Functional and Banach Space Stochastic Calculi: PathDependent Kolmogorov Equations Associated with the Frame of a Brownian Motion
Abstract
First, we revisit basic theory of functional Itô/pathdependent 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 pathdependent heat equation, for which wellposedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strongviscosity 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.
Keywords
Horizontal and vertical derivative functional Itô/pathdependent calculus Banach space stochastic calculus Strongviscosity solutions Calculus via regularization2010 Math Subject Classification:
35D35 35D40 35K10 60H05 60H10 60H301 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 strongviscosity 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, 6, 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, 13, 14, 15, 16]), including the case \(B = C([T,0])\), which concerns the applications to the pathdependent 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, 6, 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 \(u:[0,T]\times \mathscr {C}([T,0])\rightarrow {\mathbb {R}}\).
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, 15, 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 pathdependent 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 pathdependent equations associated to the window Brownian motion. For more general equations we refer to [9] (for strict solutions) and to [10] (for strongviscosity 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 pathdependent 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 viscositytype solution, namely a solution which is not required to be differentiable.
The issue of providing a suitable definition of viscosity solutions for pathdependent PDEs has attracted a great interest, see Peng [33] and Tang and Zhang [42], Ekren et al. [18, 19, 20], Ren et al. [34]. In particular, the definition of viscosity solution provided by [18, 19, 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 nonMarkovian 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 strongviscosity solution to distinguish it from the classical notion of viscosity solution. A strongviscosity solution to a pathdependent 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 strongviscosity 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 FukushimaDirichlet decomposition). Instead, our definition of strongviscosity solution to the pathdependent 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 strongviscosity solution and that of viscosity solution, see Theorem 3.7 in [8]. We prove a uniqueness theorem for strongviscosity 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 pathdependent equation is not the pathdependent heat equation) and also for the application of strongviscosity 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 pathdependent 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 strongviscosity 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.
Definition 1
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 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 viceversa\()\) \(;\) then \([V] =[Y,V]= 0\). Moreover \(\int _0^\cdot Y d^V=\int _0^\cdot Y dV \), is the LebesgueStieltjes 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 nonanticipating 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 wellknown 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
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 \(b=0\).
Definition 4
Let \(f,g:[a,b]\rightarrow {\mathbb {R}}\) be càdlàg functions.
The notation concerning this integral is justified since when the integrator f has bounded variation the previous integrals are LebesgueStieltjes integrals on [a, b].
Proposition 3
Proof
Let us now introduce the deterministic covariation.
Definition 5
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
Proof
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 timeindependent 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 timedependent 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.
Remark 3
 (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).
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.
 (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 \(K\subset [T,0[\).
Remark 4
(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
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
Proof
Proposition 6
Proof
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 \(\eta \in \mathscr {C}([T,0])\).
Definition 10
 (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 mapis continuous on \([0,\infty [\times {\mathbb {R}}\) \(;\)$$ (\varepsilon ,a)\longmapsto D^H\tilde{u}(\gamma (\cdot \varepsilon ),a), \qquad (\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}$$
 (iii)
\(D^V u\) and \(D^{VV}u\) exist everywhere on \(\mathscr {C}([T,0])\) and are continuous.
Remark 5
 (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 mapis continuous on \([0,\delta (\gamma )[\times {\mathbb {R}}\).$$\begin{aligned} (\varepsilon ,a)\longmapsto D^H\tilde{u}(\gamma (\cdot \varepsilon ),a), \qquad (\varepsilon ,a)\in [0,\infty [\times {\mathbb {R}}\end{aligned}$$(18)
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.
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
Proof
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
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 mapis continuous on \(I\times [0,\infty [\times {\mathbb {R}}\) \(;\)$$ (t,\varepsilon ,a)\longmapsto D^H\tilde{u}(t,\gamma (\cdot \varepsilon ),a), \qquad (t,\varepsilon ,a)\in I\times [0,\infty [\times {\mathbb {R}}$$
 (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
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 firstorder (resp. secondorder) 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, 15, 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 \({\mathscr {U}}\).
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.
(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 socalled reasonable crossnorms \(\alpha \) on \(E\otimes F\), verifying \(\alpha (e\otimes f)=\Vert e\Vert _E\Vert f\Vert _F\).
Definition 16
 (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.
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 Chisubspace (of \((E\hat{\otimes }_\pi ^2)^*\) \()\).
Remark 11
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)of \({\mathscr {U}}\) and \(D_\cdot ^{\text {ac}}{\mathscr {U}}\), respectively.$$ u:\mathscr {C}([T,0])\rightarrow {\mathbb {R}}, \qquad \qquad D_\cdot ^{\text {ac}}u:\mathscr {C}([T,0])\rightarrow BV([T,0]) $$
Proof
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
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.
Proposition 9
 (i)
\(D_x^{2,Diag}{\mathscr {U}}(\eta )\), the diagonal element of the secondorder 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\()\) \(:\)of \({\mathscr {U}}\) and \(D_{dx\,dy}^2{\mathscr {U}}\), respectively.$$ u:\mathscr {C}([T,0])\rightarrow {\mathbb {R}}, \qquad \qquad D_{dx\,dy}^2u:\mathscr {C}([T,0])\rightarrow \chi _0 $$
 (iii)
The horizontal derivative \(D^H{\mathscr {U}}(\eta )\) exists at \(\eta \in C([T,0])\).
Proof
3 StrongViscosity Solutions to PathDependent PDEs
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 thatWe simply write \({\mathbb {H}}^p(t,T)\) when \(d=1\).$$ \Vert Z\Vert _{_{{\mathbb {H}}^p(t,T)^d}}^p := \ {\mathbb {E}}\bigg [\bigg (\int _t^T Z_s^2 ds\bigg )^{\frac{p}{2}}\bigg ] \ < \ \infty . $$
 \({\mathbb {A}}^{+,2}(t,T)\), \(0 \le t \le T\), 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
Theorem 4
Proof
We conclude this subsection with an existence result for the pathdependent 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

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

\(\varphi _1,\ldots ,\varphi _N\in C^2([0,T])\).
Proof
3.2 Towards a Weaker Notion of Solution: A Significant Hedging Example
Lemma 1
Proof
See Proposition 3.7, Chapter III, in [35]. \(\Box \)
Lemma 2
Proof
We also have the following regularity result regarding the function f.
Lemma 3
Proof
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
Proof
We provide two distinct proofs.
3.3 StrongViscosity Solutions
Motivated by previous subsection and following [10], we now introduce a concept of weak (viscosity type) solution for the pathdependent Eq. (43), which we call strongviscosity 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 strongviscosity 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,for all \((t,\eta )\in [0,T]\times C([T,0])\), \(y,y'\in {\mathbb {R}}\), and \(z,z'\in {\mathbb {R}}\).$$\begin{aligned} F_n(t,\eta ,y,z)  F_n(t,\eta ,y',z')&\le \ C(yy' + zz'), \\ {\mathscr {U}}_n(t,\eta ) + H_n(\eta ) + F_n(t,\eta ,0,0)&\le \ C\big (1 + \Vert \eta \Vert _\infty ^m\big ), \end{aligned}$$
 (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 nonlocally 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 strongviscosity solution holds.
Theorem 6
Proof
We now prove an existence result for strongviscosity solutions to the pathdependent heat equation, namely to Eq. (43) in the case \(F\equiv 0\). To this end, we need the following stability result for strongviscosity 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,for all \((t,\eta )\in [0,T]\times C([T,0])\), \(y,y'\in {\mathbb {R}}\), and \(z,z'\in {\mathbb {R}}\).$$\begin{aligned} F_{n,k}(t,\eta ,y,z)  F_{n,k}(t,\eta ,y',z')&\le \ C(yy' + zz'), \\ {\mathscr {U}}_{n,k}(t,\eta ) + H_{n,k}(\eta ) + F_{n,k}(t,\eta ,0,0)&\le \ C\big (1 + \Vert \eta \Vert _\infty ^m\big ), \end{aligned}$$
 (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 strongviscosity 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 strongviscosity solution was used, see Remark 12(i); however, proceeding along the same lines we can prove the present result.
Theorem 7
Proof
Notes
Acknowledgments
The present work was partially supported by the ANR Project MASTERIE 2010 BLAN 0121 01. The second named author also benefited partially from the support of the “FMJH Program Gaspard Monge in optimization and operation research” (Project 20141607H).
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