_{x\,y}^{2,L^2} u(\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}) \\&- D_{x\,y}^{2,L^2} {\mathscr {U}}(\eta ) \Big ) \frac{(\eta (x)-\eta (x-\varepsilon )) (\eta (y)-\eta (y-\varepsilon ))}{\varepsilon } dxdy \bigg ) d\lambda \bigg | \\&\le \ \int _0^1(1-\lambda )\big \Vert D_{\cdot \,\cdot }^{2,L^2}u(\eta +\lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0]}) \\&- D_{\cdot \,\cdot }^{2,L^2}{\mathscr {U}}(\eta )\big \Vert _{L^2([-T,0]^2)}\sqrt{\int _{[-T,0]^2}\frac{(\eta (x)-\eta (x-\varepsilon ))^2 (\eta (y)-\eta (y-\varepsilon ))^2}{\varepsilon ^2}dxdy}d\lambda \\&\le \ \int _0^1(1-\lambda )\rho _{\phi _\eta }(\varepsilon ) \bigg (\int _{[-T,0]} \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \bigg )d\lambda \\&= \ \frac{1}{2}\rho _{\phi _\eta }(\varepsilon ) \int _{[-T,0]} \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0. \end{aligned}$$

Finally, we consider the term $I_5(\varepsilon )$ in (37). Define $\psi _\eta :[0,1]^2\rightarrow L^\infty ([-T,0])$ as follows:

$$ \psi _\eta (\varepsilon ,\lambda )(\cdot ) \ = \ D_\cdot ^{2,Diag} u\big (\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}\big ). $$

We see that $\psi _\eta$ is uniformly continuous. Let $\rho _{\psi _\eta }$ denote the continuity modulus of $\psi _\eta$, then

$$\begin{aligned}&\big \Vert D_\cdot ^{2,Diag} u\big (\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}\big ) - D_\cdot ^{2,Diag} {\mathscr {U}}(\eta )\big \Vert _{L^\infty ([-T,0])} \\&= \ \Vert \psi _\eta (\varepsilon ,\lambda ) - \psi _\eta (0,\lambda )\Vert _{L^\infty ([-T,0])} \ \le \ \rho _{\psi _\eta }(\varepsilon ). \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\bigg |\int _0^1 (1-\lambda )\bigg ( \int _{[-T,0]} \Big ( D_x^{2,Diag} u(\eta + \lambda (\eta (\cdot -\varepsilon )-\eta (\cdot ))1_{[-T,0[}) \\&- D_x^{2,Diag} {\mathscr {U}}(\eta ) \Big ) \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \bigg ) d\lambda \bigg | \\&\le \ \int _0^1(1-\lambda )\bigg (\int _{[-T,0]}\rho _{\psi _\eta }(\varepsilon ) \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \bigg ) d\lambda \\&= \ \frac{1}{2}\rho _{\psi _\eta }(\varepsilon )\int _{[-T,0]} \frac{(\eta (x)-\eta (x-\varepsilon ))^2}{\varepsilon } dx \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ 0. \end{aligned}$$

In conclusion, we have proved that all the integral terms in the right-hand side of (37), unless $I_1(\varepsilon )$, admit a limit when $\varepsilon$ goes to zero. Since the left-hand side admits a limit, namely $D^H{\mathscr {U}}(\eta )$, we deduce that the backward integral

$$ I_1(\varepsilon ) \ = \ \int _{[-T,0]} D_{dx}^\perp {\mathscr {U}}(\eta ) \frac{\eta (x) - \eta (x-\varepsilon )}{\varepsilon } \ \overset{\varepsilon \rightarrow 0^+}{\longrightarrow } \ \int _{[-T,0]} D_{dx}^\perp {\mathscr {U}}(\eta ) d^+\eta (x) $$

exists and it is finite, which concludes the proof. $\Box$

3 Strong-Viscosity Solutions to Path-Dependent PDEs

In the present section we study the semilinear parabolic path-dependent equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t{\mathscr {U}}+ D^H{\mathscr {U}}+ \frac{1}{2}D^{VV}{\mathscr {U}}+ F(t,\eta ,{\mathscr {U}},D^V{\mathscr {U}}) \ = \ 0, \;\;\; &{}\forall \,(t,\eta )\in [0,T[\times C([-T,0]), \\ {\mathscr {U}}(T,\eta ) \ = \ H(\eta ), &{}\forall \,\eta \in C([-T,0]). \end{array}\right. } \end{aligned}$$

(43)

We refer to ${\mathscr {L}}{\mathscr {U}}=\partial _t{\mathscr {U}}+ D^H{\mathscr {U}}+ \frac{1}{2}D^{VV}{\mathscr {U}}$ as the path-dependent heat operator. The results of this section are generalized in [9, 10], where more general path-dependent equations will be considered. Here we shall impose the following assumptions on H and F.

(A)   $H:C([-T,0])\rightarrow {\mathbb {R}}$ and $F:[0,T]\times C([-T,0])\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ are Borel measurable functions and satisfy, for some positive constants C and m,

$$\begin{aligned} |F(t,\eta ,y,z) - F(t,\eta ,y',z')|&\le \ C(|y-y'| + |z-z'|), \\ |H(\eta )| + |F(t,\eta ,0,0)|&\le \ C\big (1 + \Vert \eta \Vert _\infty ^m\big ), \end{aligned}$$

for all $(t,\eta )\in [0,T]\times C([-T,0])$, $y,y'\in {\mathbb {R}}$, and $z,z'\in {\mathbb {R}}$.

3.1 Strict Solutions

In the present subsection, we provide the definition of strict solution to Eq. (43) and we state an existence and uniqueness result.

Definition 19

A function ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ in $C^{1,2}(([0,T[\times \text {past})\times \text {present})\,\cap \,C([0,T]\times C([-T,0]))$, which solves Eq. (43), is called a strict solution to (43).

We now introduce some additional notations. Let $(\varOmega ,{\mathscr {F}},{\mathbb {P}})$ be a complete probability space on which a real Brownian motion $W=(W_t)_{t\ge 0}$ is defined. Let ${\mathbb {F}}=({\mathscr {F}}_t)_{t\ge 0}$ denote the completion of the natural filtration generated by W.

• ${\mathbb {S}}^p(t,T)$, $p\ge 1$, $0 \le t \le T$, the set of real càdlàg ${\mathbb {F}}$-adapted processes $Y=(Y_s)_{t\le s\le T}$ such that

  $$ \Vert Y\Vert _{_{{\mathbb {S}}^p(t,T)}}^p := \ {\mathbb {E}}\Big [ \sup _{t\le s\le T} |Y_s|^p \Big ] \ < \ \infty . $$

• ${\mathbb {H}}^p(t,T)^d$, p $\ge$ 1, $0 \le t \le T$, the set of ${\mathbb {R}}^d$-valued predictable processes $Z=(Z_s)_{t\le s\le T}$ such that

  $$ \Vert Z\Vert _{_{{\mathbb {H}}^p(t,T)^d}}^p := \ {\mathbb {E}}\bigg [\bigg (\int _t^T |Z_s|^2 ds\bigg )^{\frac{p}{2}}\bigg ] \ < \ \infty . $$

  We simply write ${\mathbb {H}}^p(t,T)$ when $d=1$.

• ${\mathbb {A}}^{+,2}(t,T)$, $0 \le t \le T$, the set of real nondecreasing predictable processes K $=$ $(K_s)_{t\le s\le T}\in {\mathbb {S}}^2(t,T)$ with $K_t$ $=$ 0, so that

  $$ \Vert K\Vert _{_{{\mathbb {S}}^2(t,T)}}^2 := \ {\mathbb {E}}\big [|K_T|^2\big ]. $$

• ${\mathbb {L}}^p(t,T;{\mathbb {R}}^m)$, $p\ge 1$, $0 \le t \le T$, the set of ${\mathbb {R}}^m$-valued ${\mathbb {F}}$-predictable processes $\phi = (\phi _s)_{t \le s \le T}$ such that

  $$ \Vert \phi \Vert _{_{{\mathbb {L}}^p(t,T;{\mathbb {R}}^m)}}^p := \ {\mathbb {E}}\bigg [\int _t^T |\phi _s|^p ds\bigg ] \ < \ \infty . $$

Definition 20

Let $t\in [0,T]$ and $\eta \in C([-T,0])$. Then, we define the stochastic flow

$$ \mathbb W_s^{t,\eta }(x) \ = \ {\left\{ \begin{array}{ll} \eta (x+s-t), &{}-T \le x \le t-s, \\ \eta (0) + W_{x+s} - W_t, \qquad &{}t-s < x \le 0, \end{array}\right. } $$

for any $t \le s \le T$.

Theorem 4

Suppose that Assumption (A) holds. Let ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ be a strict solution to Eq. (43), satisfying the polynomial growth condition

$$\begin{aligned} |{\mathscr {U}}(t,\eta )| \ \le \ C\big (1 + \Vert \eta \Vert _\infty ^m\big ), \qquad \forall \,(t,\eta )\in [0,T]\times C([-T,0]). \end{aligned}$$

(44)

for some positive constants C and m. Then, we have

$$ {\mathscr {U}}(t,\eta ) \ = \ Y_t^{t,\eta }, \qquad \forall \,(t,\eta )\in [0,T]\times C([-T,0]), $$

where $(Y_s^{t,\eta },Z_s^{t,\eta })_{s\in [t,T]} = ({\mathscr {U}}(s,\mathbb W_s^{t,\eta }),D^V{\mathscr {U}}(s,\mathbb W_s^{t,\eta })1_{[t,T[}(s))_{s\in [t,T]}\in {\mathbb {S}}^2(t,T)\times {\mathbb {H}}^2(t,T)$ is the solution to the backward stochastic differential equation: ${\mathbb {P}}$-a.s.,

$$ Y_s^{t,\eta } \ = \ H(\mathbb W_T^{t,\eta }) + \int _s^T F(r,\mathbb W_r^{t,\eta },Y_r^{t,\eta },Z_r^{t,\eta }) dr - \int _s^T Z_r^{t,\eta } dW_r, \qquad t \le s \le T. $$

In particular, there exists at most one strict solution to Eq. (43).

Proof

Fix $(t,\eta )\in [0,T[\times C([-T,0])$ and set, for all $t \le s \le T$,

$$ Y_s^{t,\eta } \ = \ {\mathscr {U}}(s,\mathbb W_s^{t,\eta }), \qquad Z_s^{t,\eta } \ = \ D^V{\mathscr {U}}(s,\mathbb W_s^{t,\eta })1_{[t,T[}(s). $$

Then, for any $T_0\in [t,T[$, applying Itô formula (28) to ${\mathscr {U}}(s,\mathbb W_s^{t,\eta })$ and using the fact that ${\mathscr {U}}$ solves Eq. (43), we find, ${\mathbb {P}}$-a.s.,

$$\begin{aligned} Y_s^{t,\eta } \ = \ Y_{T_0}^{t,\eta } + \int _s^{T_0} F(r,\mathbb W_r^{t,\eta },Y_r^{t,\eta },Z_r^{t,\eta }) dr - \int _s^{T_0} Z_r^{t,\eta } dW_r, \qquad t \le s \le T_0. \end{aligned}$$

(45)

The claim would follow if we could pass to the limit in (45) as $T_0\rightarrow T$. To do this, we notice that it follows from Proposition B.1 in [10] that there exists a positive constant c, depending only on T and the constants C and m appearing in the statement of the present Theorem 4, such that

$$ {\mathbb {E}}\int _t^{T_0} |Z_s^{t,\eta }|^2 ds \ \le \ c\Vert Y^{t,\eta }\Vert _{{\mathbb {S}}^2(t,T)}^2 + c{\mathbb {E}}\int _t^T |F(r,\mathbb W_r^{t,\eta },0,0)|^2 dr, \qquad \forall \,T_0\in [t,T[. $$

We recall that, for any $q\ge 1$,

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{t\le s\le T}\Vert \mathbb W_s^{t,\eta }\Vert _\infty ^q\Big ] \ < \ \infty . \end{aligned}$$

(46)

Notice that from (44) and (46) we have $\Vert Y^{t,\eta }\Vert _{{\mathbb {S}}^2(t,T)}<\infty$, so that $Y\in {\mathbb {S}}^2(t,T)$. Then, from the monotone convergence theorem we find

$$ {\mathbb {E}}\int _t^T |Z_s^{t,\eta }|^2 ds \ \le \ c\Vert Y^{t,\eta }\Vert _{{\mathbb {S}}^2(t,T)}^2 + c{\mathbb {E}}\int _t^T |F(r,\mathbb W_r^{t,\eta },0,0)|^2 dr. $$

Therefore, it follows from the polynomial growth condition of F and (46) that $Z\in {\mathbb {H}}^2(t,T)$. This implies, using the Lipschitz character of F in (y, z), that ${\mathbb {E}}\int _t^T |F(r,\mathbb W_r^{t,\eta },Y_r^{t,\eta },Z_r^{t,\eta })|^2 dr$ $<\infty$, so that we can pass to the limit in (45) and we get the claim. $\Box$

We conclude this subsection with an existence result for the path-dependent heat equation, namely for Eq. (43) with $F\equiv 0$, for which we provide an ad hoc proof. For more general cases we refer to [9].

Theorem 5

Suppose that Assumption (A) holds. Let $F\equiv 0$ and H be given by, for all $\eta \in C([-T,0])$, (the deterministic integrals are defined according to Definition 4(i))

$$\begin{aligned} H(\eta ) \ = \ h\bigg (\int _{[-T,0]}\varphi _1(x+T)d^-\eta (x),\ldots ,\int _{[-T,0]}\varphi _N(x+T)d^-\eta (x)\bigg ), \end{aligned}$$

(47)

where

• h belongs $C^2({\mathbb {R}}^N)$ and its second order partial derivatives satisfy a polynomial growth condition,

• $\varphi _1,\ldots ,\varphi _N\in C^2([0,T])$.

Then, there exists a unique strict solution ${\mathscr {U}}$ to the path-dependent heat Eq. (43), which is given by

$$ {\mathscr {U}}(t,\eta ) \ = \ {\mathbb {E}}\big [H(\mathbb W_T^{t,\eta })\big ], \qquad \forall \,(t,\eta )\in [0,T]\times C([-T,0]). $$

Proof

Let us consider the function ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ given by, for all $(t,\eta )\in [0,T]\times C([-T,0])$,

$$\begin{aligned} {\mathscr {U}}(t,\eta )&= \ {\mathbb {E}}\big [H(\mathbb W_T^{t,\eta })\big ] \\&= \ {\mathbb {E}}\bigg [h\bigg (\int _{[-t,0]}\varphi _1(x+t)d^-\eta (x) + \int _t^T\varphi _1(s)dW_s,\ldots \bigg )\bigg ] \\&= \ \varPsi \bigg (t,\int _{[-t,0]}\varphi _1(x+t)d^-\eta (x),\ldots ,\int _{[-t,0]}\varphi _N(x+t)d^-\eta (x)\bigg ), \end{aligned}$$

where

$$ \varPsi (t,x_1,\ldots ,x_N) \ = \ {\mathbb {E}}\bigg [h\bigg (x_1+\int _t^T\varphi _1(s)dW_s,\ldots ,x_N+\int _t^T\varphi _N(s)dW_s\bigg )\bigg ], $$

for any $(t,x_1,\ldots ,x_N)\in [0,T]\times {\mathbb {R}}^N$. Notice that, for any $i,j=1,\ldots ,N$,

$$\begin{aligned} D_{x_i}\varPsi (t,x_1,\ldots ,x_N)&= \ {\mathbb {E}}\bigg [D_{x_i} h\bigg (x_1+\int _t^T\varphi _1(s)dW_s,\ldots ,x_N+\int _t^T\varphi _N(s)dW_s\bigg )\bigg ], \\ D_{x_ix_j}^2\varPsi (t,x_1,\ldots ,x_N)&= \ {\mathbb {E}}\bigg [D_{x_ix_j}^2 h\bigg (x_1+\int _t^T\varphi _1(s)dW_s,\ldots ,x_N+\int _t^T\varphi _N(s)dW_s\bigg )\bigg ], \end{aligned}$$

so that $\varPsi$ and its first and second spatial derivatives are continuous on $[0,T]\times {\mathbb {R}}^N$. Let us focus on the time derivative $\partial _t\varPsi$ of $\varPsi$. We have, for any $\delta >0$ such that $t+\delta \in [0,T]$,

$$\begin{aligned}&\frac{\varPsi (t+\delta ,x_1,\ldots ,x_N)-\varPsi (t,x_1,\ldots ,x_N)}{\delta } \\&= \ \frac{1}{\delta }{\mathbb {E}}\bigg [h\bigg (x_1+\int _{t+\delta }^T\varphi _1(s)dW_s,\ldots \bigg )-h\bigg (x_1+\int _t^T\varphi _1(s)dW_s,\ldots \bigg )\bigg ]. \end{aligned}$$

Then, using a standard Taylor formula, we find

$$\begin{aligned}&\frac{\varPsi (t+\delta ,x_1,\ldots ,x_N)-\varPsi (t,x_1,\ldots ,x_N)}{\delta } \\&= \ -\frac{1}{\delta }{\mathbb {E}}\bigg [\int _0^1 \sum _{i=1}^N D_{x_i}h\bigg (x_1+\int _t^T\varphi _1(s)dW_s - \alpha \int _t^{t+\delta }\varphi _1(s)dW_s,\ldots \bigg )\int _t^{t+\delta }\varphi _i(s)dW_sd\alpha \bigg ]. \nonumber \end{aligned}$$

(48)

Now, it follows from the integration by parts formula of Malliavin calculus, see, e.g., formula (1.42) in [31] (taking into account that Itô integrals are Skorohod integrals), that, for any $i=1,\ldots ,N$,

$$\begin{aligned}&{\mathbb {E}}\bigg [D_{x_i}h\bigg (x_1+\int _t^T\varphi _1(s)\big (1-\alpha 1_{[t,t+\delta ]}(s)\big )dW_s,\ldots \bigg )\int _t^{t+\delta }\varphi _i(s)dW_s\bigg ] \\&= \ (1-\alpha ){\mathbb {E}}\bigg [\sum _{j=1}^N D_{x_ix_j}^2 h\bigg (x_1+\int _t^T\varphi _1(s)\big (1-\alpha 1_{[t,t+\delta ]}(s)\big )dW_s,\ldots \bigg )\int _t^{t+\delta }\varphi _i(s)\varphi _j(s)ds\bigg ]. \nonumber \end{aligned}$$

(49)

Then, plugging (49) into (48) and letting $\delta \rightarrow 0^+$, we get (recalling that $D_{x_ix_j}^2 h$ has polynomial growth, for any i, j)

$$\begin{aligned} \partial _t^+\varPsi (t,x_1,\ldots ,x_N) \ = \ -\frac{1}{2}{\mathbb {E}}\bigg [\sum _{i,j=1}^N D_{x_ix_j}^2 h\bigg (x_1+\int _t^T\varphi _1(s)dW_s,\ldots \bigg )\varphi _i(t)\varphi _j(t)\bigg ], \end{aligned}$$

(50)

for any $(t,x_1,\ldots ,x_N)\in [0,T[\times {\mathbb {R}}^N$, where $\partial _t^+\varPsi$ denotes the right-time derivative of $\varPsi$. Since $\varPsi$ and $\partial _t^+\varPsi$ are continuous, we deduce that $\partial _t\varPsi$ exists and is continuous on [0, T[ (see for example Corollary 1.2, Chap. 2, in [32]). Moreover, from the representation formula (50) we see that $\partial _t\varPsi$ exists and is continuous up to time T. Furthermore, from the expression of $D_{x_ix_j}^2 \varPsi$, we see that

$$ \partial _t\varPsi (t,x_1,\ldots ,x_N) \ = \ -\frac{1}{2}\sum _{i,j=1}^N \varphi _i(t)\varphi _j(t) D_{x_ix_j}^2 \varPsi (t,x_1,\ldots ,x_N). $$

Therefore, $\varPsi \in C^{1,2}([0,T]\times {\mathbb {R}}^N)$ and is a classical solution to the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \varPsi (t,\mathbf x) + \frac{1}{2}\sum _{i,j=1}^N \varphi _i(t)\varphi _j(t) D_{x_ix_j}^2 \varPsi (t,\mathbf x) = 0, \qquad \qquad &{}\forall \,(t,\mathbf x)\in [0,T[\times {\mathbb {R}}^N, \\ \varPsi (T,\mathbf x) = h(\mathbf x), &{}\forall \,\mathbf x\in {\mathbb {R}}^N. \end{array}\right. } \end{aligned}$$

(51)

Now we express the derivatives of ${\mathscr {U}}$ in terms of $\varPsi$. We begin noting that, taking into account Proposition 4, we have

$$ \int _{[-t,0]}\varphi _i(x\,+\,t)d^-\eta (x) \ = \ \eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x)\dot{\varphi }_i(x\,+\,t)dx, \qquad \forall \,\eta \in C([-T,0]). $$

This in turn implies that ${\mathscr {U}}$ is continuous with respect to the topology of $\mathscr {C}([-T,0])$. Therefore, ${\mathscr {U}}$ admits a unique extension $u:\mathscr {C}([-T,0])\rightarrow {\mathbb {R}}$, which is given by

$$ u(t,\eta ) \ = \ \varPsi \bigg (t,\int _{[-t,0]}\varphi _1(x+t)d^-\eta (x),\ldots ,\int _{[-t,0]}\varphi _N(x+t)d^-\eta (x)\bigg ), $$

for all $(t,\eta )\in [0,T]\times \mathscr {C}([-T,0])$. We also define the map $\tilde{u}:[0,T]\times \mathscr {C}([-T,0[)\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ as in (13):

$$\begin{aligned} \tilde{u}(t,\gamma ,a) \ = \ u(t,\gamma 1_{[-T,0[}+a1_{\{0\}}) \ = \ \varPsi \bigg (t,\ldots ,a\varphi _i(t) - \int _{-t}^0 \gamma (x)\dot{\varphi }_i(x+t)dx,\ldots \bigg ), \end{aligned}$$

for all $(t,\gamma ,a)\in [0,T]\times \mathscr {C}([-T,0[)\times {\mathbb {R}}$. Let us evaluate the time derivative $\partial _t{\mathscr {U}}(t,\eta )$, for a given $(t,\eta )\in [0,T[\times C([-T,0])$:

$$\begin{aligned} \partial _t{\mathscr {U}}(t,\eta )&= \ \partial _t\varPsi \bigg (t,\int _{[-t,0]}\varphi _1(x+t)d^-\eta (x),\ldots ,\int _{[-t,0]}\varphi _N(x+t)d^-\eta (x)\bigg ) \\&\quad \ + \sum _{i=1}^N D_{x_i}\varPsi \bigg (t,\ldots ,\int _{[-t,0]}\varphi _i(x+t)d^-\eta (x),\ldots \bigg )\partial _t\bigg (\int _{[-t,0]}\varphi _i(x+t)d^-\eta (x)\bigg ). \end{aligned}$$

Notice that

$$\begin{aligned} \partial _t\bigg (\int _{[-t,0]}\varphi _i(x+t)d^-\eta (x)\bigg )&= \ \partial _t\bigg (\eta (0)\varphi (t) - \int _{-t}^0\eta (x)\dot{\varphi }_i(x+t)dx\bigg ) \\&= \ \eta (0)\dot{\varphi }(t) - \eta (-t)\dot{\varphi }_i(0^+) - \int _{-t}^0 \eta (x)\ddot{\varphi }_i(x+t) dx. \end{aligned}$$

Let us proceed with the horizontal derivative. We have

$$\begin{aligned}&D^H{\mathscr {U}}(t,\eta ) \ = \ D^H u(t,\eta ) \ = \ D^H\tilde{u}(t,\eta _{|[-T,0[},\eta (0)) \\&= \ \lim _{\varepsilon \rightarrow 0^+} \frac{\tilde{u}(t,\eta _{|[-T,0[}(\cdot ),\eta (0)) - \tilde{u}(t,\eta _{|[-T,0[}(\cdot -\varepsilon ),\eta (0))}{\varepsilon } \\&= \ \lim _{\varepsilon \rightarrow 0^+} \bigg (\frac{1}{\varepsilon }\varPsi \left( t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x)\dot{\varphi }_i(x+t)dx,\ldots \right) \\&\quad \ - \frac{1}{\varepsilon }\varPsi \left( t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x-\varepsilon )\dot{\varphi }_i(x+t)dx,\ldots \right) \bigg ). \end{aligned}$$

From the fundamental theorem of calculus, we obtain

$$\begin{aligned}&\frac{1}{\varepsilon }\varPsi \left( t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x)\dot{\varphi }_i(x+t)dx,\ldots \right) \\&- \frac{1}{\varepsilon }\varPsi \left( t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x-\varepsilon )\dot{\varphi }_i(x+t)dx,\ldots \right) \\&= \ \frac{1}{\varepsilon }\int _0^\varepsilon \sum _{i=1}^N D_{x_i}\varPsi \left( t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x-y)\dot{\varphi }_i(x+t)dx,\ldots \right) \partial _y\bigg (\eta (0)\varphi _i(t) \\&\quad \ - \int _{-t}^0 \eta (x-y)\dot{\varphi }_i(x+t)dx\bigg ) dy. \end{aligned}$$

Notice that

$$\begin{aligned}&\partial _y\bigg (\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x-y)\dot{\varphi }_i(x+t)dx\bigg ) \ = \ -\partial _y\bigg (\int _{-t-y}^{-y}\eta (x)\dot{\varphi }_i(x+y+t)dx\bigg ) \\&= \ - \bigg (\eta (-y)\dot{\varphi }_i(t) - \eta (-t-y)\dot{\varphi }_i(0^+) + \int _{-t-y}^{-y}\eta (x)\ddot{\varphi }_i(x+y+t)dx\bigg ). \end{aligned}$$

Therefore

$$\begin{aligned}&D^H{\mathscr {U}}(t,\eta ) \\&= \ -\lim _{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon }\int _0^\varepsilon \sum _{i=1}^N D_{x_i}\varPsi \bigg (t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x-y)\dot{\varphi }_i(x+t)dx,\ldots \bigg )\bigg (\eta (-y)\dot{\varphi }_i(t) \\&\quad \ - \eta (-t-y)\dot{\varphi }_i(0^+) + \int _{-t-y}^{-y}\eta (x)\ddot{\varphi }_i(x+y+t)dx\bigg ) dy \\&=\ - \sum _{i=1}^N D_{x_i}\varPsi \bigg (t,\ldots ,\eta (0)\varphi _i(t) - \int _{-t}^0 \eta (x)\dot{\varphi }_i(x+t)dx,\ldots \bigg )\bigg (\eta (0)\dot{\varphi }(t) - \eta (-t)\dot{\varphi }_i(0^+) \\&\quad \ - \int _{-t}^0 \eta (x)\ddot{\varphi }_i(x+t) dx\bigg ). \end{aligned}$$

Finally, concerning the vertical derivative we have

$$\begin{aligned} D^V{\mathscr {U}}(t,\eta ) \ = \ D^Vu(t,\eta )&= \ \partial _a\tilde{u}(t,\eta 1_{[-T,0[}+\eta (0)1_{\{0\}}) \\&= \ \sum _{i=1}^N D_{x_i}\varPsi \bigg (t,\int _{[-t,0]}\varphi _1(x+t)d^-\eta (x),\ldots \bigg )\varphi _i(t) \end{aligned}$$

and

$$\begin{aligned} D^{VV}{\mathscr {U}}(t,\eta ) \ = \ D^{VV}u(t,\eta )&= \ \partial _{aa}^2\tilde{u}(t,\eta 1_{[-T,0[}+\eta (0)1_{\{0\}}) \\&= \ \sum _{i,j=1}^N D_{x_ix_j}^2\varPsi \bigg (t,\int _{[-t,0]}\varphi _1(x+t)d^-\eta (x),\ldots \bigg )\varphi _i(t)\varphi _j(t). \end{aligned}$$

From the regularity of $\varPsi$ it follows that ${\mathscr {U}}\in C^{1,2}(([0,T]\times \text {past})\times \text {present})$. Moreover, since $\varPsi$ satisfies the Cauchy problem (51), we conclude that $\partial _t{\mathscr {U}}(t,\eta ) + D^H{\mathscr {U}}(t,\eta ) + \frac{1}{2}D^{VV}{\mathscr {U}}(t,\eta )=0$, for all $(t,\eta )\in [0,T[\times C([-T,0])$, therefore ${\mathscr {U}}$ is a classical solution to the path-dependent heat Eq. (43).

3.2 Towards a Weaker Notion of Solution: A Significant Hedging Example

In the present subsection, we consider Eq. (43) in the case $F\equiv 0$. This situation is particularly interesting, since it arises, for example, in hedging problems of path-dependent contingent claims. More precisely, consider a real continuous finite quadratic variation process X on $(\varOmega ,{\mathscr {F}},{\mathbb {P}})$ and denote ${\mathbb {X}}$ the window process associated to X. Let us assume that $[X]_t = t$, for any $t\in [0,T]$. The hedging problem that we have in mind is the following: given a contingent claim’s payoff $H(\mathbb X_T)$, is it possible to have

$$\begin{aligned} H(\mathbb X_T) \ = \ H_0 + \int _0^T Z_t \, d^- X_t, \end{aligned}$$

(52)

for some $H_0\in {\mathbb {R}}$ and some ${\mathbb {F}}$-adapted process $Z = (Z_t)_{t\in [0,T]}$ such that $Z_t=v(t,\mathbb X_t)$, with $v:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$? When X is a Brownian motion W and $\int _0^T|Z_t|^2dt<\infty$, ${\mathbb {P}}$-a.s., the previous forward integral is an Itô integral. If H is regular enough and it is cylindrical in the sense of (47), we know from Theorem 5 that there exists a unique classical solution ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ to Eq. (43).

Then, we see from Itô’s formula (28) that ${\mathscr {U}}$ satisfies, ${\mathbb {P}}$-a.s.,

$$\begin{aligned} {\mathscr {U}}(t,\mathbb X_t) \ = \ {\mathscr {U}}(0,\mathbb X_0) + \int _0^t D^V{\mathscr {U}}(s,\mathbb X_s)\,d^- X_s, \qquad 0 \le t \le T. \end{aligned}$$

(53)

In particular, (52) holds with $Z_t=D^V{\mathscr {U}}(t,\mathbb X_t)$, for any $t\in [0,T]$, $H_0 = {\mathscr {U}}(0,\mathbb X_t)$.

However, a significant hedging example is the lookback-type payoff

$$ H(\eta ) \ = \ \sup _{x\in [-T,0]}\eta (x), \qquad \forall \,\eta \in C([-T,0]). $$

We look again for ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ which verifies (53), at least for X being a Brownian motion W. Since ${\mathscr {U}}(t,\mathbb W_t)$ has to be a martingale, a candidate for ${\mathscr {U}}$ is ${\mathscr {U}}(t,\eta )={\mathbb {E}}[H(\mathbb W_T^{t,\eta })]$, for all $(t,\eta )\in [0,T]\times C([-T,0])$. However, this latter ${\mathscr {U}}$ can be shown not to be regular enough in order to be a classical solution to Eq. (43), even if it is “virtually” a solution to the path-dependent semilinear Kolmogorov equation (43). This will lead us to introduce a weaker notion of solution to Eq. (43). To characterize the map ${\mathscr {U}}$, we notice that it admits the probabilistic representation formula, for all $(t,\eta )\in [0,T]\times C([-T,0])$,

$$\begin{aligned} {\mathscr {U}}(t,\eta )&= \ {\mathbb {E}}\big [H(\mathbb W_T^{t,\eta })\big ] \ = {\mathbb {E}}\Big [\sup _{-T\le x\le 0}\mathbb W_T^{t,\eta }(x)\Big ] \\&= \ {\mathbb {E}}\Big [\Big (\sup _{-t\le x\le 0}\eta (x)\Big )\vee \Big (\sup _{t\le x\le T}\big (W_x-W_t+\eta (0)\big )\Big )\Big ] \ = \ f\Big (t,\sup _{-t\le x\le 0}\eta (x),\eta (0)\Big ), \end{aligned}$$

where the function $f:[0,T]\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} f(t,m,x) \ = \ {\mathbb {E}}\big [m \vee (S_{T-t} + x)\big ], \qquad \forall \,(t,m,x)\in [0,T]\times {\mathbb {R}}\times {\mathbb {R}}, \end{aligned}$$

(54)

with $S_t = \sup _{0 \le s \le t} W_s$, for all $t\in [0,T]$. Recalling Remark 3, it follows from the presence of $\sup _{-t\le x\le 0}\eta (x)$ among the arguments of f, that ${\mathscr {U}}$ is not continuous with respect to the topology of $\mathscr {C}([-T,0])$, therefore it can not be a classical solution to Eq. (43). However, we notice that $\sup _{-t\le x\le 0}\eta (x)$ is Lipschitz on $(C([-T,0]),\Vert \cdot \Vert _\infty )$, therefore it will follow from Theorem 7 that ${\mathscr {U}}$ is a strong-viscosity solution to Eq. (43) in the sense of Definition 21. Nevertheless, in this particular case, even if ${\mathscr {U}}$ is not a classical solution, we shall prove that it is associated to the classical solution of a certain finite dimensional PDE. To this end, we begin computing an explicit form for f, for which it is useful to recall the following standard result.

Lemma 1

(Reflection principle) For every $a>0$ and $t>0$,

$$ {\mathbb {P}}(S_t \ge a) \ = \ {\mathbb {P}}(|B_t| \ge a). $$

In particular, for each t, the random variables $S_t$ and $|B_t|$ have the same law, whose density is given by:

$$ \varphi _t(z) \ = \ \sqrt{\frac{2}{\pi t}} e^{-\frac{z^2}{2t}}1_{[0,\infty [}(z), \qquad \forall \,z\in {\mathbb {R}}. $$

Proof

See Proposition 3.7, Chapter III, in [35]. $\Box$

From Lemma 1 it follows that, for all $(t,m,x)\in [0,T[\times {\mathbb {R}}\times {\mathbb {R}}$,

$$\begin{aligned} f(t,m,x) \ = \ \int _0^\infty m \vee (z + x) \, \varphi _{T-t}(z) dz \ = \ \int _0^\infty m \vee (z + x) \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{z}{\sqrt{T-t}}\Big ) dz, \end{aligned}$$

where $\varphi (z) = \exp (z^2/2)/\sqrt{2\pi }$, $z\in {\mathbb {R}}$, is the standard Gaussian density.

Lemma 2

The function f defined in (54) is given by, for all $(t,m,x)\in [0,T[\times {\mathbb {R}}\times {\mathbb {R}}$,

$$\begin{aligned} f(t,m,x) \ = \ 2m \Big (\varPhi \Big (\frac{m-x}{\sqrt{T-t}}\Big ) - \frac{1}{2}\Big ) + 2x \Big (1 - \varPhi \Big (\frac{m-x}{\sqrt{T-t}}\Big )\Big ) + \sqrt{\frac{2(T-t)}{\pi }} e^{-\frac{(m-x)^2}{2(T-t)}}, \end{aligned}$$

for $x \le m$, and

$$ f(t,x,m) \ = \ x + \sqrt{\frac{2(T-t)}{\pi }}, $$

for $x > m$, where $\varPhi (y) = \int _{-\infty }^y \varphi (z)dz$, $y\in {\mathbb {R}}$, is the standard Gaussian cumulative distribution function.

Proof

First case: $x \le m$. We have

$$\begin{aligned} f(t,m,x) \ = \ \int _0^{m-x} m \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{z}{\sqrt{T-t}}\Big ) dz + \int _{m-x}^\infty (z+x) \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{z}{\sqrt{T-t}}\Big ) dz. \end{aligned}$$

(55)

The first integral on the right-hand side of (55) becomes

$$ \int _0^{m-x} m \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{z}{\sqrt{T-t}}\Big ) dz \ = \ 2m \int _0^{\frac{m-x}{\sqrt{T-t}}} \varphi (z) dz \ = \ 2m \Big (\varPhi \Big (\frac{m-x}{\sqrt{T-t}}\Big ) - \frac{1}{2}\Big ), $$

where $\varPhi (y) = \int _{-\infty }^y \varphi (z)dz$, $y\in {\mathbb {R}}$, is the standard Gaussian cumulative distribution function. Concerning the second integral in (55), we have

$$\begin{aligned} \int _{m-x}^\infty (z+x) \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{z}{\sqrt{T-t}}\Big ) dz&= \ 2\sqrt{T-t}\int _{\frac{m-x}{\sqrt{T-t}}}^\infty z\varphi (z) dz + 2x \int _{\frac{m-x}{\sqrt{T-t}}}^\infty \varphi (z) dz \\&= \ \sqrt{\frac{2(T-t)}{\pi }} e^{-\frac{(m-x)^2}{2(T-t)}} + 2x \Big (1 - \varPhi \Big (\frac{m-x}{\sqrt{T-t}}\Big )\Big ). \end{aligned}$$

Second case: $x>m$. We have

$$\begin{aligned} f(t,m,x)&= \ \int _0^\infty (z + x) \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{z}{\sqrt{T-t}}\Big ) dz \\&= \ 2\sqrt{T-t}\int _0^\infty z\varphi (z)dz + 2x\int _0^\infty \varphi (z)dz \ = \ \sqrt{\frac{2(T-t)}{\pi }} + x. \end{aligned}$$

We also have the following regularity result regarding the function f.

Lemma 3

The function f defined in (54) is continuous on $[0,T]\times {\mathbb {R}}\times {\mathbb {R}}$, moreover it is once (resp. twice) continuously differentiable in (t, m) (resp. in $x$ $)$ on $[0,T[\times \overline{Q}$, where $\overline{Q}$ is the closure of the set $Q := \{(m,x)\in {\mathbb {R}}\times {\mathbb {R}}:m > x \}$. In addition, the following Itô formula holds:

$$\begin{aligned} f(t,S_t,B_t)&= \ f(0,0,0) + \int _0^t \Big (\partial _t f(s,S_s,B_s) + \frac{1}{2}\partial _{xx}^2 f(s,S_s,B_s)\Big ) ds \\&\quad \ + \int _0^t \partial _m f(s,S_s,B_s) dS_s + \int _0^t \partial _x f(s,S_s,B_s) dB_s, \qquad 0 \le t \le T,\,{\mathbb {P}}\text {-a.s.} \nonumber \end{aligned}$$

(56)

Proof

The regularity properties of f are deduced from its explicit form derived in Lemma 2, after straightforward calculations. Concerning Itô’s formula (56), the proof can be done along the same lines as the standard Itô formula. We simply notice that, in the present case, only the restriction of f to $\overline{Q}$ is smooth. However, the process $((S_t,B_t))_t$ is $\overline{Q}$-valued. It is well-known that if $\overline{Q}$ would be an open set, then Itô’s formula would hold. In our case, $\overline{Q}$ is the closure of its interior Q. This latter property is enough for the validity of Itô’s formula. In particular, the basic tools for the proof of Itô’s formula are the following Taylor expansions for the function f:

$$\begin{aligned} f(t',m,x)&= \ f(t,m,x) + \partial _t f(t,m,x) (t'-t) \\&\quad \ + \int _0^1 \partial _t f(t + \lambda (t'-t),m,x)(t' - t) d\lambda , \\ f(t,m',x)&= \ f(t,m,x) + \partial _m f(t,m,x) (m'-m) \\&\quad \ + \int _0^1 \partial _m f(t,m + \lambda (m'-m),x)(m' - m) d\lambda , \\ f(t,m,x')&= \ f(t,m,x) + \partial _x f(t,m,x)(x'-x) + \frac{1}{2}\partial _{xx}^2 f(t,m,x)(x'-x)^2 \\&\quad \ + \int _0^1(1-\lambda )\big (\partial _{xx}^2 f(t,m,x+\lambda (x'-x)) - \partial _{xx}^2 f(t,m,x)\big )(x'-x)^2d\lambda , \end{aligned}$$

for all $(t,m,x)\in [0,T]\times \overline{Q}$. To prove the above Taylor formulae, note that they hold on the open set Q, using the regularity of f. Then, we can extend them to the closure of Q, since f and its derivatives are continuous on $\overline{Q}$. Consequently, Itô’s formula can be proved in the usual way. $\Box$

Even though, as already observed, ${\mathscr {U}}$ does not belong to $C^{1,2}(([0,T[\times \text {past})\times \text {present})\cap C([0,T]\times C([-T,0]))$, so that it can not be a classical solution to Eq. (43), the function f is a solution to a certain Cauchy problem, as stated in the following proposition.

Proposition 10

The function f defined in (54) solves the backward heat equation:

$$ {\left\{ \begin{array}{ll} \partial _t f(t,m,x) + \frac{1}{2}\partial _{xx}^2 f(t,m,x) \ = \ 0, \qquad \;\, &{}\forall \,(t,m,x)\in [0,T[\times \overline{Q}, \\ f(T,m,x) \ = \ m, &{}\forall \,(m,x)\in \overline{Q}. \end{array}\right. } $$

Proof

We provide two distinct proofs.

Direct proof. Since we know the explicit expression of f, we can derive the form of $\partial _t f$ and $\partial _{xx}^2 f$ by direct calculations:

$$\begin{aligned} \partial _t f(t,m,x) \ = \ - \frac{1}{\sqrt{T-t}} \varphi \Big (\frac{m-x}{\sqrt{T-t}}\Big ), \qquad \partial _{xx}^2 f(t,m,x) \ = \ \frac{2}{\sqrt{T-t}} \varphi \Big (\frac{m-x}{\sqrt{T-t}}\Big ), \end{aligned}$$

for all $(t,m,x)\in [0,T[\times \overline{Q}$, from which the claim follows.

Probabilistic proof. By definition, the process $(f(t,S_t,B_t))_{t\in [0,T]}$ is given by:

$$\begin{aligned} f(t,S_t,B_t) \ = \ {\mathbb {E}}\big [S_T\big |{\mathscr {F}}_t\big ], \end{aligned}$$

so that it is a uniformly integrable ${\mathbb {F}}$-martingale. Then, it follows from Itô’s formula (56) that

$$ \int _0^t \Big (\partial _t f(s,S_s,B_s) + \frac{1}{2}\partial _{xx}^2 f(s,S_s,B_s)\Big ) ds + \int _0^t \partial _m f(s,S_s,B_s) dS_s \ = \ 0, $$

for all $0 \le t \le T$, ${\mathbb {P}}$-almost surely. As a consequence, the claim follows if we prove that

$$\begin{aligned} \int _0^t \partial _m f(s,S_s,B_s) dS_s \ = \ 0. \end{aligned}$$

(57)

By direct calculation, we have

$$ \partial _m f(t,m,x) \ = \ 2\varPhi \Big (\frac{m-x}{\sqrt{T-t}}\Big ) - 1, \qquad \forall (t,m,x)\in [0,T[\times \overline{Q}. $$

Therefore, (57) becomes

$$\begin{aligned} \int _0^t \bigg (2\varPhi \bigg (\frac{S_s-B_s}{\sqrt{T-s}}\bigg ) - 1\bigg ) dS_s \ = \ 0. \end{aligned}$$

(58)

Now we observe that the local time of $S_s-B_s$ is equal to $2S_s$, see Exercise 2.14 in [35]. It follows that the measure $dS_s$ is carried by $\{s:S_s-B_s=0\}$. This in turn implies the validity of (58), since the integrand in (58) is zero on the set $\{s:S_s-B_s=0\}$. $\Box$

3.3 Strong-Viscosity Solutions

Motivated by previous subsection and following [10], we now introduce a concept of weak (viscosity type) solution for the path-dependent Eq. (43), which we call strong-viscosity solution to distinguish it from the classical notion of viscosity solution.

Definition 21

A function ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ is called strong-viscosity solution to Eq. (43) if there exists a sequence $({\mathscr {U}}_n,H_n,F_n)_n$ of Borel measurable functions ${\mathscr {U}}_n:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$, $H_n:C([-T,0])\rightarrow {\mathbb {R}}$, $F_n:[0,T]\times C([-T,0])\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, satisfying the following.

• (i) For all $t\in [0,T]$, the functions ${\mathscr {U}}_n(t,\cdot )$, $H_n(\cdot )$, $F_n(t,\cdot ,\cdot ,\cdot )$ are equicontinuous on compact sets and, for some positive constants C and m,

  $$\begin{aligned} |F_n(t,\eta ,y,z) - F_n(t,\eta ,y',z')|&\le \ C(|y-y'| + |z-z'|), \\ |{\mathscr {U}}_n(t,\eta )| + |H_n(\eta )| + |F_n(t,\eta ,0,0)|&\le \ C\big (1 + \Vert \eta \Vert _\infty ^m\big ), \end{aligned}$$

  for all $(t,\eta )\in [0,T]\times C([-T,0])$, $y,y'\in {\mathbb {R}}$, and $z,z'\in {\mathbb {R}}$.

• (ii) ${\mathscr {U}}_n$ is a strict solution to

  $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t{\mathscr {U}}_n + D^H{\mathscr {U}}_n + \frac{1}{2}D^{VV}{\mathscr {U}}_n + F_n(t,\eta ,{\mathscr {U}}_n,D^V{\mathscr {U}}_n) = 0, \;\; &{}\forall \,(t,\eta )\in [0,T[\times C([-T,0]), \\ {\mathscr {U}}_n(T,\eta ) = H_n(\eta ), &{}\forall \,\eta \in C([-T,0]). \end{array}\right. } \end{aligned}$$

• (iii) $({\mathscr {U}}_n,H_n,F_n)$ converges pointwise to $({\mathscr {U}},H,F)$ as n tends to infinity.

Remark 12

(i) Notice that in [8], Definition 3.4, instead of the equicontinuity on compact sets we supposed the local equicontinuity, i.e., the equicontinuity on bounded sets (see Definition 3.3 in [8]). This latter condition is stronger when ${\mathscr {U}}$ (as well as the other coefficients) is defined on a non-locally compact topological space, as for example $[0,T]\times C([-T,0])$.

(ii) We observe that, for every $t\in [0,T]$, the equicontinuity on compact sets of $({\mathscr {U}}_n(t,\cdot ))_n$ together with its pointwise convergence to ${\mathscr {U}}(t,\cdot )$ is equivalent to requiring the uniform convergence on compact sets of $({\mathscr {U}}_n(t,\cdot ))_n$ to ${\mathscr {U}}(t,\cdot )$. The same remark applies to $(H_n(\cdot ))_n$ and $(F_n(t,\cdot ,\cdot ,\cdot ))_n$, $t\in [0,T]$. $\Box$

The following uniqueness result for strong-viscosity solution holds.

Theorem 6

Suppose that Assumption (A) holds. Let ${\mathscr {U}}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ be a strong-viscosity solution to Eq. (43). Then, we have

$$ {\mathscr {U}}(t,\eta ) \ = \ Y_t^{t,\eta }, \qquad \forall \,(t,\eta )\in [0,T]\times C([-T,0]), $$

where $(Y_s^{t,\eta },Z_s^{t,\eta })_{s\in [t,T]}\in {\mathbb {S}}^2(t,T)\times {\mathbb {H}}^2(t,T)$, with $Y_s^{t,\eta }={\mathscr {U}}(s,\mathbb W_s^{t,\eta })$, solves the backward stochastic differential equation, ${\mathbb {P}}$-a.s.,

$$ Y_s^{t,\eta } \ = \ H(\mathbb W_T^{t,\eta }) + \int _s^T F(r,\mathbb W_r^{t,\eta },Y_r^{t,\eta },Z_r^{t,\eta }) dr - \int _s^T Z_r^{t,\eta } dW_r, \qquad t \le s \le T. $$

In particular, there exists at most one strong-viscosity solution to Eq. (43).

Proof

Consider a sequence $({\mathscr {U}}_n,H_n,F_n)_n$ satisfying conditions (i)-(iii) of Definition 21. For every $n\in \mathbb {N}$ and any $(t,\eta )\in [0,T]\times C([-T,0])$, we know from Theorem 4 that $(Y_s^{n,t,\eta },Z_s^{n,t,\eta })_{s\in [t,T]} = ({\mathscr {U}}_n(s,\mathbb W_s^{t,\eta }),D^V{\mathscr {U}}_n(s,\mathbb W_s^{t,\eta }))_{s\in [t,T]}\in {\mathbb {S}}^2(t,T)\times {\mathbb {H}}^2(t,T)$ is the solution to the backward stochastic differential equation, ${\mathbb {P}}$-a.s.,

$$\begin{aligned} Y_s^{n,t,\eta } \ = \ H_n(\mathbb W_T^{t,\eta }) + \int _s^T F_n(r,\mathbb W_r^{t,\eta },Y_r^{n,t,\eta },Z_r^{n,t,\eta }) dr - \int _s^T Z_r^{n,t,\eta } dW_r, \qquad t \le s \le T. \end{aligned}$$

Our aim is to pass to the limit in the above equation as $n\rightarrow \infty$, using Theorem C.1 in [10]. From the polynomial growth condition of $({\mathscr {U}}_n)_n$ and estimate (46), we see that

$$ \sup _n\Vert Y^{n,t,\eta }\Vert _{{\mathbb {S}}^p(t,T)} \ < \ \infty , \qquad \text {for any }p\ge 1. $$

This implies, using standard estimates for backward stochastic differential equations (see, e.g., Proposition B.1 in [10]) and the polynomial growth condition of $(F_n)_n$, that

$$ \sup _n\Vert Z^{n,t,\eta }\Vert _{{\mathbb {H}}^2(t,T)} \ < \ \infty . $$

Let $Y_s^{t,\eta }={\mathscr {U}}(s,\mathbb W_s^{t,\eta })$, for any $s\in [t,T]$. Then, we see that all the requirements of Theorem C.1 in [10] follow by assumptions and estimate (46), so the claim follows. $\Box$

We now prove an existence result for strong-viscosity solutions to the path-dependent heat equation, namely to Eq. (43) in the case $F\equiv 0$. To this end, we need the following stability result for strong-viscosity solutions.

Lemma 4

Let $({\mathscr {U}}_{n,k},H_{n,k},F_{n,k})_{n,k}$, $({\mathscr {U}}_n,H_n,F_n)_n$, $({\mathscr {U}},H,F)$ be Borel measurable functions such that the properties below hold.

• (i) For all $t\in [0,T]$, the functions ${\mathscr {U}}_{n,k}(t,\cdot )$, $H_{n,k}(\cdot )$, and $F_{n,k}(t,\cdot ,\cdot ,\cdot )$, $n,k\in \mathbb {N}$, are equicontinuous on compact sets and, for some positive constants C and m,

  $$\begin{aligned} |F_{n,k}(t,\eta ,y,z) - F_{n,k}(t,\eta ,y',z')|&\le \ C(|y-y'| + |z-z'|), \\ |{\mathscr {U}}_{n,k}(t,\eta )| + |H_{n,k}(\eta )| + |F_{n,k}(t,\eta ,0,0)|&\le \ C\big (1 + \Vert \eta \Vert _\infty ^m\big ), \end{aligned}$$

  for all $(t,\eta )\in [0,T]\times C([-T,0])$, $y,y'\in {\mathbb {R}}$, and $z,z'\in {\mathbb {R}}$.

• (ii) ${\mathscr {U}}_{n,k}$ is a strict solution to

  $$ {\left\{ \begin{array}{ll} \partial _t{\mathscr {U}}_{n,k} + D^H{\mathscr {U}}_{n,k} + \frac{1}{2}D^{VV}{\mathscr {U}}_{n,k} \\ + \, F_{n,k}(t,\eta ,{\mathscr {U}}_{n,k},D^V{\mathscr {U}}_{n,k}) \ = \ 0, &{}\qquad \forall \,(t,\eta )\in [0,T[\times C([-T,0]), \\ {\mathscr {U}}_{n,k}(T,\eta ) \ = \ H_{n,k}(\eta ), &{}\qquad \forall \,\eta \in C([-T,0]). \end{array}\right. } $$

• (iii) $({\mathscr {U}}_{n,k},H_{n,k},F_{n,k})$ converges pointwise to $({\mathscr {U}}_n,H_n,F_n)$ as k tends to infinity.

• (iv) $({\mathscr {U}}_n,H_n,F_n)$ converges pointwise to $({\mathscr {U}},H,F)$ as n tends to infinity.

Then, there exists a subsequence $({\mathscr {U}}_{n,k_n},H_{n,k_n},F_{n,k_n})_n$ which converges pointwise to $({\mathscr {U}},H,$ F) as n tends to infinity. In particular, ${\mathscr {U}}$ is a strong-viscosity solution to Eq. (43).

Proof

See Lemma 3.4 in [8] or Lemma 3.1 in [10]. We remark that in [8] a slightly different definition of strong-viscosity solution was used, see Remark 12(i); however, proceeding along the same lines we can prove the present result.

Theorem 7

Suppose that Assumption (A) holds. Let $F\equiv 0$ and H be continuous. Then, there exists a unique strong-viscosity solution ${\mathscr {U}}$ to the path-dependent heat Eq. (43), which is given by

$$ {\mathscr {U}}(t,\eta ) \ = \ {\mathbb {E}}\big [H(\mathbb W_T^{t,\eta })\big ], \qquad \forall \,(t,\eta )\in [0,T]\times C([-T,0]). $$

Proof

Let $(e_i)_{i\ge 0}$ be the orthonormal basis of $L^2([-T,0])$ composed by the functions

$$\begin{aligned} e_0=\frac{1}{\sqrt{T}}, \qquad e_{2i-1}(x)=\sqrt{\frac{2}{T}}\sin \bigg (\frac{2\pi }{T}(x+T)i\bigg ), \qquad e_{2i}(x)=\sqrt{\frac{2}{T}}\cos \bigg (\frac{2\pi }{T}(x+T)i\bigg ), \end{aligned}$$

for all $i\in \mathbb {N}\backslash \{0\}$. Let us define the linear operator $\varLambda :C([-T,0])\rightarrow C([-T,0])$ by

$$ (\varLambda \eta )(x) \ = \ \frac{\eta (0)-\eta (-T)}{T}x, \qquad x\in [-T,0],\,\eta \in C([-T,0]). $$

Notice that $(\eta -\varLambda \eta )(-T) = (\eta -\varLambda \eta )(0)$, therefore $\eta -\varLambda \eta$ can be extended to the entire real line in a periodic way with period T, so that we can expand it in Fourier series. In particular, for each $n\in \mathbb {N}$ and $\eta \in C([-T,0])$, consider the Fourier partial sum

$$\begin{aligned} s_n(\eta -\varLambda \eta ) \ = \ \sum _{i=0}^n (\eta _i-(\varLambda \eta )_i) e_i, \qquad \forall \,\eta \in C([-T,0]), \end{aligned}$$

(59)

where (denoting $\tilde{e}_i(x) = \int _{-T}^x e_i(y) dy$, for any $x\in [-T,0]$), by Proposition 4,

$$\begin{aligned} \eta _i \ = \ \int _{-T}^0 \eta (x)e_i(x) dx&= \ \eta (0)\tilde{e} _i(0) - \int _{[-T,0]} \tilde{e}_i(x)d^-\eta (x) \nonumber \\&= \ \int _{[-T,0]} (\tilde{e}_i(0) - \tilde{e}_i(x)) d^- \eta (x), \end{aligned}$$

(60)

since $\eta (0) = \int _{[-T,0]}d^-\eta (x)$. Moreover we have

$$\begin{aligned} (\varLambda \eta )_i&= \ \int _{-T}^0 (\varLambda \eta )(x)e_i(x) dx \ = \ \frac{1}{T} \int _{-T}^0 xe_i(x) dx \bigg (\int _{[-T,0]} d^-\eta (x) - \eta (-T)\bigg ). \end{aligned}$$

(61)

Define

$$ \sigma _n \ = \ \frac{s_0 + s_1 + \cdots + s_n}{n+1}. $$

Then, by (59),

$$ \sigma _n(\eta -\varLambda \eta ) \ = \ \sum _{i=0}^n \frac{n+1-i}{n+1} (\eta _i-(\varLambda \eta )_i) e_i, \qquad \forall \,\eta \in C([-T,0]). $$

We know from Fejér’s theorem on Fourier series (see, e.g., Theorem 3.4, Chapter III, in [44]) that, for any $\eta \in C([-T,0])$, $\sigma _n(\eta -\varLambda \eta )\rightarrow \eta -\varLambda \eta$ uniformly on $[-T,0]$, as n tends to infinity, and $\Vert \sigma _n(\eta -\varLambda \eta )\Vert _\infty \le \Vert \eta -\varLambda \eta \Vert _\infty$. Let us define the linear operator $T_n:C([-T,0])\rightarrow C([-T,0])$ by (denoting $e_{-1}(x) = x$, for any $x\in [-T,0]$)

$$\begin{aligned} T_n\eta \ = \ \sigma _n(\eta -\varLambda \eta ) + \varLambda \eta&= \ \sum _{i=0}^n \frac{n+1-i}{n+1} (\eta _i-(\varLambda \eta )_i) e_i + \frac{\eta (0) - \eta (-T)}{T}e_{-1} \nonumber \\&= \ \sum _{i=0}^n \frac{n+1-i}{n+1} x_ie_i + x_{-1}e_{-1}, \end{aligned}$$

(62)

where, using (60) and (61),

$$\begin{aligned} x_{-1}&= \int _{[-T,0]} \frac{1}{T} d^-\eta (x) - \frac{1}{T}\eta (-T), \\ x_i&= \ \int _{[-T,0]}\bigg (\tilde{e}_i(0)-\tilde{e}_i(x)- \frac{1}{T}\int _{-T}^0 xe_i(x)dx\bigg )d^-\eta (x) + \frac{1}{T}\int _{-T}^0 xe_i(x)dx\,\eta (-T), \end{aligned}$$

for $i=0,\ldots ,n$. Then, for any $\eta \in C([-T,0])$, $T_n\eta \rightarrow \eta$ uniformly on $[-T,0]$, as n tends to infinity. Furthermore, there exists a positive constant M such that

$$\begin{aligned} \Vert T_n\eta \Vert _\infty \ \le \ M\Vert \eta \Vert _\infty , \qquad \forall \,n\in \mathbb {N},\,\forall \,\eta \in C([-T,0]). \end{aligned}$$

(63)

In particular, the family of linear operators $(T_n)_n$ is equicontinuous. Now, let us define $\bar{H}_n:C([-T,0])\rightarrow {\mathbb {R}}$ as follows

$$ \bar{H}_n(\eta ) \ = \ H(T_n\eta ), \qquad \forall \,\eta \in C([-T,0]). $$

We see from (63) that the family $(\bar{H}_n)_n$ is equicontinuous on compact sets. Moreover, from the polynomial growth condition of H and (63) we have

$$ |\bar{H}_n(\eta )| \ \le \ C(1 + \Vert T_n\eta \Vert _\infty ^m) \ \le \ C(1 + M^m\Vert \eta \Vert _\infty ^m), \qquad \forall \,n\in \mathbb {N},\,\forall \,\eta \in C([-T,0]). $$

Now, we observe that since $\{e_{-1},e_0,e_1,\ldots ,e_n\}$ are linearly independent, then we see from (62) that $T_n\eta$ is completely characterized by the coefficients of $e_{-1},e_0,e_1,\ldots ,e_n$. Therefore, the function $\bar{h}_n:{\mathbb {R}}^{n+2}\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \bar{h}_n(x_{-1},\ldots ,x_n) \ = \ \bar{H}_n(\eta ) \ = \ H\bigg (\sum _{i=0}^n \frac{n+1-i}{n+1} x_ie_i + x_{-1}e_{-1}\bigg ), \quad \forall \,(x_{-1},\ldots ,x_n)\in {\mathbb {R}}^{n+2}, \end{aligned}$$

completely characterizes $\bar{H}_n$. Moreover, fix $\eta \in C([-T,0])$ and consider the corresponding coefficients $x_{-1},\ldots ,x_n$ with respect to $\{e_{-1},\ldots ,e_n\}$ in the expression (62) of $T_n\eta$. Set

$$\begin{aligned} \varphi _{-1}(x)&= \ \frac{1}{T}, \qquad \varphi _i(x) \ = \ \tilde{e}_i(0)-\tilde{e}_i(x-T)- \frac{1}{T}\int _{-T}^0 xe_i(x)dx, \qquad x\in [0,T], \\ a_{-1}&= \ -\frac{1}{T}, \qquad \;\; a_i \ = \ \frac{1}{T}\int _{-T}^0 xe_i(x)dx. \end{aligned}$$

Notice that $\varphi _{-1},\ldots ,\varphi _n\in C^\infty ([0,T])$. Then, we have

$$\begin{aligned} \bar{H}_n(\eta ) = \bar{h}_n\bigg (\int _{[-T,0]}\!\varphi _{-1}(x\,+\,T)d^-\eta (x) + a_{-1}\eta (-T),\ldots ,\int _{[-T,0]}\!\varphi _n(x\,+\,T)d^-\eta (x)\,+\,a_n\eta (-T)\bigg ). \end{aligned}$$

Let $\phi (x) = c\exp (1/(x^2 - T^2))1_{[0,T[}(x)$, $x\ge 0$, with $c>0$ such that $\int _0^\infty \phi (x)dx=1$. Define, for any $\varepsilon >0$, $\phi _\varepsilon (x) = \phi (x/\varepsilon )/\varepsilon$, $x\ge 0$. Notice that $\phi _\varepsilon \in C^\infty ([0,\infty [)$ and (denoting $\tilde{\phi }_\varepsilon (x) = \int _0^x\phi _\varepsilon (y)dy$, for any $x\ge 0$),

$$\begin{aligned} \int _{-T}^0\eta (x)\phi _\varepsilon (x+T)dx&= \ \eta (0)\tilde{\phi }_\varepsilon (T) - \int _{[-T,0]}\tilde{\phi }_\varepsilon (x+T)d^-\eta (x) \\&= \ \int _{[-T,0]}\big (\tilde{\phi }_\varepsilon (T)-\tilde{\phi }_\varepsilon (x+T)\big )d^-\eta (x). \end{aligned}$$

Therefore

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+}\int _{[-T,0]}\big (\tilde{\phi }_\varepsilon (T)\,-\,\tilde{\phi }_\varepsilon (x+T)\big )d^-\eta (x) \ = \ \lim _{\varepsilon \rightarrow 0^+} \int _{-T}^0\eta (x)\phi _\varepsilon (x+T)dx \ = \ \eta (-T). \end{aligned}$$

For this reason, we introduce the function $H_n:C([-T,0])\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} H_n(\eta ) \ = \ \bar{h}_n\bigg (\ldots ,\int _{[-T,0]}\varphi _i(x+T)d^-\eta (x) + a_i\int _{[-T,0]}\big (\tilde{\phi }_n(T)-\tilde{\phi }_n(x+T)\big )d^-\eta (x),\ldots \bigg ). \end{aligned}$$

Now, for any $n\in \mathbb {N}$, let $(h_{n,k})_{k\in \mathbb {N}}$ be a locally equicontinuous sequence of $C^2({\mathbb {R}}^{n+2};{\mathbb {R}})$ functions, uniformly polynomially bounded, such that $h_{n,k}$ converges pointwise to $h_n$, as k tends to infinity. Define $H_{n,k}:C([-T,0])\rightarrow {\mathbb {R}}$ as follows:

$$\begin{aligned} H_{n,k}(\eta ) = h_{n,k}\bigg (\ldots ,\int _{[-T,0]}\varphi _i(x+T)d^-\eta (x) + a_i\int _{[-T,0]}\big (\tilde{\phi }_n(T)-\tilde{\phi }_n(x+T)\big )d^-\eta (x),\ldots \bigg ). \end{aligned}$$

Then, we know from Theorem 5 that the function ${\mathscr {U}}_{n,k}:[0,T]\times C([-T,0])\rightarrow {\mathbb {R}}$ given by

$$ {\mathscr {U}}_{n,k}(t,\eta ) \ = \ {\mathbb {E}}\big [H_{n,k}(\mathbb W_T^{t,\eta })\big ], \qquad \forall \,(t,\eta )\in [0,T]\times C([-T,0]) $$

is a classical solution to the path-dependent heat Eq. (43). Moreover, the family $({\mathscr {U}}_{n,k})_{n,\varepsilon ,k}$ is equicontinuous on compact sets and uniformly polynomially bounded. Then, using the stability result Lemma 4, it follows that ${\mathscr {U}}$ is a strong-viscosity solution to the path-dependent heat Eq. (43). $\Box$