Infinite horizon backward stochastic Volterra integral equations and discounted control problems

Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted $L^2$-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted $L^2$-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin's maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.


Introduction
Backward stochastic Volterra integral equations (BSVIEs for short) are Volterra-type extensions of backward stochastic differential equations (BSDEs for short). A class of BSVIEs introduced by Lin [19] and Yong [41,43] is the following: where T ∈ (0, ∞) is a given terminal time, W (·) is a Brownian motion, ψ(·) is a given (not necessarily adapted) stochastic process called the free term, and g is a given map called the driver. The unknown we are looking for is the pair (Y (·), Z(·, ·)), where Y (·) = (Y (s)) s∈[0,T ] and Z(t, ·) = (Z(t, s)) s∈[t,T ] are adapted for each t ∈ [0, T ]. Sometimes a BSVIE of the form (1.1) is called a Type-I BSVIE. If the free term ψ(·) and the driver g of Type-I BSVIE (1.1) does not depend on t, then the second term Z(t, s) of the solution becomes independent of t, and (1.1) becomes the standard BSDE: which can be written in the differential form: dY (t) = −g(t, Y (t), Z(t)) dt + Z(t) dW (t), t ∈ [0, T ], Y (T ) = ψ.
BSDEs have very interesting applications in stochastic control, mathematical finance, PDE theory, and many other fields. Relevant theory of BSDEs can be found in the textbook of Zhang [44] and references cited therein. Type-I BSVIEs are important tools to study stochastic control and mathematical finance taking into account the time-inconsistency. For example, time-inconsistent dynamic risk measures and time-inconsistent recursive utilities can be modelled by the solutions to Type-I BSVIEs (see [42,18,2,31]). Time-inconsistent stochastic control problems related to Type-I BSVIEs were studied by Wang and Yong [32], Hernández and Possamaï [14], and the author [13]. Also, Beissner and Rosazza Gianin [5] applied Type-I BSVIEs to arbitrage-free asset pricing via a path of EMMs called an EMM-string. As a further extension of Type-I BSVIEs, Yong [41,43] considered the following equation: . This kind of solution concept was introduced by Yong [43] and applied to the duality principle appearing in stochastic control problems for (forward) stochastic Volterra integral equations (SVIEs for short). Since then, Type-II BSVIEs have been found important to study stochastic control problems of SVIEs (see [7,29,30,37,35]).
We emphasize that, in all the previous works mentioned above, BSVIEs and the related stochastic control problems are considered on a finite horizon [0, T ] with a given (deterministic) terminal time T ∈ (0, ∞). In contrast, in this paper, we are interested in BSVIEs and stochastic control problems with T = ∞. More precisely, we consider an infinite horizon BSVIE of the following form: where λ ∈ R represents a given (not necessarily positive) "discount rate" which determines the long-time behavior of the solution. This form of BSVIE appears for the first time in the literature. As in the finite horizon case, we often call equation (1.3) an infinite horizon Type-II BSVIE and the equation (1.3) with the driver g(t, s, y, z 1 , z 2 ) being independent of z 2 an infinite horizon Type-I BSVIE. Infinite horizon BSVIE (1.3) can be seen as an extension of an infinite horizon BSDE of the following form: dY (t) = − g(t, Y (t), Z(t)) − λY (t) dt + Z(t) dW (t), t ≥ 0, (1.4) which was studied in, for example, [26,27,11,10,40,39]. Compared with the finite horizon case, in the infinite horizon BSDE (1.4), the terminal condition for Y (·) is replaced by a (given) growth condition at infinity. The main difficulty appearing in the infinite horizon case is to identify the condition for the discount rate λ which ensures the existence and uniqueness of the solution in a given weighted space. A similar difficulty also arises in the study of infinite horizon BSVIE (1.3). Specifically, consider the situation where we are given a free term ψ(·) satisfying E ∞ 0 e 2ηt |ψ(t)| 2 dt < ∞ for some fixed weight η ∈ R. Then the adapted M-solution (Y (·), Z(·, ·)) to the infinite horizon BSVIE (1.3) should satisfy a similar growth condition at infinity: |Z(t, s)| 2 ds dt < ∞.
A delicate question is the condition which ensures the existence and uniqueness of the adapted M-solution for any free term ψ(·) in a given weighted L 2 -space. In this paper, we mainly focus on this problem. Intuitively speaking, in order to ensure the well-posedness in a large space (i.e., the weight η ∈ R is a large negative number), the driver g has to be discounted sufficiently at infinity (i.e., the discount rate λ ∈ R should be a sufficiently large positive number). A criterion for this trade-off relationship between the weight η and the discount rate λ is determined by the driver g.
With the above intuitive discussions in mind, we prove the well-posedness of infinite horizon BSVIE (1.3) (see Theorem 3.7). We also show that the adapted M-solutions of finite horizon BSVIEs converge to the adapted M-solution of the original infinite horizon BSVIE (see Theorem 3.9). Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for an infinite horizon linear Type-I BSVIE (see Theorem 3.12), which is an extension of the results of Hu and Øksendal [16] and Wang, Yong, and Zhang [33]. We also show a duality principle between a linear SVIE and an infinite horizon linear Type-II BSVIE in a weighted L 2 -space (see Theorem 3.15), which is an extension of that of Yong [43]. We remark that our results mentioned above are not only extensions of the corresponding known results to infinite horizon cases, but also generalizations to unbounded coefficients cases. Indeed, in [16,33] and [43], they considered linear (finite horizon) BSVIEs with bounded coefficients, while in this paper we treat linear infinite horizon BSVIEs with unbounded coefficients. These generalizations are important to apply the results to stochastic control problems for a forward SVIEs with singular coefficients.
As an application of infinite horizon BSVIEs, we investigate an infinite horizon optimal control problem for a forward SVIE. There are many applications of SVIEs in different research areas concerned with the effects of memory and delay, such as stochastic production management models (see [37]), epidemic models (see [8,25]), and rough volatility models in mathematical finance (see [12]). In this paper, we define the state process X u (·) corresponding to a given control process u(·) by the solution to a controlled SVIE of the form X u (t) = ϕ(t) + t 0 b(t, s, X u (s), u(s)) ds + t 0 σ(t, s, X u (s), u(s)) dW (s), t ≥ 0.
The objective is to minimize a discounted cost functional J λ (u(·)) := E ∞ 0 e −λt h(t, X u (t), u(t)) dt over all admissible control processes u(·). The coefficients b and σ are allowed to be singular at the diagonal t = s. The state process X u (·) is in general neither Markovian nor semimartingales and includes the fractional Brownian motion with Hurst parameter smaller than 1/2 as a special case. In this framework, by using the duality principle mentioned above, we provide a necessary condition for optimality by means of Pontryagin's maximum principle (see Theorem 4.3). The corresponding adjoint equation can be expressed as an infinite horizon linear Type-II BSVIE with unbounded coefficients. We also provide sufficient conditions for optimality in terms of the related Hamiltonian (see Theorem 4.5). These results can be applied to infinite horizon stochastic control problems of linear-quadratic types (see Example 4.7), fractional stochastic differential equations (see Example 4.10), and stochastic integro-differential equations (see Example 4.12).
Let us further discuss some novelties of this paper compared with the related works. As we mentioned above, this paper is the first to investigate BSVIEs on the infinite horizon. Furthermore, our method to show the well-posedness is completely different from the ones in the literature of (finite horizon) BSVIEs. A main technique used in the literature is a fixed point argument based on some estimates for auxiliary BSDE systems (see, for example, [43]). In contrast, our method to show the well-posedness of infinite horizon BSVIEs does not rely on the theory of BSDEs or Itô's calculus. Instead, we first show a priori estimates for the adapted M-solutions by some very simple calculations and then prove the well-posedness by using the so-called method of continuation which is known as a fundamental technique in the theory of forward-backward SDEs (see the textbook of Ma and Yong [21]).
Infinite horizon stochastic control problems for stochastic differential equations (SDEs for short) with discounted cost functional have been studied in, for example, [11,23,24]. In these papers, infinite horizon BSDEs play central roles in different ways. Fuhrman and Tessitore [11] investigated a relationship between an infinite horizon BSDE and an elliptic Hamilton-Jacobi-Bellman equation to obtain the optimal control. Maslowski and Veverka [23] obtained a sufficient maximum principle, and Orrieri and Veverka [24] established a necessary maximum principle for an SDE with dissipative coefficients. Our results are extensions of [23,24] to the Volterra setting. Indeed, in the SDEs setting, the adjoint equation which we obtain in the general Volterra setting is reduced to the infinite horizon BSDE considered in the aforementioned papers (see Example 4.9).
Concerned with stochastic control problems for SVIEs, Yong [43] and Wang [35] obtained a necessary condition for optimality by means of the stochastic maximum principle, where the (first order) adjoint equation can be described as a finite horizon BSVIE with bounded coefficients. Shi, Wang, and Yong [29] considered a mean-field type stochastic control problem and obtained a necessary condition by using a mean-field BSVIE. Agram and Øksendal [3] proposed a Malliavin calculus technique to obtain the optimal control. Chen and Yong [7] and Wang [34] studied linear-quadratic control problems for SVIEs. The aforementioned papers are restricted to finite horizon cases. Also, since some kinds of regularity are required for the coefficients, they exclude the case of singular coefficients such as the fractional Brownian motion with Hurst parameter H < 1/2. Concerned with the later problem, Abi Jaber, Miller, and Pham [1] considered a finite horizon linear-quadratic control problem for an SVIE of convolution type, which includes a singular coefficients case. They used a technique of the so-called Markovian lift of a stochastic Volterra process and obtained the optimal feedback control, depending on non-standard Banach space-valued Riccati equations. Their results heavily rely on the special structure of the state process and the cost functional. Compared with the above papers, we consider an infinite horizon stochastic control problem with a discounted cost functional of a general form including the quadratic cost, where the controlled dynamics of the state process is described as an SVIE with nonlinear, non-convolution type, and singular coefficients. Therefore, apart from the extension to the infinite horizon case, our results give a new insight at this general setting.
This paper is organized as follows. In Section 2, we briefly discuss the well-posedness of (forward) SVIEs with singular coefficients. In Section 3, we prove the well-posedness of infinite horizon BSVIEs. We also provide a convergence result for finite horizon BSVIEs, a variation of constant formula for an infinite horizon linear Type-I BSVIE, and a duality principle between a linear SVIE and an infinite horizon linear Type-II BSVIE in a weighted L 2 -space. In Section 4, as an application of infinite horizon BSVIEs, we study an infinite horizon stochastic control problem for an SVIE with singular coefficients. Some examples including controlled fractional stochastic differential equations and controlled stochastic integro-differential equations are presented in Section 4.3.

Notation
For each d 1 , d 2 ∈ N, we denote the space of (d 1 ×d 2 )-matrices by R d1×d2 , which is endowed with the Frobenius norm denoted by | · |. We define R d1 := R d1×1 , that is, each element of R d1 is understood as a column vector. We denote by ·, · the usual inner product in a Euclidean space. For each matrix A, A ⊤ denotes the transpose of A.
We define Let U be a nonempty Borel subset of a Euclidean space, and let p ∈ [1, ∞) and β ∈ R be fixed. L p F∞ (Ω; U ) denotes the set of U -valued and F ∞ -measurable L p -random variables. Define Also, we define If U is a Euclidean space, these spaces are Banach spaces with the norms, for example, , and L p, * F (0, ∞; U ) are defined by similar manners as before.
For each z(·) ∈ L 2 which is a d 1 -dimensional square-integrable martingale. Here and elsewhere, for each z ∈ R d1×d , z k ∈ R d1 denotes the k-th column vector for k = 1, . . . , d.
We say that a process X(·) is a solution to SVIE (2.1) if X(·) is in L 2, * F (0, ∞; R n ) and satisfies the equality for a.e. t ≥ 0, a.s. A delicate problem is the identification of the weight of the space where the solution uniquely exists. The following lemma is useful for the analysis of SVIEs.
Proof. By using the assumptions, together with Young's convolution inequality, we have Thus the estimate (2.2) holds. Similarly, we have Thus the estimate (2.3) holds.
Observe that, if K b ∈ L 1, * (0, ∞; R + ) \ L 1 (0, ∞; R + ) or K σ ∈ L 2, * (0, ∞; R + ) \ L 2 (0, ∞; R + ), then the number µ in the above lemma has to be strictly positive, and this is indeed a typical case. Motivated by the above lemma, we define The number ρ b,σ gives a criterion for the weight of the space in which the solution to SVIE (2.1) uniquely exists for any ϕ(·). Now we prove the well-posedness of SVIE (2.1).
Clearly, X(·) is the unique solution of the original SVIE (2.1).

Infinite horizon BSVIEs
In this section, we investigate infinite horizon BSVIE (1.3), which we rewrite for readers' convenience: where the free term ψ(·) ∈ L 2, * F∞ (0, ∞; R m ) with m ∈ N, the driver g, and the discount rate λ ∈ R are given. We impose the following assumptions on the driver g: (iii) There exist K g,y ∈ L 1, * (0, ∞; R + ) and K g,z1 , K g,z2 ∈ L 2, * (0, ∞; R + ) such that |g(t, s, y, z 1 , for any (y, z 1 , z 2 ), (y ′ , z ′ 1 , z ′ 2 ) ∈ R m × R m×d × R m×d , for a.e. (t, s) ∈ ∆[0, ∞), a.s. Note that, as in the SVIEs case, the Lipschitz coefficients are allowed to be singular at the diagonal t = s and the infinity. We give a definition of the solution.
Definition 3.1. We call a pair (Y (·), Z(·, ·)) the adapted M-solution of infinite horizon BSVIE (3.1) if (Y (·), Z(·, ·)) is in M 2, * F (0, ∞; R m × R m×d ) and satisfies the equality for a.e. t ≥ 0, a.s. The above definition is a natural extension of that of [43] to the infinite horizon setting. The "M"-solution is named after the martingale representation theorem Recall that this relation is a part of the definition of the space

Well-posedness of infinite horizon BSVIEs
In this subsection, we prove the well-posedness of infinite horizon BSVIE (3.1) under Assumption 2. A delicate problem is the condition which ensures the existence and uniqueness of the adapted M-solution for any free term ψ(·) in a given weighted L 2 -space. As we mentioned in the introductory section, in order to ensure the well-posedness in a large space, the driver g has to be discounted sufficiently at infinity. The following lemma provides a useful criterion for this trade-off relationship.
With the above lemma in mind, we introduce the following domain: which gives a criterion for the trade-off relationship between the weight of the space of the solution and the discount rate. Note that, since K g,y ∈ L 1, * (0, ∞; R + ) and K g,z1 , K g,z2 ∈ L 2, * (0, ∞; R + ), the set R g is nonempty.
(ii) If the driver g(t, s, y, z 1 , z 2 ) does not depend on z 1 , then we can take K g,z1 = 0. In this case, we have (iii) If the driver g(t, s, y, z 1 , z 2 ) does not depend on (y, z 2 ), then we can take K g,y = K g,z2 = 0. In this case, we have We prove the well-posedness of infinite horizon BSVIE (3.1) by using the so-called method of continuation. For this purpose, we first provide a priori estimates for infinite horizon BSVIEs, which is important on its own right. Proposition 3.4. Suppose that Assumption 2 holds, and fix (η, λ) ∈ R g . Let ψ(·) ∈ L 2,η F∞ (0, ∞; R m ) be given, and let (Y (·), Z(·, ·)) ∈ M 2,η F (0, ∞; R m × R m×d ) be an adapted M-solution of infinite horizon BSVIE (3.1).

Then it holds that
for a.e. t ≥ 0. Thus, by using Minkowski's inequality, we get By Lemma 3.2, the second term in the right-hand side is estimated as Noting that (η, λ) ∈ R g with R g defined by (3.2), by the estimates (3.5) and (3.6), we get On the other hand, since (Y (·), Z(·, ·)) ∈ M 2,η for a.e. t ≥ 0. Therefore, we obtain , and thus the estimate (3.3) holds. Next, we prove the estimate Note that ψ(·) ∈ L 2,η F∞ (0, ∞; R m ), and the driver g satisfies Assumption 2 with the functions K g,y , K g,z1 , and K g,z2 . By estimate (3.3), it holds that Thus the estimate (3.4) holds.
Next, we consider a trivial infinite horizon BSVIE whose generator g vanishes.
has a unique adapted M-solution (Y (·), Z(·, ·)) ∈ M 2,η F (0, ∞; R m × R m×d ). Proof. The uniqueness is trivial. We prove the existence. We should consider the issues of (joint) measurability and the integrability carefully. We split the proof into three steps.
Step 1. Assume that t → ψ(t) ∈ L 2 F∞ (Ω; R m ) is continuous. By the martingale representation theorem, for any t ≥ 0, there exists a unique adapted process . Then it holds that for any t ≥ 0. By the continuity of the map t → ψ(t) ∈ L 2 F∞ (Ω; R m ), it is easy to see that the maps t → Y (t) ∈ L 2 F∞ (Ω; R m ) and t → Z(t, ·) ∈ L 2 F (0, ∞; R m×d ) are continuous. Thus, there exist (jointly) measurable versions of Y (·) and Z(·, ·). Furthermore, we have Thus, the measurable version (Y (·), Z(·, ·)) is in M 2,η F (0, ∞; R m × R m×d ), and it is the adapted M-solution of (3.7).
By the dominated convergence theorem, we see that , and the limit is the adapted M-solution of (3.7). This completes the proof.
Lemma 3.6. Suppose that Assumption 2 holds, and fix (η, λ) ∈ R g . Assume that the property (P γ0 ) holds for some γ 0 ∈ [0, 1). Then (P γ ) holds for any γ for i ∈ N as the unique adapted M-solution of the following infinite horizon BSVIE: By the assumption, the sequence {(Y (i) (·), Z (i) (·, ·))} i∈N can be defined inductively. Noting that γ 0 ∈ [0, 1), by the stability estimate (3.4), together with Lemma 3.2, for each i ∈ N, we have for a.e. t ≥ 0, a.s., and thus the pair ( is an adapted M-solution of the infinite horizon BSVIE (3.8) with the parameter γ. Furthermore, by Proposition 3.4, the adapted M-solution is unique. Since ψ(·) ∈ L 2,η F∞ (0, ∞; R m ) is arbitrary, we see that the property (P γ ) holds. This completes the proof. Now we are ready to prove the well-posedness of infinite horizon BSVIE (3.1).
• For a.e. t ∈ [0, ∞), define Φ(t, ·) as the solution of the SDE (3.11) where I m ∈ R m×m is the identity matrix.

12)
where, for each A ∈ R m×m , |A| op denotes the operator norm of A as a linear operator on R m .
where, in the first inequality, we used the Cauchy-Schwarz inequality for the conditional expectation E t [·]; in the second inequality, we used the estimate (3.14); in the third inequality, we used the Cauchy-Schwarz inequality for the integral ∞ t · · · ds; in the last inequality, we used Young's convolution inequality. Observe that     B k (t, r)Z k (t, r) dr for (t, s) ∈ ∆[0, ∞). For each (t, s) ∈ ∆[0, ∞), by taking the conditional expectation E s [·] on both sides of (3.9), we have For a.e. t ≥ 0, by using Itô's formula for the product Φ(t, ·)Y (t, ·), we obtain for any s ≥ t, a.s. Note that the left-hand side tends to Thus, the stochastic integral in the right-hand side of (3.15) is a uniformly integrable martingale (with respect to the time parameter s ∈ [t, ∞)). By letting s → ∞ and taking the conditional expectation E t [·] on both sides of (3.15), we obtain Hence, by using the notation Ξ 1 (t, s) := Φ(t, s)A(t, s), for a.e. t ≥ 0, a.s. We shall show that, for each N ∈ N, for a.e. t ≥ 0, a.s. Thus the equality (3.17) holds for N = 1. Let M ∈ N be fixed, and assume that (3.17) holds for N = M . Again by using (3.16) and Fubini's theorem, we have Noting that the process Y (·) is in L 2,η F (0, ∞; R m ), by Lemma 3.11 (ii), the process On the other hand, by the estimate (3.12), This implies that the process (E t [Φ(t, ∞)ψ(t)]) t≥0 is in L 2,η F (0, ∞; R m ), and thus the process for a.e. t ≥ 0, a.s. This completes the proof.

Then it holds that
(3.21) Proof. By the (forward) SVIE (3.18), the definition of adapted M-solutions, and Fubini's theorem, we have On the other hand, by the infinite horizon BSVIE (3.19), for a.e. t ≥ 0, a.s. Noting that X(·) is adapted, we see that Thus, the equality (3.21) holds.

Discounted control problems for SVIEs with singular coefficients
As an application of the results on infinite horizon BSVIEs, we consider an infinite horizon stochastic control problem where the dynamics of the state process is described as a (forward) SVIE with singular coefficients. The duality principle which we proved in the previous section plays a crucial role.
For each µ ∈ R, define the set of control processes by U −µ := L 2,−µ F (0, ∞; U ), where U is a convex body (i.e., U is convex and has a nonempty interior) in R ℓ with ℓ ∈ N. For each control process u(·) ∈ U −µ , define the corresponding state process X u (·) as the solution of the following controlled SVIE: where ϕ(·) is a given process, and b, σ are given deterministic maps. In order to measure the performance of u(·) and X u (·), we consider the following discounted cost functional : where h is a given R-valued deterministic function and λ ∈ R is a discount rate. The stochastic control problem is a problem to seek a control processû(·) ∈ U −µ such that If it is the case, we callû(·) an optimal control.
Proof. Let an arbitrary u(·) ∈ U −µ be fixed. We apply the duality principle (Theorem 3.15) to the variational SVIE (4.6) and the adjoint equation (4.10). By using the definition of adapted M-solutions and Fubini's theorem, we have Thus, by the variational inequality (4.5), we get Since u(·) ∈ U −µ is arbitrary, we see that (4.11) holds for a.e. t ≥ 0, a.s. This completes the proof.

Sufficient conditions for optimality
In this subsection, we provide sufficient conditions for optimality in terms of the Hamiltonian H λ defined by (4.12).
Proof. Take an arbitrary u(·) ∈ U −µ , and denote the corresponding state process by X(·) := X u (·). Observe that By using Fubini's theorem and the definition of adapted M-solutions, we see that  Noting the controlled SVIE (4.1) and the equation (4.13), we see that the last term in the above equalities is equal to Consequently, we obtain Therefore, by the assumptions, it holds that Since u(·) ∈ U −µ is arbitrary, we see that and thusû(·) is optimal. This completes the proof.
Remark 4.6. The above arguments can be easily generalized to the case where b(t, s, 0, 0) and σ(t, s, 0, 0) are nonzero (see Remarks 2.4 and 3.8). Also, the case of random coefficients b and σ (with suitable measurability and integrability conditions) can be treated by the same way as above.

Examples
In this subsection, we apply the above results to some concrete examples. First, we investigate an infinite horizon linear-quadratic (LQ for short) control problem for an SVIE.
Remark 4.8. In finite horizon settings, LQ control problems for SVIEs with non-singular coefficients were studied by Chen and Yong [7] and Wang [34]. Also, Abi Jaber, Miller, and Pham [1] investigated a finite horizon LQ control problem for an SVIE with singular coefficients of convolution type. Compared with these papers, the above example provides a characterization of the optimal control of an infinite horizon LQ control problem for an SVIE with singular coefficients of non-convolution type. However, the optimality condition (4.17) alone dose not give a (causal) state-feedback representation of the optimal control, and further analysis would be needed. We hope to report some further results on this problem in the near future.
Next, we observe a classical SDEs case.
The next example is concerned with a controlled dynamics with a fractional time derivative.
Remark 4.11. The above observations are also valid in the cases of other fractional derivatives such as the Riemann-Liouville fractional derivative (see [17]) and the Hilfer fractional derivative (see [15]). We remark that Lin and Yong [20] established Pontryagin's (necessary) maximum principle for a finite horizon control problem of a deterministic singular Volterra equation and applied it to fractional order ordinary differential equations of the Riemann-Liouville and Caputo types. Our result shown in the above example is a stochastic and infinite horizon version of [20].
Lastly, we consider a controlled stochastic integro-differential equation of an Itô-Volterra type, which can be seen as an SDE with unbounded delay. Since many systems arising from realistic models can be described as differential equations with unbounded delay (see [9] and references cited therein), optimal control problems for integro-differential systems are important problems. Example 4.12 (Discounted control problems for stochastic integro-differential equations). Consider a minimization problem of a discounted cost functional J λ (u(·)) defined by (4.2). Suppose that the state process X u (·) solves the following controlled stochastic integro-differential equation: where x 0 ∈ R n is a given initial state, are given measurable maps. Similar equations without control were studied by, for example, Appleby [4] and Mao and Riedle [22]. Also, Sakthivel, Nieto, and Mahmudov [28] investigated approximate controllability of both deterministic and stochastic integro-differential equations, which include delay of the state process alone (without delay of the control process). We note that the controlled stochastic integro-differential equation (4.22) includes not only delay of the state process X u (·), but also delay of the control process u(·).