Abstract
We consider the valuation of contingent claims with delayed dynamics in a Samuelson complete market model. We find a pricing formula that can be decomposed into terms reflecting the current market values of the past and the future, showing how the valuation of prospective cashflows cannot abstract away from the contribution of the past. As a practical application, we provide an explicit expression for the market value of human capital in a setting with wage rigidity. The formula we derive has successfully been used to explicitly solve the infinite dimensional stochastic control problems addressed in Biffis et al. (SIAM J Control Optim 58(4):1906–1938, 2020), Djeiche et al. (Stoch Process Appl 145:48–85, 2022) and Biagini et al. (SIAM J Financial Math 13(3):1004–1039, 2022).
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1 Introduction
It is a standard result in asset pricing theory that the absence of arbitrage opportunities is “essentially” equivalent to the existence of an equivalent probability measure under which the price of any contingent claim is a local martingale after deflation by the money market account; see [17, 24, 25]. In this paper, we preserve the standard formulation of arbitrage pricing in a complete market model with security prices evolving as geometric Brownian motions (GBM), in the spirit of Samuelson’s contribution [36] (see [37] for an overview). The main novelty of our work is that we consider contingent claims that have dynamics described by a stochastic delay differential equation (SDDE).
It is perhaps surprising that using the noarbitrage pricing machinery we are able to derive an explicit valuation formula for dynamics with memory, a case that is notoriously difficult to handle. However, we find a particularly appealing solution showing that the price can be decomposed into a term related to the ‘current market value of the past’ (in a sense to be made precise below), and the more traditional term reflecting the ‘current market value of the future’ (in the spirit of a discounted cashflow approach). In our setting the contribution of the past is represented by the portion of a contingent claim’s past trajectory that shapes its dynamics going forward.^{Footnote 1} Our pricing formula demonstrates that the market consistent valuation of future cashflows cannot ignore the contribution of the past. This is important for a number of applications in which pathdependency is a key feature of the state variables, as we now discuss.
As a practical application of our results, we consider in detail contingent claims representing stochastic wages received by an agent over her lifetime (e.g., [7, 20]). It is well known that when labor income is spanned by the assets available for trade, the market value of human capital can be easily derived via riskneutral valuation in a setting with labor income driven by a Stochastic Differential Equation (SDE); see [13] for an overview. However, the empirical literature on labor income dynamics widely relies on autoregressive moving average (ARMA) processes (e.g., [1, 26, 31, 33]).
The contributions by [19, 29, 35] show how SDDEs can be understood as the weak limit of discrete time ARMA processes. Therefore specific classes of SDDEs can be used to model labor income, thereby providing the continuous time counterpart of ARMA models supported by the empirical literature. We therefore consider the introduction of delayed drift and volatility coefficients in a GBM labor income model to provide a tractable example of wage dynamics that adjusts slowly to financial market shocks. We obtain a closed form solution for human capital, which makes explicit the contributions of the market value of the past and the future. Our results demonstrate that SDDEs are valuable modeling tools that can address the findings of a large body of empirical literature on wage rigidity (e.g., [3, 15, 28, 30]). Moreover, the results open the way to finding explicit solutions to interesting classes of lifecycle portfolio choice problems with state costraints (see [6, 7, 16]), as discussed in Sect. 3.
Although in this paper we discuss the application to human capital more extensively, the results can be used for other applications of interest. For instance, we provide some references to the literature on counterparty risk and derivatives valuation, in which analogous dynamics arise in the context of collateralization procedures entailing a delay in the markingtomarket procedure of overthecounter derivatives (e.g., [11, 12]).
It should be noted that noarbitrage pricing in the case of delayed price dynamics has been recently studied by several authors (see [2, 32], for example). Their focus, however, is on proving completeness of a market with security prices with delayed dynamics and hence very different from ours. On the other hand, their work suggests the opportunity to explore extensions of our results to broader settings in which market completeness is preserved, including those in which tradable assets have delayed drift and volatility terms.
The paper is organized as follows. In the next section, we introduce the setup and state our main result. In Section 3 we discuss in detail an application to human capital valuation and briefly outline other areas of application. In Sect. 4, we prove the main result of the paper. Finally, Sect. 5 offers concluding remarks. We relegate to an “Appendix” the proofs of some more technical results.
2 Setup and statement of the main result
We consider a filtered probability space \((\Omega , {\mathcal {F}}, {\mathbb {F}}, {\mathbb {P}})\) and an \({\mathbb {F}}\)adapted vector valued process \((S_0,S)\), representing the price evolution of a money market account, \(S_0\), and n risky assets, \(S=(S_1,\ldots ,S_n)^\top \), whose dynamics is given by
where \(\mu \in {\mathbb {R}}^n\), and the matrix \(\sigma \in {\mathbb {R}}^{n \times n}\) is assumed to be invertible. Here Z is an ndimensional Brownian motion and we assume that \({\mathbb {F}}:=\{{\mathcal {F}}_t\}_{t\ge 0}\) is the filtration generated by Z, and enlarged with the \({\mathbb {P}}\)null sets. In the following, for given vectors \(a, b \in \mathbb {R}^n\), we denote by \(a \cdot b\) the scalar product in \(\mathbb {R}^n\) and by \(\Vert \cdot \Vert \) the corresponding norm.
We are interested in the valuation of a payment stream represented by an \({\mathbb {F}}\)adapted process \((y_t)_{t\ge 0}\). The payment stream can be thought of as capturing the marktomarket process of a trading account, the flow of profits and losses generated from a trading strategy, the collateral flows arising from an overthecounter derivative transaction, or the labor income received by an agent over time. The latter is the application we will consider in detail in the next section.
We assume that the payment stream y obeys the following SDDE with delay in both the drift and the diffusion terms:
where \(\mu _y \in {\mathbb {R}}\), \(\sigma _y \in {\mathbb {R}}^n\), the signed measures \(\phi ,\varphi _i\) (for \(i=1,\dots ,n\)) have bounded variation on \(\left[ d,0\right] \) and \(x_0 \in {\mathbb {R}}\), \(x_1 \in L^2 \big ( [d,0]; {\mathbb {R}}\big ) \), where \(L^2 \big ( [d,0]; {\mathbb {R}}\big )\) denotes the space of Lebesgue square integrable (deterministic) functions on \([d,0]\). In the following, we denote by \(\phi \) the total variation of a signed measure of bounded variation \(\phi \) on \([d,0]\). Existence of a unique solution to SDDE (2) is ensured by the following result, which is proved in “Appendix A.1”.
Proposition 2.1
For any given initial datum \((x_0,x_1)\in \mathbb {R} \times L^2(d,0;\mathbb {R})\) equation (2) admits a unique strong (in the probabilistic sense) solution \(y\in L^2(\Omega \times [0,T] )\), for all \(T>0\), with \(\mathbb {P}\)a.s. continuous paths.
Assuming the process y to represent an agent’s income stream, we see that (2) provides a compelling example (which will be shown to be tractable) of wage dynamics adjusting slowly to financial market shocks. In particular, the measures \(\phi ,\varphi _1,\ldots \varphi _n\) can be thought as modulating the impact of past income’s realizations on wages going forward. As discussed in [7], this setting can also be related to the literature on ‘learning your income’ (e.g., [22]), whereby an agent learns about her earning potential based on past wages. Here, moving averages can be used as a tractable substitute for fully fledged Bayesian filters and can be justified on the grounds of bounded rationality.
As the market is complete and the stream process y is spanned by the stock S, we can find uniquely the current value of the future uncertain income stream by considering the expected discounted value under the unique stochastic discount factor \(\xi \) (see [18]):
where the process \(\xi \) is \({\mathbb {F}}\)adapted and evolves according to the following SDE,
with the constant \(\kappa \) representing the market price of risk, which is given by
with \({\textbf{1}} = (1,\dots , 1)^\top \) denoting the unitary vector in \({\mathbb {R}}^n\).
Our main goal now is to provide an explicit formula for the quantity \(H(t_0)\) given in (3). The challenging aspect of the problem lies in the fact that we consider a payment stream whose dynamics is path dependent, thus making the problem considerably different from and harder to prove than the cases discussed in the extant literature. Before stating the main result of the paper, we introduce the following conditions that will be assumed to hold throughout the paper.
Assumption 2.2
The following conditions are assumed to apply:

(i)
The measure \(\Phi \) on \([d,0]\) defined as
$$\begin{aligned} \Phi (\cdot ):=\phi (\cdot )(\varphi _1(\cdot ),\ldots ,\varphi _n(\cdot )) \cdot \kappa \end{aligned}$$(6)is a signed measure of bounded variation.

(ii)
The quantity
$$\begin{aligned} r\mu _0+\sigma _y \cdot \kappa  \int _{d}^0 e^{r\tau }\Phi (\textrm{d} \tau ) \end{aligned}$$(7)is assumed to be strictly positive.
Under assumption 2.2, which will be discussed in details in Sect. 4.5, the following result holds.
Theorem 2.3
Under Assumption 2.2, for any \(t_0\ge 0\), the quantity \(H(t_0)\) defined in (3) admits the following explicit expression:
where \(y(t_0)\) denotes the solution at time \(t_0\) of equation (2) and the constant K and function G are given by:
In expression (8), we can recognize an annuity factor, \(K^{1}\), multiplying a term representing the current market value of labor income, \(y(t_0)\), and a term representing the current market value of the past trajectory of y over the time window \((t_0d,t_0)\). The ‘market value of the past’ trades off the returns on the payment stream against its exposure to financial risk, as can be seen from expression (6). When the delay terms in the drift and volatility coefficients vanish, the valuation of the payment stream reduces to \(K^{1}y(t_0)\), in line with [20], among others.
Remark 2.4
The setup described above can be extended to the case of payments over a bounded horizon in some interesting situations. When the payment stream is received until an exogenous Poisson stopping time \(\tau _\delta \) (representing death or an irreversible unemployment shock when y represents labor income), expression (8) still applies, provided discounting is carried out at rate \(r+\delta \) instead of r, where \(\delta >0\) represents the Poisson parameter; see [6, 7, 16]. The case in which payments are received until a finite, deterministic time (capturing irreversible retirement when y represents labor income) can also be considered, at the price of additional technical work; see [5].
3 Applications
3.1 Optimal portfolio problems with path dependent labor income
We now consider in detail the case in which the contingent claim y in (2) represents stochastic wages received by an agent over her lifetime. As discussed in the Introduction, SDDEs allow us to rely on continuous time labor income dynamics better matching some of the salient features documented in the empirical literature. The pathdependency of (2) captures the slow adjustment of labor income to financial market shocks and provides a counterpart to discrete time ARMA models used to model wage dynamics. See [7] for a comprehensive list of references and [5] for the complexities brought about by considering unspanned shocks in labor income dynamics.
In the context of lifecycle portfolio choice, obtaining an explicit expression for human capital is often crucial not only to reveal the structure of optimal solutions (e.g., [13]), but also to handle state constraints (e.g., [7, 20]). We will discuss these points by making reference to some applications in Sects. 3.1.1–3.1.3. Before proceeding, we note that assumption 2.2 is all we need to provide the explicit valuation result of Theorem 2.3, but the particular application to human capital requires labor income to be positive almost surely. A sufficient condition for this to be the case is as follows (see [7, Proposition 2.7] for a proof).
Remark 3.1
When \(\varphi _i=0\) for all \(i=1,ldots,n\) (i.e., when the delay term in the volatility coefficient of (2) vanishes), the variation of constants formula yields the following explicit representation:
where
One can then see that in this case \(y(t)>0\) \(\mathbb {P}\)a.s. if \(x_0>0\), \(x_1 \ge 0\) a.s. and \(\phi \ge 0\) a.s..
The results of Theorem 2.3 and the solution approach followed in this paper show how tools from infinitedimensional analysis can be successfully used to address nonMarkovian valuation problems, which are beyond the reach of conventional approaches. In the following, we illustrate some of those applications to show how the findings of Theorem 2.3 (or suitable generalization of it) can be successfully used to identify explicit solutions in some interesting situations.
3.1.1 Infinite horizon lifecycle portfolio choice
In [7] the authors solve an infinite horizon portfolio problem with borrowing constraints in which an agent receives labor income affected by financial market shocks in a path dependent way. The framework is the complete market model described by (1). An agent is endowed with initial wealth \(w\ge 0\) and receives wages until death, which coincides with the first jump time \(\tau _{\delta }\) of a Poisson process with parameter \(\delta >0\). The financial wealth of the agent at time \(t\ge 0\), denoted by W(t), can be invested in the riskless and risky assets. The agent receives wages at rate y(t) and consumes at rate \( c(t)\ge 0\). The wealth amount allocated to the risky assets is \( \theta (t)\in {\mathbb {R}}^n\) at each time \(t\ge 0\). The agent sets a bequest target \(B(\tau _\delta )\ge 0\) at death. In line with [20], the agent can purchase life insurance at the (actuarially fair) premium rate \(\delta ( B(t)W(t))\), for \(t<\tau _\delta \), to reach the bequest target and hence cover any shortfall \(B(\tau _{\delta })W(\tau _{\delta })\) at death. The agent’s wealth (before death) is assumed to obey the standard dynamic budget constraint given by the SDE
In line with the empirical evidence on labour income dynamics recalled in the Introduction, labor income is modelled via the following SDDE:
where \(\mu _y \in {\mathbb {R}}\), \(\sigma _y \in {\mathbb {R}}^n\) and the functions \(\phi (\cdot ), x_1(\cdot )\) belong to \(L^2\left( d,0; {\mathbb {R}}\right) \) (thus making (11) a particular case of (2)).
Denoting by \(k>0\) the intensity of preference for leaving a bequest, by \(\gamma \in (0,1) \cup (1, +\infty )\) the riskaversion coefficient and by \(\rho >0\) the subjective discount rate, the objective is to maximize the expected utility from lifetime consumption and bequest, which on \(\{\tau ^\delta >t\}\) takes the form:
Maximization is carried out over all triplets \(\left( c,\theta ,B\right) \in \Big \{{\mathbb {F}}\text{ predictable } \ c(\cdot ), B(\cdot ), \theta (\cdot ) :c(\cdot ), B(\cdot ) \in L^1 (\Omega \times [0, +\infty );{\mathbb {R}}_{+}),\theta (\cdot ) \in L^2(\Omega \times {\mathbb {R}}; {\mathbb {R}}^n)\Big \} \) subject to the state constraint
which is a ‘noborrowingwithoutrepayment’ constraint [20]. As the second term appearing in (13) represents the agent’s human capital at time t, constraint (13) captures the situation in which human capital can be pledged as collateral and represents the agent’s maximum borrowinge capacity. In line with [7], we note that the triplets \(\left( c,\theta ,B\right) \) must be understood as representing the predeath counterparts of the controls predictable relative to the reference filtration \({\mathbb {F}}\). Similarly, on the trace of \({\mathbb {F}}\) on \(\{\tau ^\delta >t\}\), the process \(\xi \) satisfies equation (4) with a drift of the form \(\xi (t)(r+\delta )\), as explained in Remark 2.4.
In [7], the authors find an explicit solution to the optimization problem under power utility. The proof of the result relies on the resolution of an infinitedimensional HamiltonJacobiBellman (HJB) equation, which can be considered as an infinitedimensional version of the classical Merton problem. From a technical point of view, the key idea is to extend the state space so as to include the past trajectory of y over \([d,0]\). In this way, the problem becomes infinite dimensional and Markovian (in the current wealth level and path of y over the time window \([d,0]\)). In this infinitedimensional reformulation of the problem, it becomes essential to rewrite the constraint (13) by exploiting the explicit expression given in Theorem 2.3. Importantly, the optimal risky asset allocation found in [7],
reveals that the decomposition of human capital into a past and future component is essential in understanding the agent’s hedging demand, as we now discuss. We first note that the solution follows the logic of Merton’s, in that the agent chooses constant fractions of total wealth (given by financial wealth and human capital). In line with [9, 20], the agent considers the capitalized value of future wages as if they were a traded asset. As the agent’s labor income is instantaneously perfectly correlated with the risky assets, a negative income hedging demand arises (the term \(\sigma ^{1}\sigma _y K^{1}y(t)\)), and the allocation to risky assets is reduced by a term proportional to the regression coefficient of labor income shocks on risky asset returns ([13]). The key difference with the benchmark model with no delay is that the hedging demand only depends on the ‘future component’ of human capital, and not on the capitalized market value of the labor income’s past trajectory. The intuition is that when wages respond to market shocks with a delay, then human capital is more predictable and only the portion of human capital exposed to future market shocks drives the negative income hedging demand.
3.1.2 Extensions: robustness and finite horizon
The authors in [6] study a robust version of the lifecycle optimal portfolio choice problem presented in Sect. 3.1.1. Again, the result in Theorem 2.3 is essential in obtaining an explicit solution of the problem considered.
The authors in [5] consider the same problem as in the previous section, but now with a finite time horizon, which can be interpreted as the agent’s fixed retirement date. The authors extend the results of Theorem 2.3 to the finite horizon case, obtaining again a decomposition of human capital into two components pertaining to the past and the future evolution of labor income. They then proceed to solve the lifecycle portfolio choice model, which entails timedependent state constraints, as horizon effects are now material during the working life of the agent. This appears to be a novel type of problem in the infinite dimensional stochastic control literature, which is again successfully solved by exploiting the structure of the explicit solution for human capital.
3.1.3 Meanfield games
Another generalization of the problem addressed in [7] is considered in [16]. Similarly to [7], the authors consider a lifecycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky securities and a riskless asset subject to a borrowing constraint. However, in addition to assuming the dynamics of labor income to adjust slowly to financial market shocks, they also assume it to be benchmarked to the wages of a population of agents with comparable tasks and/or ranks. Specifically, each agent i’s labor income \(y_i\) is benchmarked against the wages \(y^n:=(y_1,y_2,\ldots ,y_n)\) of a population of n agents. As n grows larger, the problem falls into the family of optimal control of infinite dimensional McKeanVlasov dynamics type. By adding a suitable new variable, the authors simplify the problem and are able to find explicitly the solution of the associated infinitedimensional HJB equation and the optimal feedback controls. A necessary step to solve the problem is to provide a suitable reformulation of the noborrowing without repayment constraint (13), where now labor income obeys an SDDE with a drift containing not only a pathdependent term but also a mean reverting term. This is carried out in [16, Section 3], where the authors provide a generalization of formula (8) by carefully adapting the techniques used in Sect. 4 of this paper.
3.2 Counterparty risk and derivatives valuation
As a simple example of application of our setup to the context of overthecounter derivatives, let us consider in equation (2) the case of \(n=1\), \(\mu _0=0\), \(\phi =0\), \(\sigma _0=0\), and \( \varphi (s) =\delta _{d}(s)\), where \(\delta _{a}(s)\) indicates the deltaDirac measure at a, so that equation (2) now reads
Then, for \(t \in [0,d )\) we have
In this case \(X_0(t)\) is Gaussian, and dynamics (15) could be used to model, for example, the variation margin of an overthecounter swap, when the collateralization procedure relies on a delayed marktomarket value of the instrument (see [11], page 316, or [8, 12] for some examples).
4 Proof of the result
Within this section we consider the unique continuous \({\mathbb {F}}\)adapted solution y of (2) given in Proposition 2.1. The proof of Theorem 2.3 can be divided in the following steps:

we incorporate the discount factor \(\xi \) in the equivalent riskneural probability measure \(\tilde{\mathbb {P}}\) and rewrite the dynamics of y under \(\tilde{\mathbb {P}}\). Derive the deterministic delayed equation satisfies by the quantity \(\tilde{\mathbb {E}}[y(t)\mathcal {F}_{t_0}]\). (Sect. 4.1).

We rewrite the delayed equation for \(\tilde{\mathbb {E}}[y(t)\mathcal {F}_{t_0}]\) as a deterministic evolution equation, which takes values in a suitable Hilbert space incorporatingthe past and the present in its structure. We will appeal to the socalled productspace framework for pathdependent equations (Sect. 4.2).

We exploit suitable spectral properties of the operator that appears in the above mentioned infinitedimensional formulation of the problem in order to obtain expression (8) for \(H(t_0)\) (Sects. 4.3 and 4.4).

We clarify the relationship between the spectral properties used and our Assumption 2.2 (Sect. 4.5).
The above first three steps will lead to Proposition 4.8, whereas the last step will be formalized in Lemmas 4.10 and 4.11. Theorem 2.3 will then follow as an immediate consequence of these results. For readability, we relegate to an “Appendix” the proofs of some technical lemmas.
4.1 Equivalent probability measure
We find it more convenient to change perspective from a valuation formula relying on the use of the stochastic discount factor \(\xi \) under the measure \({\mathbb {P}}\) to the one discounting at the riskfree rate r the equivalent martingale measure \(\tilde{\mathbb {P}}\), which is therefore called ‘riskneutral measure’. It is well know that the two approaches are equivalent under no arbitrage in our setting.
We start by considering the equivalent probability measure \({\tilde{{\mathbb {P}}}}\) on \(\mathcal {F}_s\) such that^{Footnote 2}
By [27, Lemma 3.5.3] we can write
and thus^{Footnote 3}
The idea is to now understand what kind of differential equation the quantity \(\tilde{{\mathbb {E}}}\left[ y(s)\mid {\mathcal {F}}_{t_0} \right] ={\tilde{{\mathbb {E}}}}_s\left[ y(s)\mid {\mathcal {F}}_{t_0} \right] \) satisfies. Let \(\tilde{{\mathbb {P}}}\) the measure such that \(\left. \tilde{{\mathbb {P}}}\right _{{\mathcal {F}}_s}=\tilde{{\mathbb {P}}}(s)\) for all \(s\ge 0\). By the Girsanov Theorem the process \({\tilde{Z}}(t) = Z(t) + \kappa t\) is an ndimensional Brownian motion under \( \tilde{{\mathbb {P}}}\). The dynamics of y under \( \tilde{{\mathbb {P}}}\) is then
where \(\Phi \) is defined in (6). Integrating between \(t_0\) and t we obtain
and therefore, by taking the conditional expected value on both sides, we get
The following Lemma guarantees that the stochastic integral with respect to \({\tilde{Z}} \) is a martingale, and has zero mean. The proof is provided in “Appendix A.2”.
Lemma 4.1
It holds that
We thus obtain that
and, by definition of conditional mean and by Fubini’s Theorem, the expression in (21) reduces to
Therefore, defining
we have that \(M_{t_0}\) satisfies for \(t\ge t_0\) the equation (with random initial conditions)
Existence of a unique solution of the above system is guaranteed by the following generalization of [4, Part II, Chapter 4, Theorem3.2] to random initial conditions.
Lemma 4.2
Given any fixed \(\mathcal {F}_{t_0}\)measurable \(\mathbb {R} \times L^2([d,0];\mathbb {R})\)valued random variable \(m=(m_0,m_1)\), the Cauchy problem
admits a unique absolutely continuous solution. Moreover, system (25) is equivalent to (24) when we choose \((m_0,m_1)=(y(t_0),y(t_0+\cdot ))\).
4.2 Reformulation of the problem in an infinitedimensional framework
We now reformulate the differential equation with delay (24) as an evolution equation with values in the so called DelfourMitter Hilbert space, defined as
whose elements are denoted as \( x=(x_0,x_1)\). \(\mathcal {H}\) is a Hilbert space when endowed with the inner product \(\langle (x_0,x_1),(y_0,y_1)\rangle _{\mathcal {H}}=x_0y_0+\langle x_1,y_1\rangle \), the latter being the usual inner product of \(L^2(d,0;{\mathbb {R}})\).
We define the operator \(A:\mathcal {D}(A) \subset \mathcal {H} \rightarrow \mathcal {H}\) as
with \(\Phi \) defined in (6). Here by \(W^{1,2}\big ( [d, 0]; {\mathbb {R}} \big )\) we denote the Sobolev space of weakly differentiable squareintegrable functions.
We can then reformulate equation (24) as an evolution equation in \(\mathcal {H}\).
Consider, again for any fixed \(\mathcal {F}_{t_0}\)measurable \(\mathcal {H}\)valued random variable \({\textbf{m}}=\left( m_0,m_1\right) \), the \(\mathcal {H}\)valued process \({\textbf{M}}(t_0;\cdot )\) that is the solution on \([t_0,+\infty )\) of
We collect in the following Proposition some useful results about the above equation (for more details see e.g. [14, AppendixA]).
Proposition 4.3

(i)
The operator A generates a strongly continuous semigroup \(\left\{ S(t)\right\} _{t\ge 0}\) in \(\mathcal {H}\).

(ii)
S(t) is a compact operator for every \(t\ge d\).

(iii)
For every \(\mathcal {F}_{t_0}\)measurable \(\mathcal {H}\)valued random variable m the process
$$\begin{aligned} S(tt_0){\textbf{m}} ; \end{aligned}$$(28)is the unique weak (in distributional sense) solution of (27); in particular
$$\begin{aligned} {\textbf{M}}(t_0;t)={\textbf{M}}(0;tt_0)\ . \end{aligned}$$(29)  (iv)
Proof
See “Appendix A.3”. \(\square \)
As an immediate consequence of the above result we obtain the desired equivalence between equations (27) and (24).
Corollary 4.4
Let y be a solution of (2) on \([0,t_0]\); when choosing \({\textbf{m}}\) as \((m_0,m_1)= (y(t_0), y(t_0+\cdot ))\), (27) is equivalent to (24) and in this case we have
From now on we thus will work with formulation (27). The spectral properties of the operator A, that appears in this infinitedimensional formulation, will be crucial to prove our result. We devote the next section to the analysis of these properties.
4.3 Spectral properties of A
In the present section we collect some technical results concerning the spectral properties of the operator A. Proof of Theorem 2.3 is based on the Lemmas presented here. The technical proofs are postponed to the “Appendix”.
Lemma 4.5
The spectrum of the operator A is given by
where
The spectrum \(\sigma \left( A\right) \) is a countable set and every \(\lambda \in \sigma \left( A\right) \) is an isolated eigenvalue of finite multiplicity.
The spectral bound of A is
Proof
See [23, Chapter 7, Lemma 2.1 and Theorem4.2]\(\square \)
We ca explicitly compute the resolvent operator of A.
Lemma 4.6
Let \(\rho (A)\) denote the resolvent set of A and let \(\lambda \in {\mathbb {R}}\cap \rho \left( A\right) \). The resolvent operator of A at \(\lambda \), denoted by \(R(\lambda ,A)\) is given by
with
Proof
See “Appendix A.4”. \(\square \)
Lemma 4.7
For every real \(\lambda \) such that \(\lambda >\lambda _0\) and every \({\textbf{m}}=(m_0,m_1)\in \mathcal {H}\) we have
Proof
Identity (34) is well known to hold for all real \(\lambda \) larger than the type of S(t). Since S(t) is compact for every \(t\ge d\), its type is actually equal to its spectral radius \(\lambda _0\). For a reference see e.g. [4, Part II, Chapter 1, Corollary 2.5]. \(\square \)
4.4 Deriving the explicit formula for H
In this section we exploit the results derived in the above sections to prove the following.
Proposition 4.8
Assume \(r>\lambda _0\), then for any \(t_0\ge 0\), the quantity \(H(t_0)\) defined in (3) has the following explicit form
where \(y(t_0)\) denotes the solution at time \(t_0\) of equation (2),
and G is given by
Remark 4.9
Notice that the statement of the above result is the same of Theorem 2.3, but the assumptions here are different: we assume \(r>\lambda _0\) instead of Assumption 2.2. An explanation of why we do actually consider Assumption 2.2 in Theorem 2.3, will be provided in the next section.
Proof
Let \({\textbf{m}}=(m_0,m_1)= (y(t_0), y(t_0+\cdot ))\). We denote here by \(\Pi \) the projection on the first (finitedimensional) component of \(\mathcal {H}\), i.e. \( \Pi [ {\textbf{m}}] =\Pi [(m_0,m_1)]=m_0\).
Starting from (18), we have
From the above equalities we infer, in particular, the \(\mathbb {P}\)integrability of \( \int _{t_0}^{+\infty } {\mathbb {E}} [ \xi (s) y(s) \mid {\mathcal {F}}_{t_0} ]\,\textrm{d}s\), which justifies the equality
In fact, by the characteristic property of the conditional mean, and Fubini’s Theorem we have that, for any \(F \in {\mathcal {F}}_{t_0}\)
Defining now \(K:=K(r)\) and recalling (3), (35) and (36), by (37) and (38), the result immediately follows. \(\square \)
4.5 Motivations for Assumption 2.2
In Proposition 4.8 we proved our main result under the Assumption \(r>\lambda _0\). This requirement is difficult to check in practice, since it requires an explicit computation of the spectral bound \(\lambda _0\). In the present section we therefore look for some sufficient conditions easier to check.
Set for \(\lambda \in \mathbb {C}\)
where by \(\Phi \) we denote the total variation measure of \(\Phi \). Set
We note that \(\widetilde{\lambda _0}\) is the spectral radius of the operator \(\widetilde{A}: \mathcal {D}(\widetilde{A})\subset \mathcal {H} \rightarrow \mathcal {H}\) defined as follows:
Lemma 4.10
The function \({\widetilde{K}}\), restricted to the real numbers, is strictly increasing and the spectral bound \(\widetilde{\lambda _0}\) is the only real root of the equation \({\widetilde{K}}(\xi )=0\). In particular,
Proof
See “Appendix A.5”. \(\square \)
Recall the definition of K given in (30) and the definition of the spectral bound of A, \(\lambda _0\), given (31).
Lemma 4.11
It holds
Proof
See “Appendix A.6”. \(\square \)
Thanks to the above two Lemmas it becomes now clear why we work under Assumption 2.2 in Theorem 2.3. It provides a sufficient condition for the condition \(r>\lambda _0\), imposed in Proposition 4.8, to hold. In fact, assume \({\widetilde{K}}>0\) as in Assumption 2.2, then by Lemmas 4.10 and 4.11 we immediately get \(r>\lambda _0\).
Remark 4.12
Notice that, if \(\Phi \) is a positive measure, then \({\widetilde{K}} \equiv K\), \(\lambda _0 \equiv \widetilde{\lambda _0}\) and the condition \(K>0\) becomes also necessary, that is \(K>0 \iff r>\lambda _0\).
5 Conclusion
In this paper we have considered a complete market model in the spirit of Samuelson’s original contribution [36], in which security prices evolve as geometric Brownian motions. Despite working within a classical setup, we have obtained a novel explicit pricing formula for stream of payments with delayed dynamics, by using techniques from infinitedimensional stochastic analysis. Our valuation formula results in an explicit expression demonstrating the importance of appreciating the past to quantify the current market value of the future. The approach followed in this paper highlights how tools from infinitedimensional analysis can be successfully used to address valuation problems that are nonMarkovian, and hence beyond the reach of conventional approaches. As highlighted in the applications discussed in Sect. 3, it is apparent how our results and the techniques developed here can be used successfully to explicitly solve interesting classes of infinite dimensional stochastic optimal control problems with nontrivial state constraints.
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Notes
Recall that the state price density \(\xi \) characterizes the RadonNikodym derivative that defines the change of probability measure from the objective probability measure \(\mathbb {P}\) to the riskneutral measure \(\tilde{\mathbb {P}}\) via the relationship \(\xi (s)=e^{rs}\rho (s)= e^{rs}\frac{\textrm{d}\tilde{\mathbb {P}}}{\textrm{d}\mathbb {P}}(s)\).
Recall (see (3)) that our aim is to evaluate the expectation \({\mathbb {E}} \left[ \int _{t_0}^{+\infty } \xi (s) y(s) \text {d}s \mid {\mathcal {F}}_{t_0} \right] \). We will prove that \(\int _{t_0}^{+\infty } {\mathbb {E}} \left[ \xi (s) y(s)\mid {\mathcal {F}}_{t_0} \right] \text {d}s\) is equal to the r.h.s. of (8) and then justify the equality \({\mathbb {E}} \left[ \int _{t_0}^{+\infty } \xi (s) y(s) \text {d}s \mid {\mathcal {F}}_{t_0} \right] =\int _{t_0}^{+\infty } {\mathbb {E}} \left[ \xi (s) y(s)\mid {\mathcal {F}}_{t_0} \right] \text {d}s\).
For more details on the estimates, the interested reader can consult that paper.
Notice that this condition require to work with \(p>2\).
This condition is made in order to guarantee the convergence of the integrals that will appear in what follows. Notice that this condition require to work with \(p>4\).
References
Abowd, J.M., Card, D.: On the covariance structure of earnings and hours changes. Econometrica 57(2), 411–445 (1989)
Arriojas, M., Hu, Y., Mohammed, S.A., Pap, G.: A delayed Black and Scholes formula. Stoch. Anal. Appl. 25(2), 471–492 (2007)
Barattieri, A., Basu, S., Gottschalk P.: Some evidence on the importance of sticky wages, NBER Working Paper No. 16130 (2010)
Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and control of infinite dimensional systems, Second Edition, Birkhauser (2007)
Biagini, S., Biffis, E., Gozzi, F., Zanella, M.: Wage rigidity and retirement in optimal portfolio choice, Working Paper: arXiv:2101.09732 (2022)
Biagini, S., Gozzi, F., Zanella, M.: Robust portfolio choice with sticky wages. SIAM J. Financial Math. 13(3), 1004–1039 (2022)
Biffis, E., Gozzi, F., Prosdocimi, C.: Optimal portfolio choice with path dependent labor income: the infinite horizon case. SIAM J. Control Optim. 58(4), 1906–1938 (2020)
Biffis, E., Blake, D., Pitotti, L., Sun, A.: The cost of counterparty risk and collateralization in longevity swaps. J. Risk Insur. 83(2), 387–419 (2016)
Bodie, Z., Merton, R.C., Samuelson, W.F.: Labor supply exibility and portfolio choice in a life cycle model. J. Econ. Dyn. Control 16(3), 427–449 (1992)
Boucekkine, R., Licandro, O., Puch, L., Del Rio, F.: Vintage capital and the dynamics of the AK model. J. Econ. Theory 120(1), 39–72 (2005)
Brigo, D., Morini, M., Pallavicini, A.: Counterparty Credit Risk, Collateral and Funding: With Pricing Cases for all Asset Classes. Wiley, Collateral and Funding (2013)
Brigo, D., Pallavicini, A.: Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrongway risks. J. Financial Eng. 1(1), 1–60 (2014)
Campbell, J.Y., Viceira, L.M.: Strategic Asset Allocation: Portfolio Choice for LongTerm Investors. Oxford University Press, Oxford (2001)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Dickens, W.T., Goette, L., Groshen, E.L., Holden, S., Messina, J., Schweitzer, M.E., Turunen, J., Ward, M.E.: How wages change: micro evidence from the international wage flexibility project. J. Econ. Perspect. 21(2), 195–214 (2007)
Djeiche, B., Gozzi, F., Zanco, G., Zanella, M.: Optimal portfolio choice with path dependent benchmarked labor income: a mean field model. Stoch. Process. Appl. 145, 48–85 (2022)
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(1), 463–520 (1994)
Duffie, D.: Dynamic Asset Pricing Theory. Princeton University Press, Princeton (2001)
Dunsmuir, W., Goldys, B., Tran, C.V.: Stochastic delay differential equations as weak limits of autoregressive moving average time series, Working Paper, University of New South Wales and University of Sydney (2015)
Dybvig, P., Liu, H.: Lifetime consumption and investment: retirement and constrained borrowing. J. Econ. Theory 145(3), 885–907 (2010)
Fabbri, G., Gozzi, F.: Solving optimal growth models with vintage capital: the dynamic programming approach. J. Econ. Theory 143(1), 331–373 (2008)
Guvenen, F.: Learning your earning: Are labor income shocks really very persistent? Am. Econ. Rev. 97(3), 687–712 (2007)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, Berlin (1993)
Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20(3), 381–408 (1979)
Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11(3), 215–260 (1981)
Hubbard, R.G., Skinner, J., Zeldes, S.P.: Precautionary saving and social insurance. J. Polit. Econ. 103(2), 360–399 (1995)
Karatzsas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)
Khan, S.: Evidence of nominal wage stickiness from microdata. Am. Econ. Rev. 87(5), 993–1008 (1997)
Lorenz, R.: Weak approximation of stochastic delay differential equations with bounded memory by discrete time series, Ph.D. Thesis, Humboldt University, Berlin (2005)
Le Bihan, H., Montornès, J., Heckel, T.: Sticky wages: evidence from quarterly microeconomic data. Am. Econ. J. Macroecon. 4(3), 1–32 (2012)
MaCurdy, T.E.: The use of time series processes to model the error structure of earnings in a longitudinal data analysis. J. Econ. 18(1), 83–114 (1982)
Mao, X., Sabanis, S.: Delay geometric Brownian motion in financial option valuation. Stochastics 85(2), 295–320 (2013)
Meghir, C., Pistaferri, L.: Income variance dynamics and heterogeneity. Econometrica 72(1), 1–32 (2004)
Mohammed, S.A.: Stochastic Differential Systems with Memory: Theory. Southern Illinois University Carbondale, Examples and Applications (1998)
Reiss, M.: Nonparametric estimation for stochastic delay differential equations, Ph.D. Thesis, Humboldt University, Berlin (2002)
Samuelson, P.A.: Rational theory of warrant pricing. Ind. Manag. Rev. 6, 13–32 (1965)
Samuelson, P.A.: Mathematics of speculative price. SIAM Rev. 15(1), 1–42 (1973)
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A Further technical details
A Further technical details
1.1 A.1 Proof of Proposition 2.1
The existence and uniqueness result for (2) is not covered by the extant literature. When the initial datum \(x = (x_0,x_1)\), seen as a function on \([d,0]\), is continuous, existence and uniqueness of the strong solution to the SDDE for y is proved by [34, Theorem I.2]. When the initial datum \(x\in {\mathbb {R}}\times L^2([d,0),\textrm{d}t ;{\mathbb {R}})\), with the additional requirement that \(\phi \) and \((\varphi _1,\ldots ,\varphi _n)\) are absolutely continuous w.r.t the Lebesgue measure, that is \(\textrm{d}\phi = \varphi \textrm{d}t\), \(\textrm{d}(\varphi _1,\ldots ,\varphi _n)= (\phi _1,\ldots ,\phi _n)\,\textrm{d}t\) and the RadonNikodym densities \(\varphi , \phi _1,\ldots ,\phi _n \in L^2([d,0), \textrm{d}t)\), the existence and uniqueness result follows by [34, Remark I.3(iv)]. We need to extend this latter result to the case in which \(\phi \) and \((\varphi _1,\ldots ,\varphi _n)\) are signed measures of bounded variation on \(\left[ d,0\right] \). We will prove the result by means of the same procedure employed for the proof of [6, Proposition B.2]. There the authors prove the existence and uniqueness of the solution for an equation similar to (2), under more general assumptions on the measure \(\phi \), but with no delay in the diffusion term.
Let us start by introducing the standard notation for the past path at t of a (deterministic) function \(h:[d,T] \rightarrow \mathbb {R}\), for \(0\le t\le T\), that is the function \(h_t\)
The past path of y at t for the realization \(\omega \) is thus \(y_t(s,\omega ): =y(t+s,\omega ) \, \, s \in [d,0]\). As usual, we hide the dependence of the process on \(\omega \) and write the delay terms in the drift and in the diffusion as follows:
and
The delay parts in (2), given by (42) and (43) can be then expressed in terms of (an extension of) the following linear operators ok kernel type:
and
Since the operators L and G are welldefined only on the space of continuous functions \(C([d,0]; \mathbb {R})\), when the initial datum does not belongs to \(C([d,0]; \mathbb {R})\) but just to \(L^2([d,0); \mathbb {R})\) problems may arise. In fact, consider the initial datum \((x_0, x_1) \in \mathbb {R} \times L^2(d,0;\mathbb {R})\) and proceed by assuming that the solution to (2) exists. Denote the past path on the window \([td,t]\) by \(y_t:[d,0] \rightarrow \mathbb {R}\), \(y_t(s):=y(t+s)\) a.e. \(t \ge 0\), \(s \in [d,0]\). Then, for \(0 \le t <d\), the past path is
which, in general, is not a continuous function, but only square integrable. Therefore, the operators L and G introduced in (44) and (45) cannot be applied to \(y_t\) since the integrals in (42) and (43) may not be well defined. The first issue is thus to show that L and G admit continuous extensions to the square integrable functions on \([d,0]\), as made precise in the following lemma.
Lemma 5.1
Let \(L:C([d,0];\mathbb {R}) \rightarrow \mathbb {R}\) and \(G: C([d,0];\mathbb {R}) \rightarrow \mathbb {R}^n\) be the continuous and linear maps given in (44) and (45) respectively. Fix \(T>0\) and define on \(C([d,T];\mathbb {R})\) the delay operators
Then,

1.
the maps \(\mathcal {L}:C([d,T];\mathbb {R}) \rightarrow L^2([0,T];\mathbb {R})\) and \(\mathcal {G}:C([d,T];\mathbb {R}) \rightarrow L^2([0,T];\mathbb {R}^n)\) satisfy, respectively, the inequalities
$$\begin{aligned} \Vert \mathcal {L}(y)\Vert _{L^2([0,T];\mathbb {R})}\le \phi ([d,0]) \Vert y\Vert _{L^2([d,T];\mathbb {R})}, \quad \forall y \in C([d,T];\mathbb {R}). \nonumber \\ \end{aligned}$$(46)$$\begin{aligned} \Vert \mathcal {G}(y)\Vert _{L^2([0,T];\mathbb {R}^n)}\le \left( \sum _{i=1}^n [\varphi _i([d,0])]^2\right) ^{\frac{1}{2}} \Vert y\Vert _{L^2([d,T];\mathbb {R})}, \quad \forall y \in C([d,T];\mathbb {R}).\nonumber \\ \end{aligned}$$(47) 
2.
\(\mathcal {L}\) and \(\mathcal {G}\) have \(L^2\)norm continuous, linear extensions (still denoted by \(\mathcal {G}\) and \(\mathcal {L}\), respectively) to \(L^2([d,T];\mathbb {R})\).
Proof
The proof follows the lines of [6, Lemma B.1] (see also [4, Part II, Chapter 4, Theorem3.3], but for the sake of completeness, we prove the result for the operator \(\mathcal {G}\). For the operator \(\mathcal {L}\) one follows the same reasoning.

1.
$$\begin{aligned} \Vert \mathcal {G}(y)\Vert _{L^2([0,T];\mathbb {R}^n)}&=\Vert Gy_{\cdot }\Vert _{L^2([0,T];\mathbb {R}^n)} = \sup _{h\in L^2([0,T];\mathbb {R}^n), \Vert h\Vert _{L^2}\le 1}\int _0^TGy_r \cdot h(r)\, \textrm{d}r\\&=\sup _{h\in L^2([0,T];\mathbb {R}^n), \Vert h\Vert _{L^2}\le 1}\int _0^T \begin{pmatrix} \int _{d}^0 y_r(s) \varphi _1(\text {d}s) \\ \vdots \\ \int _{d}^0y_r(s) \varphi _n(\text {d}s) \end{pmatrix} \cdot h(r)\, \textrm{d}r\\&= \sup _{h\in L^2([0,T];\mathbb {R}^n), \Vert h\Vert _{L^2}\le 1}\sum _{i=1}^{n} \int _0^T h_i(r)\int _{d}^0 y_r(s)\varphi _i(\textrm{d}s)\, \textrm{d}r. \end{aligned}$$
We estimate the ith component (\(i=1,\ldots ,n\)) of the above expression exploiting the Fubini Theorem and the Hölder inequality.
$$\begin{aligned} \int _0^Th_i(r) \int _{d}^0y_r(s)\, \varphi _i(\textrm{d}s)\, \textrm{d}r&\le \int _0^Th_i(r) \int _{d}^0y(r+s)\, \varphi _i(\textrm{d}s)\, \textrm{d} r\\&= \int _{d}^0\int _0^Th_i(r)y(r+s)\, \textrm{d}r\, \varphi _i(\textrm{d}s) \\&\le \int _{d}^0\Vert h_i\Vert _{L^2([0,T];\mathbb {R})}\Vert y\Vert _{L^2([s,s+T];\mathbb {R})}\, \varphi _i(\textrm{d}s)\\&\le \varphi _i([d,0])\Vert h_i\Vert _{L^2([0,T];\mathbb {R})}\Vert y\Vert _{L^2([d,T];\mathbb {R})}, \end{aligned}$$where for the last inequality we exploit the inclusion \([s, s+T] \subseteq [d,T]\). Therefore, by means of the Hölder inequality we obtain
$$\begin{aligned}&\Vert \mathcal {G}(y)\Vert _{L^2([0,T];\mathbb {R}^n)} \le \sup _{h\in L^2([0,T];\mathbb {R}^n), \Vert h\Vert _{L^2}\le 1}\\&\quad \sum _{i=1}^{n}\varphi _i([d,0])\Vert h_i\Vert _{L^2([0,T];\mathbb {R})}\Vert y\Vert _{L^2([d,T];\mathbb {R})}\\&\quad \le \Vert y\Vert _{L^2([d,T];\mathbb {R})}\sup _{h\in L^2([0,T];\mathbb {R}^n), \Vert h\Vert _{L^2}\le 1} \\&\quad \times \left( \sum _{i=1}^n [\varphi _i([d,0])]^2\right) ^{\frac{1}{2}}\left( \sum _{i=1}^n \Vert h_i\Vert ^2_{L^2([0,T];\mathbb {R})}\right) ^{\frac{1}{2}}\\&\quad =\Vert y\Vert _{L^2([d,T];\mathbb {R})}\sup _{h\in L^2([0,T];\mathbb {R}^n), \Vert h\Vert _{L^2}\le 1} \left( \sum _{i=1}^n [\varphi _i([d,0])]^2\right) ^{\frac{1}{2}}\Vert h\Vert ^2_{L^2([0,T];\mathbb {R}^n)}\\&\quad \le \left( \sum _{i=1}^n [\varphi _i([d,0])]^2\right) ^{\frac{1}{2}}\Vert y\Vert _{L^2([d,T];\mathbb {R})}. \end{aligned}$$ 
2.
The existence of the bounded linear extension of \(\mathcal {L}\) and \(\mathcal {G}\) to \(L^2([d,t];\mathbb {R})\) is a consequence of inequalities (46) and (47) and the fact that \(C([d,T];\mathbb {R})\) is dense in \(L^2([d,T];\mathbb {R})\).
\(\square \)
We are now ready to prove Proposition 2.1. We will use the following notation: if functions \(a, b \ge 0\) satisfy the inequality \(a \le C(A)b\) with a constant \(C(A)>0\) depending on the expression A, we write \(a \lesssim _A b\).
Proof
(Proof of Proposition 2.1) The proof of the result relies on a contraction type argument. The same argument has been used in the proof of [6, Proposition B.2]. There the authors consider a SDDE of type (2) with no delay in the diffusion term. On the other hand they work in a more general setting considering a measure valued process \(\phi \) in the delay integral of the drift term.
We provide here a sketch of the proof referring to [6] for more details. We will give just the details of the estimates concerning the delay integral in the diffusion term that is missing in [6].
Let us fix the initial condition \((x_0, x_1) \in \mathbb {R}^2 \times L^2([d,0];\mathbb {R})\). Let \(T>0\), we introduce the space
endowed with the sup norm
We consider the space \(L^p(\Omega ;S_T)\), \(p \ge 2\), endowed with the norm
In the sequel we will denote by \(p':=\frac{p}{p1}\) the conjugate exponent to p and by \(p^*:=\frac{p}{p2}\) the conjugate exponent to \(\frac{p}{2}\). Given \(y \in L^p(\Omega ;S_T)\), let
Here \(\mathcal {L}\) and \(\mathcal {G}\) are the continuous linear operators introduced in Lemma 5.1 and \({\bar{y}}^{x_1} \in L^p(\Omega ;L^2([d,T];\mathbb {R}) )\) is defined as follows:
We aim at proving that F maps \(L^p(\Omega ;S_T)\) into itself for any \(p\ge 2\) and that it is a contraction on the same space when \(p>4\).
Let us start by proving that F maps \(L^p(\Omega , S_T)\), \(p\ge 2\), into itself. We write
The boundedness of the terms that appears in the r.h.s. of (50), except the last one, can be proved following the lines of [6, Proposition B.2]. We estimate the last term in the r.h.s. of (50) by means of the BurkholderDaviesGundy inequality
where in the last inequality we exploited (47) of Lemma 5.1.
Let us now prove that, for \(p>4\), F defines a contraction in \(L^p(\Omega , S_T)\). We endow this space by the equivalent norm
where \(\alpha >0\) will be chosen later on. Once we proved that F defines a contraction, by the Banach fixed point Theorem, we can infer the existence of a unique \(y \in L^p(\Omega ;S_T)\) such that \(y=F(y)\), i.e.
and this will conclude the proof.
Given \(y, z \in L^p(\Omega ;S_T)\), from (48) and (51), we have
We can estimate the first three terms in the r.h.s. of (52) proceeding as in [6, Proposition B.2]^{Footnote 4}. For the first term we obtain
For the second term we get
For the third term, by means of the so called factorization method (see e.g. [?, Section 5.3]), for a given \(\delta \in \left( \frac{1}{p}, \frac{1}{2}\right) \)^{Footnote 5}, we have
Let us now come to the estimate of the fourth term in (52). Exploiting the factorization method, for \(\eta \in \left( \frac{1}{p}, \frac{p2}{2p}\right) \)^{Footnote 6} we rewrite that stochastic integral as follows
with
and
Thanks to the Hölder inequality we estimate
Therefore we obtain
Now, recalling the definition of \(\mathcal {G}\) and that, when \(r<d\), \({\bar{z}}^{x_1}_r(s){\bar{y}}^{x_1}_r(s)=0\) for \(s \in [d,r)\) (see (49)), by means of the BurkholderDavisGundy (BDG) and the Hölder (H) inequalities, we obtain for all \(u \in [0,T]\),
Therefore,
where the last inequality is obtained as follows:
Putting together the above estimates we obtain
Finally, from (53), (54), (55) and (56) we infer
where \(C_i(\alpha ) \rightarrow 0\) as \(\alpha \rightarrow \infty \), for \(i=1,\ldots ,4\). So, by taking \(\alpha >0\) sufficiently large, this proves that F is a contraction, thus there exists a unique fixed point of it. In this way we prove the existence and uniqueness of the solution in the space \(L^p(\Omega ,S_T)\) for \(p >4\). Since, for such p, \(L^p(\Omega ,S_T)\subset L^2(\Omega ,S_T)\), such solution also belongs to \(L^2(\Omega ,S_T)\). To get uniqueness in the space \(L^2(\Omega ,C([0,T];\mathbb {R}))\) one can take two solutions y and \({{\tilde{y}}}\) in this space and take their difference. Using the fact that both are fixed points of F, by means of the Gronwall Lemma one gets \(\sup _{t\in [0,T]} \mathbb {E}\left[ y(t){{\tilde{y}}}(t)^2\right] =0\) and this concludes the proof.\(\square \)
1.2 A.2 Proof of Lemma 4.1
Proof
Let us denote with \(\sigma _y^i\) the ith component of \(\sigma _y\), and let us show that
By the trivial inequality \((a+b)^2 \le 2( a^2 + b^2)\), it is sufficient to show that
and
We immediately see that (57) holds true thanks to Proposition 2.1.
To show (58), we use the Hölder inequality and the Fubini Theorem to estimate
which is finite, thanks to Proposition 2.1. \(\square \)
1.3 A.3 Proof of Proposition 4.3
Proof

(i)
The operator A can be written in the form
$$\begin{aligned} A\left( x_0,x_1\right) =\left( \int _{d}^0 x_1(\theta )a(\text {d}\theta ), \frac{\text {d}}{\text {d}s}x_1\right) \,, \end{aligned}$$(59)where
$$\begin{aligned}a(\text {d}\theta )=\mu _y\delta _0(\text {d}\theta )+\Phi (\text {d}\theta )\,,\end{aligned}$$and \(\delta _0\) is the deltaDirac measure at zero. The measure a defines a finite measure on \(\left[ d,0\right] \). The result is thus an immediate consequence of [14, PropositionA.27].

(ii)
See e.g. [23, Chapter 7, Lemma 1.2].

(iii)
Existence and uniqueness of a weak solution given by (28) for deterministic \({\textbf{m}}\) is a classical result (see [14, Proposition A.5]). One can then easily generalize the result to random \({\textbf{m}}\). Property (29) follows from uniqueness of the solution.

(iv)
If \(m(t_0;\cdot )\) is the unique solution to (25) then the \(\mathcal {H}\)valued process \(\left( m(t_0;t),m(t_0;t+\cdot )\right) _{t\ge t_0}\) solves (27) by [4, Part II,Chapter 4, Theorem 4.3]. Since also the latter has a unique solution, its first component must be the solution to (27).
\(\square \)
1.4 A.4 Proof of Lemma 4.6
Proof
If \(\lambda \in \mathbb {R} \cap \rho (A)\) then \(K(\lambda ) \ne 0\) by Lemma 4.5. To compute \(R(\lambda , A)\), we will consider for a fixed \({\textbf{m}}=\left( m_0,m_1\right) \in \mathcal {H}\) the equation
in the unknown \((u_0,u_1)\in \mathcal {D}(A)\), that by definition of A is equivalent to
Then
and \(u_0\) is determined by the equation
yielding
Then the result immediately follows. \(\square \)
1.5 A.5 Proof of Lemma 4.10
Proof
It is immediate to check that the function \({\widetilde{K}}:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuous and differentiable with
and that
Equation \({\widetilde{K}}(\xi )=0\) has thus exactly one real solution \(\bar{\xi }\). Let us now show that \(\bar{\xi }=\widetilde{\lambda _0}\). By the definition of \(\widetilde{\lambda _0}\), clearly we have that \(\bar{\xi }\le \widetilde{\lambda _0}\). To show the opposite inequality, \(\bar{\xi }\ge \widetilde{\lambda _0}\), we consider an arbitrary \(\lambda =a+i b\in \mathbb {C}\) such that \({\widetilde{K}}(\lambda )=0\). Then
Since \(\widetilde{K}\) is an increasing function, we can infer Re\(\lambda \le \bar{\xi }\) and taking the supremum in the definition of \(\widetilde{\lambda _0}\) we obtain \(\widetilde{\lambda _0} \le \bar{\xi }\). By the same argument, the relation \({\widetilde{K}}(r)> 0 \iff r > \widetilde{\lambda _0}\) immediately follows. \(\square \)
1.6 A.6 Proof of Lemma 4.11
Proof
Since by Lemma 4.10 we know that \(\widetilde{K}\) is an increasing function and \(\widetilde{K}(\widetilde{\lambda _0})=0\), we just need to prove that \(\widetilde{K}(\lambda _0) \le 0\). For every \(\lambda =a+i b \in \mathbb {C}\) we have
Recalling the definition of \(\lambda _0\) it is enough to show that, for every \(\lambda =a+i b \in \mathbb {C}\) such that \(K(\lambda )=0\), it holds \(\widetilde{K}(a)\le 0\). We have that
This concludes the proof. \(\square \)
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Biffis, E., Goldys, B., Prosdocimi, C. et al. A pricing formula for delayed claims: appreciating the past to value the future. Math Finan Econ 17, 175–202 (2023). https://doi.org/10.1007/s11579022003317
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DOI: https://doi.org/10.1007/s11579022003317
Keywords
 Stochastic functional differential equations
 Delay equations
 Noarbitrage pricing
 Human capital
 Sticky wages