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Existence of optimal controls for stochastic Volterra equations

We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of control problems for stochastic Volterra equations (SVEs). Our study can be applied to rough processes which arise when the kernel appearing in the controlled SVE is singular at zero. The proof of existence of relaxed optimal policies relies on the interaction between integrability hypotheses on the kernel, growth conditions on the running cost functional and on the coefficients of the controlled SVEs, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. Under classical convexity assumptions, we also deduce the existence of optimal strict controls.


Introduction
Interest in stochastic Volterra equations (SVEs) of convolution type has been increasing rapidly because they provide suitable models for applications that benefit from the memory and the varying levels of regularity of their dynamics. Such applications include, among others, turbulence modeling in physics (Barndorff-Nielsen and Schmiegel, 2008;Barndorff-Nielsen et al., 2011), modeling of energy markets (Barndorff-Nielsen et al., 2013), and modeling of rough volatility in finance (Gatheral et al., 2018).
In this paper we consider finite-horizon control problems for SVEs of convolution type driven by a multidimensional Brownian motion with linear-growth coefficients and control policies with values on a metrizable topological space of Suslin type. We are particularly interested in singular kernels, such as fractional kernels proportional to t H− 1 2 with H ∈ (0, 1 2 ). These kernels are important because they allow modeling trajectories which are strictly less regular than those of classical Brownian motion. They have been used, for instance, in financial models with rough volatility which reproduce features of time series of estimated spot volatility (Gatheral et al., 2018) and implied volatility surfaces (Alòs et al., 2007;Bayer et al., 2016).
Several studies have investigated optimal control of SVEs. Yong (2006) uses the maximum principle method to obtain optimality conditions in terms of an adjoint backward stochastic Volterra equation. Agram and Øksendal (2015) also use the maximum principle together with Malliavin calculus to obtain the adjoint equation as a standard backward SDE. Although the kernel considered in these papers is not restricted to be of convolution type, the required conditions do not allow singularity of K at zero. Recently, an extended Bellman equation has been derived in Han and Wong (2019) for the associated controlled Volterra equation.
The particular case of linear-quadratic control problems for SVEs, with controlled drift and additive fractional noise with Hurst parameter H > 1/2, has been studied in Kleptsyna et al. (2003). Similarly, in Duncan and Pasik-Duncan (2013) the authors consider a general Gaussian noise with an optimal control expressed as the sum of the well-known linear feedback control for the associated deterministic linear-quadratic control problem and the prediction of the response of a system to the future noise process. Recently, Wang (2018) investigated the linear-quadratic problem of stochastic Volterra equations by providing characterizations of optimal control in terms of a forward-backward system, but leaving aside its solvability, and under some assumptions on the coefficients that preclude (singular) fractional kernels of interest.
Abi Jaber et al. (2021b) studied control problems for linear SVEs with quadratic cost function and kernels that are the Laplace transforms of certain signed matrix measures which are not necessarily finite. They establish a correspondence between the initial problem and an infinite dimensional Markovian problem on a certain Banach space. Using a refined martingale verification argument combined with a completion of squares technique, they prove that the value function is of linear quadratic form in the new state variables, with a linear optimal feedback control, depending on nonstandard Banach space-valued Riccati equations. They also show that the value function of the stochastic Volterra optimization problem can be approximated by conventional finite dimensional Markovian linear-quadratic problems.
We propose to study the existence problem by means of so-called relaxed controls in a weak probabilistic setting. This approach compactifies the original control system by embedding it into a framework in which control policies are probability measures on the control set, and the probability space is also part of the class of admissible controls. Thus, in this setting, the unknown is no longer only the control-state process but rather an array consisting of the stochastic basis and the control-state pair solution to the relaxed version of the controlled SVE.
The main purpose of this paper is to provide a set of conditions that ensures existence of optimal relaxed controls, see Theorem 3.1. Our main contribution is that we allow kernels that are singular at zero, for instance, fractional Kernels proportional to t H− 1 2 with H ∈ (0, 1 2 ), and coefficients that are not necessarily bounded in the control variable. Under one additional assumption on the coefficients and cost function, familiar in relaxed control theory since the work of Filippov (1962), we prove that the optimal relaxed value is attained by strict policies on the original control set, see Theorem 3.2.
The paper is structured as follows. In Section 2 we establish some preliminary results on controlled stochastic Volterra equations (CSVEs). In Section 3 we describe the weak relaxed formulation of the control problem, state our main results, namely Theorems 3.1 and 3.2, and provide some examples. Section 4 contains the proofs of the main results. In Appendix A we recall an important measurability result needed for the existence of optimal strict controls. Appendix B contains an overview of the main results on relative compactness and limit theorems for Young measures that are used in the proofs of the main theorems.

Controlled stochastic Volterra equations (CSVEs)
Let T > 0 and d, d ′ ∈ N be fixed. We consider the control problem of minimizing a cost functional of the form subject to X = (X t ) t∈[0,T ] being a R d -valued solution to the controlled stochastic Volterra equation (CSVE) of the form over a certain class of control processes (u t ) t∈[0,T ] taking values in a measurable control set M. The function K ∈ L 2 loc (0, T ; R d×d ) is a given kernel, the initial condition x 0 is a deterministic R d -valued continuous function on [0, T ], and (W t ) t∈[0,T ] is a d ′ -dimensional Brownian motion defined on a probability space (Ω, F, P) endowed with a filtration F = (F t ) t≥0 , satisfying the usual conditions. In our main existence results, we will consider solutions to (2) in a weak sense, see Definition 3.2.
Throughout, we will assume the following condition on the kernel K: Assumption I. There exist r ∈ (2, ∞) and γ ∈ (0, 2] such that K ∈ L r loc (R + ; R d×d ) and The following are examples of kernels that satisfy Assumption I: 1. Let K be locally Lipschitz. Then K satisfies Assumption I with γ = 1 and for any r ∈ (2, ∞).
2. The fractional kernel K(t) = t H− 1 2 with H ∈ (0, 1 2 ) satisfies Assumption I with r ∈ (2, 2 1−2H ) and γ = 2H. We consider, for now, a control set M which is assumed to be a Hausdorff topological space endowed with the Borel σ−algebra B(M ). We will assume later more specific conditions on M.
Assumption II.
The following result extends the a-priori estimates of Lemma 3.1 of Abi Jaber et al. (2019) to the case of CSVEs.
Theorem 2.1. Suppose that Assumption II holds and that K ∈ L r loc (R + ; R d×d ) for some r > 2. Let (u t ) t∈[0,T ] be a M -valued adapted control process such that for all m > 2 satisfying 1 m ∈ 1 p + 1 r , 1 2 , where the constant c depends on m, p, c lin , T, C B 1 , T ] . Proof. For simplicity, but without loss of generality, we take d = d ′ = 1. Let t ∈ [0, T ] be fixed. Then, for any m > 1 we have Using Burkholder-Davis-Gundy inequality, and Jensen's inequality with the measure µ(ds) = Note that κ 2 is finite since by Hölder's inequality we have A similar argument for the first term I yields For each n ∈ N set τ n = inf {t ≥ 0 : |X t | ≥ n} ∧ T. By the Corollary of Theorem II.18 in Protter (2005) we have that, 1 CB is the constant in the Burkholder-Davis-Gundy inequality, see e.g. Section 4, Chapter IV in Protter (2005).
Then, by (6) and (5) we have Using the same argument in the proof of Lemma 3.1 of Abi Jaber et al. (2019), this yields (4) with a constant that only depends on m, p, c lin , C B , Corollary 2.1. Under the same Assumptions of Theorem 2.1, suppose further Assumption I also holds with γ satisfying γ > 2/m, where 1 m = 1 p + 1 r . Then X admits a version with paths in C α (0, T ; R d ) for any α ∈ 0, γ 2 − 1 m . For this version, denoted again with X, we have the following: with c depending on m, p, c lin , T, Proof. Follows directly from the estimate (4) and Lemma 2.4 in Abi Jaber et al. (2019).
In particular, one can prove the following existence result for solutions to the CSVE (2).
Corollary 2.2. Let u be a M -valued F-predictable process. Assume that K satisfies Assumption I, b and σ satisfy Assumption II and they are Lipschitz uniformly with respect to (t, u) ∈ [0, T ] × M , and Suppose further that γ > 2 1 p + 1 r . Then there exists a unique continuous solution X to the CSVE (2).
Proof. Using Theorem 2.1 and Corollary 2.1, the proof is completely analogous to the proof of Theorem 3.3 in Abi Jaber et al. (2019).
We will also frequently use the following result in the proof of the main existence result of relaxed controls. This alternative formulation of stochastic Volterra equations, by considering the integrated process · 0 X s ds, is inspired by the martingale problem approach in Abi Jaber et al. (2021a) and facilitates the justification of convergence arguments that will be useful in our setting.
Lemma 2.1. Suppose that Assumption II holds, K ∈ L 2 loc (R + ; R d×d ) and Let X be a solution to the CSVE (2) and let Z be the controlled process Conversely, if X satisfies (8) with paths in L 2 loc then it solves the CSVE (2).

Relaxed control formulation
The use of stochastic relaxed controls is inspired by the works of El Karoui et al. (1987) and Haussmann and Lepeltier (1990). In what follows, P(M ) denotes the set of all probability measures on B(M ) endowed with the σ−algebra generated by the projection maps We associate a relaxed control system to the original control problem (1)-(2) as follows.
First, we extend the definition of coefficients and cost functionals with the convention provided that for each t ∈ [0, T ] and x ∈ R d the map F (t, x, ·) is integrable with respect to q ∈ P(M ).
Definition 3.1. A stochastic process q = (q t ) t∈[0,T ] with values in P(M ) is called a stochastic relaxed control (or relaxed control process) on M if the map is predictable. In other words, a stochastic relaxed control on M is a predictable process with values in P(M ).
Given a relaxed control process (q t ) t∈[0,T ] , the associated relaxed controlled equation now reads whereσ is defined, with a slight abuse of notation, so that the following holds: For the existence ofσ see e.g, Theorem 2.5-a in El Karoui et al. (1987). The relaxed cost functional is defined as Notice that the original system (1)-(2) controlled by a M -valued process u = (u t ) t∈[0,T ] coincides with the relaxed system controlled by the Dirac measures q t = δ ut , t ∈ [0, T ]. Moreover, since relaxed controls are just usual (strict) controls with control set P(M ), the results for strict controls in the previous section also hold for relaxed controls, with the control system defined in terms of the relaxed versions of coefficients, running cost and η 1 (t, q).

Weak formulation of optimal control problem
We study the existence of an optimal control for the stochastic relaxed control system in the following weak formulation.
such that the following hold: satisfying the usual conditions, is a F-adapted solution to the relaxed controlled equation (9).
The set of weak admissible relaxed control systems with time horizon [0, T ] and initial value x 0 will be denoted byŪ (x 0 , T ). Under this weak formulation, the relaxed cost functional is defined as The relaxed control problem (RCP) consists in minimizingJ overŪ (x 0 , T ). Namely, we seek π ∈Ū (x 0 , T ) such thatJ

Main existence result
In order to complete the set of assumptions for the main existence result, we need the following definition.
Observe that, since M is Hausdorff, for every inf-compact function η the level sets {η ≤ R} are closed. Therefore, every inf-compact function is lower semi-continuous and hence Borel-measurable. If M is compact, the converse holds too, i.e. every lower semicontinuous function is inf-compact. We will denote by IC(0, T ; M ) the class of measurable functions η : Assumption III.
1. The control set M is a metrizable Suslin space i.e. there exists a Polish space S and a continuous mapping ϕ : S → M such that ϕ(S) = M .

The running cost function
3. There exist η 2 ∈ IC(0, T ; M ) and constants C 1 ∈ R, C 2 > 0 such that h satisfies the following coercivity condition: for some p ≥ 1.
4. The final cost function g : R d → R is continuous.
The following is the main result of this paper.
Example 3.1 (Fractional kernel). For simplicity, we fix d = d ′ = 1, and consider the fractional kernel K(t) = t H− 1 2 with H ∈ 1 4 , 1 2 . Suppose the coefficients and running cost function have the form with • b i , σ i measurable and continuous in u ∈ M, uniformly with respect to t ∈ [0, T ], for i = 0, 1, • b 1 , σ 1 uniformly bounded in (t, u), • h 0 ∈ IC(0, T ; M ) and h 1 LSC and bounded from below.
Suppose further that |f (t, u)| p ≤ Ch 0 (t, u) for both f = b 0 , σ 0 , some constant C > 0 and p sufficiently large satisfying 1 p < 2H − 1 2 . Then, there exists r > 2 such that so that Assumption I holds for this choice of r and γ = 2H. Assumptions II, III hold with η 1 = η 2 = (Ch 0 ) 1/p , C 1 = −C inf h 1 and C 2 = C. Then, existence of an optimal relaxed control follows from Theorem 3.1.

Existence of strict controls
Our main result on the existence of optimal strict controls requires one additional assumption, familiar in relaxed control theory since the work of Filippov (1962), to ensure existence of an optimal strict control.
1. M is a closed subset of a Euclidean space.

For each
is a convex and closed subset of Theorem 3.2 (Existence of optimal strict controls). Suppose that Assumption IV holds. Then, for each π = (Ω, F, P, F, W, X, q) ∈Ū (x 0 , T ) there exists a M -valued F-predictable control process u = (u t ) t∈[0,T ] on the same probability space (Ω, F, P) such that 1. X satisfies the Volterra equation (2) controlled by the strict control process u = (u t ) t∈[0,T ] .
In particular, if the Assumptions of Theorem 3.1 also hold, there exists a weak optimal strict control for (1)-(2).
Suppose further that |φ(t, u)| p ≤ Ch 0 (t, u), with φ(t, u) = max{|u| 2 , |u|} for all t ∈ [0, T ], and p satisfying 1 p < 2H − 1 2 . As in the Example 3.1 there exists r > 2 such that Assumptions II, III hold with η 1 = η 2 = (Ch 0 ) 1/p , C 1 = −C inf h 1 and C 2 = C. By Theorem 3.1 there is an optimal relaxed control. Let Γ 1 (t, x) = ( u, z) : u ∈ M 2 , z ≥ h 0 (t, u) + h 1 (x) . Then Γ in (14) can be written as an affine transformation of Γ 1 . More precisely, Since h 0 is a function convex on R + the epigraph Γ 1 is a convex set, then Γ is a convex set and by Theorem 3.2 there is an optimal strict control. The following result connects random Young measures with predictable relaxed controls. For the proof, see e.g. Section 3.3 of Kushner (2012) or Section 2.4 of Cecchin and Fischer (2020).
Moreover, if F = (F t ) t∈[0,T ] is a given filtration that satisfies the usual conditions and λ([0, ·) × C) is F−adapted, for all C ∈ B(M ), then q is a F−predictable process.
Remark 4.1. We will denote the disintegration formula (16) by λ(du, dt) = q t (du) dt. Note that q t (C) can be seen as the time-derivative of λ([0, t) × C) that exists for almost every t ∈ [0, T ], for all C ∈ B(M ).
is measurable. This justifies referring to the maps considered in Lemma 4.1 as random Young measures.
For the two following Lemmas, E denotes a Euclidean space with norm |·| E and inner product ·, · .
Proof. We fix t ∈ [0, T ]. For each N ∈ N and i ∈ {1, . . . , d} define The integrand in the above expression is bounded and continuous with respect to u ∈ M . Therefore, by Lemma B.3 φ N is continuous for each N ∈ N, and by dominated convergence, Moreover, since (x n ) converges in C 0, T ; R d , it is bounded and there existsρ > 0 such that sup Therefore by (17), we have As η belongs to L 1 ([0, T ] × M ; λ), so does the right side of the above inequality. Therefore, by the Lebesgue's dominated convergence theorem we have is separable and metrisable, by Lemma 1.2.3 in  it follows that Σ t is jointly measurable.
We have the following result.
as n → ∞ and, by (17), The right side of the last inequality has finite expectation by the hypothesis about ψ and the Cauchy-Schwarz' inequality. Thus, using Lebesgue's dominated convergence theorem we get as n → ∞. Now, for each n ∈ N, define the measure µ n (du, dt, dω) = λ n (ω)(du, dt)P(dω) Using Hölder's inequality we get Ω Aε and this holds uniformly with respect to n ∈ N. Since η(t, ·) is lower semi-continuous for all t ∈ [0, T ], by Lemma B.1 and Fatou's lemma we have Therefore, the same estimates hold for λ, that is, and since ε ∈ (0, 1) is arbitrary, we conclude that as n → ∞, and (21) follows.

Proof of Theorem 3.1
Let π n = (Ω n , F n , P n , W n , q n , X n ) , n ∈ N be a minimizing sequence of weak admissible relaxed controls, that is, lim From this and Assumption III it follows that there exists R > 0 such that for all n ∈ N where E n denotes expectation with respect to P n . We will divide the proof in several steps.
Step 1. Define m by 1 m = 1 p + 1 r and let α ∈ 0, γ 2 − 1 m be fixed. By Corollary 2.1 and (22) the processes X n admit versions, which we also denote with X n , with paths in C α (0, Since C α (0, T ; R d ) is compactly embedded in C(0, T ; R d ), by Chebyshev's inequality it follows that the family of laws of {X n } n∈N is tight in C(0, T ; R d ). Using Lemma 2.1, for each n ∈ N the process X n satisfies For each n ∈ N we define the random Young measure λ n (du, dt) = q n t (du) dt.
We also claim that the family of laws of {λ n } n∈N is tight in Y(0, T ; M ). Indeed, for each ε > 0 define the set By Theorems B.1 and B.2, K ε is relatively compact in the stable topology of Y(0, T ; M ), and by Chebyshev's inequality we have and the tightness of the laws of {λ n } n∈N follows. We use now Prohorov's theorem to ensure and (X n , q n ) d = (X n ,q n ). Then, using once again Lemma 4.2, now with f = σσ ⊤ and η = η 2 1 , it follows thatM n is also a martingale with respect to the filtratioñ and quadratic variation M n t = t 0 (σσ ⊤ )(s,X n s ,q n s ) ds. Again, using continuity of the We use Lemma 4.1 one last time to ensure the existence of a relaxed control process (q t ) t∈[0,T ] defined on (Ω,F ,P) such that The filtrationF = F t t∈[0,T ] is defined bỹ We now claim thatM is aF-martingale. Indeed, From (26) Also by Theorem 2.1, and Chebyshev's inequality, the random variables in (30) are uniformly integrable. Then, by Lemma 4.11 in Kallenberg (2002) we havẽ Similarly, we haveẼ This, in conjunction with the martingale property ofM n , implies that for all 0 < s < t ≤ T and for all which implies thatM is aF−martingale.

A Auxiliary results
Let (Y, G, µ) be a measure space, k, m be natural numbers, and M a closed subset of a Euclidean space. Let be given measurable functions with u → φ(y, u) continuous and u → ψ i (y, u) lower semicontinuous, for each y ∈ Y and i = 1, 2, ...m. Define Γ(y, M ) = (φ(y, u), z) ∈ R k × R m : u ∈ M, z i ≥ ψ i (y, u) for i = 1, . . . , m .

B Relative compactness and limit theorems for Young measures
Young measures on metrisable Suslin control sets have been studied by Balder (2001) andde Fitte (2003). We refer to the book by  for more details.
Proof. For the metrisability part, see Proposition 2.3.1 in . For the Suslin part, see Proposition 2.3.3 in .
Proof. If (1) holds, the result follows from Theorem 2.1.3-Part G in . If (2) holds, the result follows from Proposition 2.1.12-Part (d) in .
It is worth mentioning that these last two results are, in fact, the main reasons why it suffices for the control set M to be only metrisable and Suslin, in contrast with the existing literature on stochastic relaxed controls. Indeed, Theorem B.2 is key to obtain tightness of the laws of random Young measures in the proof of the main existence result, and Lemma B.1 is used to prove the lower semi-continuity of the relaxed cost functionals as well as Theorem B.3 below. Proof. Use Lemma B.1 with f and −f .
Proof. Proposition 1 in Valadier (1993) implies that δ yn → δ y stably in Y(0, T ; S), and the result follows from Corollary 2.2.2 and Theorem 2.3.1 in .