Andrés Cárdenas
email: andres.cardenas@urosario.edu.co

Sergio Pulido
email: sergio.pulidonino@ensiie.fr

Rafael Serrano
email: rafael.serrano@urosario.edu.co

Existence of optimal controls for stochastic Volterra equations *

Keywords: stochastic Volterra equations, rough processes, relaxed control, Young measures, tightness, weak formulation. MSC2010 classifications: 93E20, 60G22, 60H20

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 [START_REF] Barndorff-Nielsen | Time change, volatility, and turbulence[END_REF][START_REF] Barndorff-Nielsen | Ambit processes and stochastic partial differential equations[END_REF], modeling of energy markets [START_REF] Barndorff-Nielsen | Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes[END_REF], and modeling of rough volatility in finance [START_REF] Gatheral | Volatility is rough[END_REF].

1

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 [START_REF] Gatheral | Volatility is rough[END_REF] and implied volatility surfaces [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF][START_REF] Bayer | Pricing under rough volatility[END_REF].

Several studies have investigated optimal control of SVEs. [START_REF] Yong | Backward stochastic Volterra integral equations and some related problems[END_REF] uses the maximum principle method to obtain optimality conditions in terms of an adjoint backward stochastic Volterra equation. [START_REF] Agram | Malliavin calculus and optimal control of stochastic Volterra equations[END_REF] 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 [START_REF] Han | Time-consistent feedback strategies with Volterra processes[END_REF] 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 [START_REF] Kleptsyna | About the linear-quadratic regulator problem under a fractional brownian perturbation[END_REF]. Similarly, in [START_REF] Duncan | Linear-quadratic fractional gaussian control[END_REF] 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, [START_REF] Wang | Linear quadratic control problems of stochastic Volterra integral equations[END_REF] 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.

In the stochastic case, relaxed control of finite-dimensional stochastic systems goes back to [START_REF] Fleming | On stochastic relaxed control for partially observed diffusions[END_REF]. Their approach was followed extensively by El [START_REF] El Karoui | Compactification methods in the control of degenerate diffusions: existence of an optimal control[END_REF], [START_REF] Haussmann | On the existence of optimal controls[END_REF], [START_REF] Kurtz | Existence of Markov controls and characterization of optimal Markov controls[END_REF], [START_REF] Mezerdi | Necessary conditions for optimality in relaxed stochastic control problems[END_REF] and [START_REF] Dufour | On the existence of strict optimal controls for constrained, controlled Markov processes in continuous time[END_REF]. Relaxed controls have also been used to study singular control problems [START_REF] Haussmann | Singular optimal stochastic controls i: Existence[END_REF][START_REF] Kurtz | Stationary solutions and forward equations for controlled and singular martingale problems[END_REF][START_REF] Andersson | The relaxed general maximum principle for singular optimal control of diffusions[END_REF], mean-field games [START_REF] Lacker | Mean field games via controlled martingale problems: existence of markovian equilibria[END_REF][START_REF] Fu | Mean field games with singular controls[END_REF][START_REF] Cecchin | Probabilistic approach to finite state mean field games[END_REF][START_REF] Benazzoli | Mean field games with controlled jumpdiffusion dynamics: Existence results and an illiquid interbank market model[END_REF][START_REF] Bouveret | Mean-field games of optimal stopping: a relaxed solution approach[END_REF][START_REF] Barrasso | Controlled diffusion mean field games with common noise, and McKean-Vlasov second order backward SDEs[END_REF], mean-field control problems [START_REF] Bahlali | Existence and optimality conditions for relaxed mean-field stochastic control problems[END_REF], continuous-time reinforcement learning (Wang et al., 2020;Wang and Zhou, 2020), and optimal control of piece-wise deterministic Markov processes (Costa and Dufour, 2010a,b;[START_REF] Do Valle Costa | Continuous average control of piecewise deterministic Markov processes[END_REF][START_REF] Bäuerle | MDP algorithms for portfolio optimization problems in pure jump markets[END_REF][START_REF] Bauerle | Optimal control of partially observable piecewise deterministic Markov processes[END_REF].

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 [START_REF] Filippov | On certain questions in the theory of optimal control[END_REF], 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

E T 0 h(t, X t , u t )dt + g(X T ) (1)
subject to X = (X t ) t∈[0,T ] being a R d -valued solution to the controlled stochastic Volterra equation (CSVE) of the form

X t = x 0 (t) + t 0 K(t -s)b(s, X s , u s ) ds + t 0 K(t -s)σ(s, X s , u s ) dW s , t ∈ [0, T ] (2)
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 h 0 |K(t)| 2 dt = O(h γ ), and
T 0 |K(t + h) -K(t)| 2 dt = O(h γ ).
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.

1. The coefficients b : [0, T ]×R d ×M → R d and σ : [0, T ]×R d ×M → R d×d are continuous in u ∈ M, and in (t, x) ∈ [0, T ] × R d uniformly with respect to u.

2. There exists a measurable function η 1 : [0, T ] × M → [0, +∞] and a constant c lin > 0 such that

|b(t, x, u)| + |σ(t, x, u)| ≤ c lin |x| + η 1 (t, u), (t, x, u) ∈ [0, T ] × R d × M. (3)
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

E T 0 η 1 (t, u t ) p dt < ∞ for some p satisfying 1 p + 1 r < 1 2 . Let X be a R d -valued solution to the controlled equation (2) with initial condition x 0 ∈ C(0, T ; R d ). Then, sup t∈[0,T ] E [|X t | m ] ≤ c, (4)
for all m > 2 satisfying1 m ∈ 1 p + 1 r , 1 2 , where the constant c depends on m, p, c lin , T, C B 1 ,

|x 0 | C(0,T ;R d ) , E T 0 η 1 (t, u t ) p dt and L 2 -continuously on K| [0,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

|X t | m ≤ 3 m-1 |x 0 | m + t 0 K(t -s)b(s, X s , u s ) ds m + t 0 K(t -s)σ(s, X s , u s ) dW s m = 3 m-1 [|x 0 | m + I + II] .
Using Burkholder-Davis-Gundy inequality, and Jensen's inequality with the measure µ(ds

) = K(t -s) 2 ds/ t 0 |K(t -τ )| 2 dτ, we have E[II] ≤ C B E t 0 K(t -s) 2 σ(s, X s , u s ) 2 ds m/2 ≤ C B K m-2 L 2 t 0 |σ(s, X s , u s )| m |K(t -s)| 2 ds. By condition (3) E[II] ≤ C B 2 m-1 c m lin K m-2 L 2 t 0 E |X s | m K(t -s) 2 ds + c -m lin E t 0 η 1 (s, u s ) m K(t -s) 2 ds = k 1 t 0 E |X s | m |K(t -s)| 2 ds + k 2 . (5)
Note that κ 2 is finite since by Hölder's inequality we have

κ 2 = c -m lin E t 0 η 1 (s, u s ) m K(t -s) 2 ds ≤ c -m lin T 1-m p -2 r E T 0 η 1 (s, u s ) p ds m/p K 2 L r .
A similar argument for the first term I yields

E[I] ≤ t m/2 2 m-1 c m lin K m-2 L 2 t 0 E |X s | m K(t -s) 2 ds + c -m lin E t 0 η 1 (s, u s ) m K(t -s) 2 ds = t m/2 C -1 B k 1 t 0 E |X s | m |K(t -s)| 2 ds + k 2 . (6) For each n ∈ N set τ n = inf {t ≥ 0 : |X t | ≥ n} ∧ T.
By the Corollary of Theorem II.18 in [START_REF] Protter | Stochastic differential equations[END_REF] we have that,

|X t | m 1 {t<τn} ≤ x 0 + t 0 K(t -s)(b(s, X s 1 {s<τn} , u s ) ds) + σ(s, X s 1 {s<τn} , u s ) dW s ) m . Let f n (t) = E |X t | m 1 {t<τn} .
Then, by ( 6) and ( 5) we have

f n ≤ kk 2 + k |K| 2 * f n where k = k 1 (1 + T m/2 C -1 B ).
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:

E |X -x 0 | m C α (0,T ;R d ) ≤ c, (7)
with c depending on m, p, c lin , T, C B , |x 0 | C(0,T ;R d ) , E T 0 η 1 (t, u t ) p dt and L 2 -continuously on K| [0,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

E T 0 η 1 (t, u t ) p dt < ∞ for some p satisfying 1 p + 1 r < 1 2 . 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

E T 0 η 2 1 (t, u t )dt < ∞.
Let X be a solution to the CSVE (2) and let Z be the controlled process

Z t = t 0 b(s, X s , u s ) ds+ t 0 σ(s, X s , u s ) dW s . If X has paths in L 2 loc then t 0 X s ds = t 0 x 0 (s) ds + t 0 K(t -s)Z s ds, t ∈ [0, T ]. (8)
Conversely, if X satisfies (8) with paths in L 2 loc then it solves the CSVE (2).

Proof. Follows from Lemma 3.2 of Abi Jaber et al. (2021a).

Relaxed control formulation

The use of stochastic relaxed controls is inspired by the works of El [START_REF] El Karoui | Compactification methods in the control of degenerate diffusions: existence of an optimal control[END_REF] and [START_REF] Haussmann | On the existence of optimal controls[END_REF]. 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

π C : P(M ) q → q(C) ∈ [0, 1], C ∈ B(M ).
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

F (t, x, q) = M F (t, x, u) q(du)
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

[0, T ] × Ω (t, ω) → q t (ω, •) ∈ P(M )
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

X t = x 0 (t) + t 0 K(t -s) b(s, X s , q s )ds + t 0 K(t -s)σ(s, X s , q s ) dW s , t ∈ [0, T ], ( 9
)
where σ is defined, with a slight abuse of notation, so that the following holds:

σσ (t, x, q) = M σσ (t, x, u) q(du), t ∈ [0, T ], x ∈ R d , q ∈ P(M ).
For the existence of σ see e.g, Theorem 2.5-a in El [START_REF] El Karoui | Compactification methods in the control of degenerate diffusions: existence of an optimal control[END_REF]. The relaxed cost functional is defined as

J(X, q) = E T 0 h(t, X t , q t ) dt + g (X T ) .
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 η1 (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.

Definition 3.2. Let T > 0 and x 0 ∈ C(0, T ; R d ) be fixed. A weak admissible relaxed control for (K, b, σ) is a system π = (Ω, F, P, F, W, X, q) (10)

such that the following hold:

1. (Ω, F, P) is a complete probability space endowed with a filtration F = (F t ) t∈[0,T ] , satisfying the usual conditions, 2. W = (W t ) t∈[0,T ] is a m-dimensional Brownian motion with respect to F, 3. q = (q t ) t∈[0,T ] is a F-predictable process with values in P(M ),

4. X = (X t ) t∈[0,T ] is a F-adapted solution to the relaxed controlled equation (9).

The map

[0, T ] × Ω (t, ω) → h(t, X t (ω), q t (ω)) ∈ R belongs to L 1 ([0, T ] × Ω; R) and g(X T ) ∈ L 1 (Ω; R).
The set of weak admissible relaxed control systems with time horizon [0, T ] and initial value x 0 will be denoted by Ū(x 0 , T ). Under this weak formulation, the relaxed cost functional is defined as

J(π) = E P T 0 h(s, X π s , q π s ) ds + g (X π T ) , π ∈ Ū(x 0 , T ). ( 11
)
The relaxed control problem (RCP) consists in minimizing J over Ū(x 0 , T ). Namely, we seek π ∈ Ū(x 0 , T ) such that

J(π) = inf π∈ Ū (x 0 ,T ) J(π). ( 12
)

Main existence result

In order to complete the set of assumptions for the main existence result, we need the following definition.

Definition 3.3. A function η : M → [0, +∞] is called inf-compact if for every R ≥ 0 the level set {η ≤ R} = {u ∈ M : η(u) ≤ R} is compact.
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 :

[0, T ] × M → [0, +∞] such that for all t ∈ [0, T ] the map η(t, •) is inf-compact.
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

h : [0, T ] × R d × M → (-∞, +∞] is measurable in t ∈ [0, T ] and lower semi-continuous with respect to (x, u) ∈ R d × M .
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:

η 2 (t, u) p ≤ C 1 + C 2 h(t, x, u), (t, x, u) ∈ [0, T ] × R d × M (13)
for some p ≥ 1.

4. The final cost function g : R d → R is continuous.

The following is the main result of this paper.

Theorem 3.1 (Existence of optimal relaxed controls). Let T > 0 and x 0 ∈ C(0, T ; R d ) be fixed. Suppose that Assumptions I, II and III hold with r > 2, γ ∈ (0, 2] and p satisfying 1 p + 1 r < 1 2 and γ > 2 1 p + 1 r . Suppose further that η 1 ≤ η 2 and there exists π ∈ Ū(x 0 , T ) such that J(π) < +∞, then (RCP) admits a weak optimal relaxed control.

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

b(t, x, u) = b 0 (t, u) + b 1 (t, u)x σ(t, x, u) = σ 0 (t, u) + σ 1 (t, u)x h(t, x, u) = h 0 (t, u) + h 1 (x)
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

1 2 -H < 1 r < H - 1 p
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.

Remark 3.1. Recently, Abi Jaber et al. ( 2021b) proved an existence result for Linear-Quadratic control problems for linear Volterra equations, and obtained a linear feedback characterization of optimal controls. Unlike Abi Jaber et al. (2021b), we do not assume linearity in the coefficients with respect to the control variable. Our assumptions, however, do not cover cost functions with 'quadratic growth' in the control variable, since we are forced to choose p strictly larger than 2.

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 [START_REF] Filippov | On certain questions in the theory of optimal control[END_REF], to ensure existence of an optimal strict control.

Assumption IV.

1. M is a closed subset of a Euclidean space.

2. For each

(t, x) ∈ [0, T ] × R d , the set Γ(t, x) = σσ (t, x, u), b(t, x, u), z : u ∈ M, z ≥ h(t, x, u) (14)
is a convex and closed subset of

S d × R d × R.
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 ] .

2.

T 0 h(s, X s , q s ) ds ≥ T 0 h(s, X s , u s ) ds, P-a.s.

In particular, if the Assumptions of Theorem 3.1 also hold, there exists a weak optimal strict control for (1)-( 2).

Example 3.2.

Let M = R or M = [-M , M ], with 0 < M < ∞, d = d = 1 and K(t) = t H-1 2 with H ∈ 1 4 , 1 2 . Suppose the coefficients have the form b(t, x, u) = b 0 (t, x) + b 1 (t, x)u 2 σ(t, x, u) = σ 1 (t, x)u h(t, x, u) = h 0 (t, u 2 ) + h 1 (x),
where

• b i , σ 1 measurable and continuous in x ∈ R, uniformly respect to t ∈ [0, T ], for i = 0, 1, • b 1 , σ 1 uniformly bounded in (t, x) and |b 0 (t, x)| ≤ K |x| , with K > 0,
• h 0 (t, .) is a function convex on R + , for each t ∈ [0, T ], h 0 ∈ IC(0, T ; M 2 ) and h 1 LSC bounded from below.

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, Γ(t, x) = b(t, x) + A(t, x)Γ 1 (t, x) where

b(t, x) =   0 b 0 (t, x) 0   , A(t, x) =   σ 2 1 (t, x) 0 b 1 (t, x) 0 0 1   , (t, x) ∈ [0, T ] × R.
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.

Remark 3.2. If σ does not depend on u ∈ M , Assumption IV holds if the set (b(t, x, u), z) : u ∈ M, z ≥ h(t, x, u) is convex and closed in R d × R.
λ(M × D) = Leb(D), D ∈ B([0, T ]), (15)
i.e. the marginal of λ on B([0, T ]) is equal to the Lebesgue measure Leb. We denote by Y(0, T ; M ) the set of Young measures on M. We endow Y(0, T ; M ) with the stable topology defined as the weakest topology for which the mappings

Y(0, T ; M ) λ → D M f (u) λ(du, dt) ∈ R are continuous, for every D ∈ B([0, T ]) and f ∈ C b (M ).
The following result connects random Young measures with predictable relaxed controls. For the proof, see e.g. Section 3.3 of [START_REF] Kushner | Weak convergence methods and singularly perturbed stochastic control and filtering problems[END_REF] or Section 2.4 of [START_REF] Cecchin | Probabilistic approach to finite state mean field games[END_REF].

Lemma 4.1 (Predictable disintegration of random Young measures). Let (Ω, F, P) be a probability space and let M be a Radon space. Let λ : Ω → Y(0, T ; M ) be such that, for every J ∈ B(M × [0, T ]), the mapping

Ω ω → λ(w)(J) = λ(ω, J) ∈ [0, T ]
is measurable. Then there exists a stochastic relaxed control (q t ) t∈[0,T ] on M such that for P-a.e. ω ∈ Ω we have

λ(ω, C × D) = D q t (ω, C) dt, C ∈ B(M ), D ∈ B([0, T ]). ( 16
)
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 ).

Remark 4.2. It can be proved (see e.g. Remark 3.20 [START_REF] Crauel | Random probability measures on Polish spaces[END_REF]) that if M is a separable and metrisable topological space, then λ : Ω → Y(0, T ; M ) is measurable with respect to the Borel σ-algebra generated by the stable topology if and only if for every J ∈ B(M × [0, T ]) the mapping

Ω ω → λ(w)(J) ∈ [0, T ]
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 •, • . Lemma 4.2. Let f : [0, T ] × R d × M → E be a Borel-measurable function, continuous in u ∈ M, and continuous in x ∈ R d uniformly with respect to u ∈ M , satisfying the growth condition |f (t, x, u)| E ≤ c lin |x| δ + η(t, u) (17)
with η ∈ IC(0, T ; M ), for some δ ≥ 1. For β ≥ 1 fixed, we denote

Y β (0, T ; M ) := λ ∈ Y(0, T ; M ) : η ∈ L β (λ) .
Then, for each t ∈ [0, T ], the mapping

Σ t : C(0, T ; R d ) × Y β (0, T ; M ) → R d defined by Σ t (x, λ) = t 0 M f (s, x(s), u) λ(du, ds), (18)
is Borel-measurable.

Proof. We fix t ∈ [0, T ]. For each N ∈ N and i ∈ {1, . . . , d} define

φ i N (λ) = t 0 M min{N, f i (s, x(s), u)} λ(du, ds), λ ∈ Y β (0, T ; M ).
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,

φ i N (λ) → Σ i t (x, λ) as N → ∞ for all λ ∈ Y β (0, T ; M ). Hence, Σ t (x, •) is measurable. Now, we prove that for λ ∈ Y β (0, T ; M ) fixed, the map Σ t (•, λ) is continuous. Let x n → x in C 0, T ; R d . Then, by assumption we have f (s, x(s), u) -f (s, x n (s), u) E → 0 as n → ∞, (s, u) ∈ [0, t] × M.
Moreover, since (x n ) converges in C 0, T ; R d , it is bounded and there exists ρ > 0 such that sup s∈[0,T ]

x n (s) -x(s) < ρ, ∀n ∈ N.

Therefore by ( 17), we have

f (s, x(s), u) -f (s, x n (s), u) E ≤ c lin (1 + 2 δ-1 ) |x| δ C(0,T ;R d ) + 2 δ-1 ρδ + 2η(s, u)
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

|Σ t (x, λ) -Σ t (x n , λ)| ≤ t 0 M |f (s, x(s), u) -f (s, x n (s), u)| λ(du, ds) → 0 as n → ∞, that is, Σ t (•, λ) is continuous. Since Y β (0, T ; M
) is separable and metrisable, by Lemma 1.2.3 in Castaing et al. (2004) it follows that Σ t is jointly measurable.

Recall that

φ n φ weakly in L 1 ([0, T ] × Ω; E) if E T 0 φ n (t), ψ(t) dt → E T 0 φ(t), ψ(t) dt, ∀ψ ∈ L ∞ ([0, T ] × Ω; E).
We have the following result.

Lemma 4.3. Let f : [0, T ] × R d × M → E be a Borel-measurable function, continuous in x ∈ R d
uniformly with respect to u ∈ M , satisfying the growth condition ( 17) for some δ ≥ 1, with η ∈ IC(0, T ; M ). Let (X n ) n∈N a sequence of R d -valued processes, and λ n (du, dt) = q n t (du) dt a sequence of stochastic relaxed controls defined on the same probability space (Ω, F, P) such that X n → X point-wise P-a.s. and in L βδ (Ω × [0, T ]; R d ) for some β > 1, and λ n → λ in the stable topology P-a.s., with λ(du, dt) = q t (du, dt). Suppose further

sup n∈N E P T 0 M η(t, u) β λ n (du, dt) < ∞. ( 19
)
For each n ∈ N, set

f n t = f (t, X n t , q n t ), f n t = f (t, X t , q n t ), t ∈ [0, T ]. Then 1. f n -f n → 0, (strongly) in L 1 ([0, T ] × Ω; E). 2. f n f, weakly in L 1 ([0, T ] × Ω; E). with f t = f (t, X t , q t ), t ∈ [0, T ].
Proof. We first prove

f n -f n → 0, (strongly) in L 1 ([0, T ] × Ω; E). ( 20
)
By uniform continuity with respect to u ∈ M , for each n ∈ N we have

I n t = M f (t, X n t , u) -f (t, X t , u) E q n t (du) ≤ sup u∈M f (t, X n t , u) -f (t, X t , u) E → 0
as n → ∞ for t ∈ [0, T ], P-a.s. From ( 17) and ( 19), we get

sup n∈N E T 0 |I n t | β dt < +∞.
Hence, {I n } n∈N is uniformly integrable on Ω × [0, T ]. Lemma 4.11 [START_REF] Kallenberg | Foundations of modern probability[END_REF] implies that and (20) follows. Now, we will prove that

E T 0 f n t -f n t E dt ≤ E T 0 I n t dt → 0, as n → ∞,
f n f, weakly in L 1 ([0, T ] × Ω; E). ( 21
) Let ψ ∈ L ∞ ([0, T ] × Ω; E) be fixed. We denote g(t, u) = f (t, X t , u), ψ t . Then, E T 0 f n t , ψ t dt = E T 0 M f (t, X t , u) q n t (du), ψ t dt = E T 0 M g(t, u) λ n (du, dt)
for each n ∈ N. Let ε ∈ (0, 1) be fixed and take C ε > max{ R ε , 1} with R defined as the supremum in ( 19), and let

A ε = (t, u) ∈ [0, T ] × M : η(t, u) β-1 > C ε . Then, for this choice of C ε , we have E λ n A ε = E Aε λ n (du, dt) ≤ 1 C ε E Aε η(t, u) β-1 λ n (du, dt) < ε.
We write

E T 0 M g(t, u) λ n (du, dt) = E A c ε g(t, u) λ n (du, dt) + E Aε g(t, u) λ n (du, dt)
and observe first that by Lemma B.3 we have P-a.s.

A c ε g(t, u) λ n (du, dt) → A c ε g(t, u) λ(du, dt)
as n → ∞ and, by (17),

A c ε g(t, u) λ n (du, dt) ≤ c lin |X| δ L 1 (0,T ;R d ) + C 1/(β-1) ε |ψ| L ∞ (0,T ;E) , P -a.s.
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

E A c ε g(t, u) λ n (du, dt) → E A c ε g(t, u) λ(du, dt)
as n → ∞. Now, for each n ∈ N, define the measure µ n (du, dt, dω) = λ n (ω)(du, dt)P(dω) on B(M ) ⊗ B([0, T ]) ⊗ F, so we have

E Aε g(t, u) λ n (du, dt) ≤ Ω Aε ϕ(t) µ n (du, dt, dω) + Ω Aε η(t, u) |ψ t | E µ n (du, dt, dω) with ϕ = c lin X δ |ψ| E ∈ L β ([0, T ]×Ω), since X δ ∈ L β ([0, T ]×Ω) and ψ ∈ L ∞ ([0, T ]×Ω).
Using Hölder's inequality we get

Ω Aε ϕ(t) µ n (du, dt, dω) ≤ Ω T 0 M ϕ(t) β µ n (du, dt, dω) 1/β • E λ n A ε 1-1/β < ϕ L β ([0,T ]×Ω) ε 1-1/β
and

Ω Aε η(t, u) |ψ(t)| E µ n (du, dt, dω) ≤ ψ L ∞ ([0,T ]×Ω;E) Ω Aε η(t, u) µ n (du, dt, dω) = ψ L ∞ ([0,T ]×Ω;E) E Aε η(t, u) β η(t, u) β-1 λ n (du, dt) ≤ ψ L ∞ ([0,T ]×Ω;E) 1 C ε E Aε η(t, u) β λ n (du, dt) ≤ ψ L ∞ ([0,T ]×Ω;E) R C ε < ψ L ∞ ([0,T ]×Ω;E) ε,
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

E T 0 M η(t, u) β λ(du, dt) ≤ lim inf n→∞ E T 0 M η(t, u) β λ n (du, dt) ≤ R.
Therefore, the same estimates hold for λ, that is,

E Aε g(t, u) λ(du, dt) ≤ ϕ L β ([0,T ]×Ω) ε 1-1/β + ψ L ∞ ([0,T ]×Ω;E) ε
and since ε ∈ (0, 1) is arbitrary, we conclude that

E T 0 M g(t, u) λ n (du, dt) → E T 0 M g(t, u) λ(du, dt)
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

n→∞ J(π n ) = inf π∈ Ū (x 0 ,T ) J(π).
From this and Assumption III it follows that there exists R > 0 such that for all n ∈ N

E n T 0 M η 2 (t, u) p q n t (du) dt ≤ C 1 + C 2 E n T 0 M h(t, X n (t), u) q n t (du) dt ≤ R (22)
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, T ; R d ) satisfying sup n∈N E n |X n -x 0 | m C α (0,T ;R d ) < ∞. 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 t 0 X n s ds = t 0 x 0 (s)ds + t 0 K(t -s) (ζ n s + M n s ) ds, t ∈ [0, T ],
where

ζ n t = t 0 b(s, X n s , q n s ) ds and M n t = t 0 σ(s, X n s , q n s ) dW n s .
A similar argument as in the proof of Theorem 2.1 and Corollary 2.1 with K replaced by the identity matrix of size d ensures that {ζ n } n∈N and {M n } n∈N are also tight in C([0, T ], R d ).

For each n ∈ N we define the random Young measure

λ n (du, dt) = q n t (du) dt. (23)
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

K ε = λ ∈ Y(0, T ; M ) : T 0 M η 2 (t, u) p λ(du, dt) ≤ R ε .
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

P n λ n ∈ Y(0, T ; M ) \ Kε ≤ P n (λ n ∈ Y(0, T ; M ) \ K ε ) ≤ ε R E n T 0 M η 2 (t, u) p λ n (du, dt) ≤ ε
and the tightness of the laws of {λ n } n∈N follows. We use now Prohorov's theorem to ensure existence of a probability measure µ on C [0, T ], R d 3 × Y(0, T ; M ) and a subsequence of {X n , ζ n , M n , λ n } n∈N , which we denote using the same index n ∈ N, such that

law (X n , ζ n , M n , λ n ) → µ, n → ∞. ( 24
)
Step 2. Dudley's generalization of the Skorohod representation theorem (see Theorem 4.30 in [START_REF] Kallenberg | Foundations of modern probability[END_REF]) ensures existence of a probability space ( Ω, F, P) and a sequence of random variables { Xn , ζn , M n , λn } n∈N with values in C([0, T ]; R d ) 3 × Y(0, T ; M ), defined on ( Ω, F, P), such that

( Xn , ζn , M n , λn ) d = (X n , ζ n , M n , λ n ), n ∈ N, (25)
and, on the same stochastic basis ( Ω, F, P), a random variable X, ζ, M , λ with values in

C([0, T ]; R d ) 3 × Y(0, T ; M ) such that ( Xn , ζn , Mn ) → ( X, ζ, M ), in C([0, T ]; R d ) 3 , P -a.s. ( 26
)
and λn → λ, stably in Y(0, T ; M ), P -a.s. ( 27)

Step 3. By Theorem 6.1 in [START_REF] Gripenberg | Volterra integral and functional equations[END_REF], the map Z → t 0 K(t -s)Z s ds is continuous from C(0, T ; R d ) to itself. In particular, it is measurable, so we also have

For each t ∈ [0, T ] let ϕ t denote the evaluation map C([0, T ]; R d ) ζ → ζ(t) ∈ R d , and let Γ t : C([0, T ]; R d ) 2 × Y p (0, T ; M ) → R d be defined as Γ t (x, ζ, λ) = Σ t (x, λ) -ϕ t (ζ), (x, ζ) ∈ C([0, T ]; R d ) 2 , λ ∈ Y p (
t 0 Xn s ds = t 0 x 0 (s) ds + t 0 K(t -s) ζn s + M n s ds. ( 28
)
Since M is a Suslin space, it also separable and Radon, see e.g. Ch. II in [START_REF] Schwartz | Radon measures on arbitrary topological spaces and cylindrical measures[END_REF].

In particular, Lemma 4.1 applies, so there exists a relaxed control process (q n t ) t∈[0,T ] defined on ( Ω, F, P) such that λn (du, dt) = qn t (du) dt, P -a.s. Now, M n is a F n -martingale with quadratic variation

M n t = t 0 (σσ )(s, X n s , q s ) ds, t ∈ [0, T ]
and (X n , q n ) d = ( Xn , qn ). Then, using once again Lemma 4.2, now with f = σσ and η = η 2 1 , it follows that M n is also a martingale with respect to the filtration

Fn t = σ ( Xn s , qn s ) : s ∈ [0, t] , t ∈ [0, T ]
and quadratic variation M n t = t 0 (σσ )(s, Xn s , qn s ) ds. Again, using continuity of the map Z → t 0 K(t -s)Z s ds from C(0, T ; R d ) to itself, we obtain 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 λ(du, dt) = qt (du) dt, P -a.s.

(29)

The filtration F = Ft t∈[0,T ] is defined by

Ft = σ{( Xs , qs ) : s ∈ [0, t]}, t ∈ [0, T ].
We now claim that M is a F-martingale. Indeed, From ( 26

Xn t -Xt 2 → 0, as n → ∞. (32)
Similarly, we have Ẽ sup

t∈[0,T ] M n t -Mt 2 → 0, as n → ∞. (33)
This, in conjunction with the martingale property of M n , implies that for all 0 < s < t ≤ T and for all

φ ∈ C b C(0, s; R d ) × Y(0, s; M ) we have that as n → ∞ 0 = Ẽ M n t -M n s φ( Xn , λn ) → Ẽ Mt -Ms φ( X, λ) ,
which implies that M is a F-martingale.

Step 4. We now pass to the limit to identify the process ( Xt ) t∈[0,T ] as a solution of the equation controlled by (q t ) t∈

By ( 32) and ( 33), for any ε > 0 there exists an integer m = m(ε) ≥ 1 for which

Ẽ sup t∈[0,T ] Xn t -Xt + M n t -Mt < ε, ∀n ≥ m. (36)
From ( 34 Therefore, there exist an integer N ≥ 1 and {α 1 , . . . , α N } with α i ≥ 0,

N i=1 α i = 1, such that N i=1 α i b m+i -b L 1 ([0,T ]× Ω;R d ) < ε. (37)
Let t ∈ [0, T ] be fixed. Using the α i 's and (28) we can write

t 0 x 0 (s)ds = N i=1 α i t 0 X m+i s ds - t 0 K(t -s) s 0 b m+i v dv + M m+i s ds .
Thus, we have

I = t 0 x 0 (s)ds + t 0 K(t -s) s 0 bv dv + Ms ds - t 0 Xs ds = N i=1 α i t 0 X m+i s ds - t 0 K(t -s) s 0 b m+i v dv + M m+i s ds + t 0 K(t -s) s 0 bv dv + Ms ds - t 0 Xs ds ≤ N i=1 α i t 0 X m+i s ds - t 0 Xs ds + N i=1 α i t 0 K(t -s) M m+i s ds - t 0 K(t -s) M ds + N i=1 α i t 0 K(t -s) s 0 b m+i v dv ds - t 0 K(t -s) s 0 bv dv ds = II + III + IV.
Then, by (36), we have

Ẽ(II) = Ẽ   N i=1 α i t 0 X m+i s ds - t 0 Xs ds   ≤ N i=1 α i Ẽ t 0 ( X m+i s -Xs )ds ≤ N i=1 α i Ẽ t 0 sup s∈[0,T ] X m+i s -Xs ds ≤ εT.
By Fubini's theorem and (36), it follows

Ẽ(III) = Ẽ   N i=1 α i t 0 K(t -s) M m+i s ds - t 0 K(t -s) Ms ds   ≤ N i=1 α i Ẽ t 0 K(t -s) M m+i s -Ms ds ≤ N i=1 α i Ẽ t 0 K(t -s) M m+i s -Ms ds ≤ ε K L 1 (0,T ) .
Using twice Jensen's inequality and (37),

Ẽ(IV ) = Ẽ   N i=1 α i t 0 K(t -s) s 0 b m+i v dv ds - t 0 K(t -s) s 0 bv dv ds   = Ẽ   t 0 K(t -s) s 0 N i=1 α i b m+i v -bv dv ds   ≤ Ẽ   t t 0 |K(t -s)| s 0 N i=1 α i b m+i v -bv dv ds   ≤ εT K L 1 (0,T ) .
Then, Ẽ(I) ≤ [T + K L 1 (0,T ) (T + 1)]ε. Since ε > 0 is arbitrary, (35) follows.

Step 5. Set σt = σ(t, Xt , qt ) and σn

t = σ(t, Xn t , qn t ) , t ∈ [0, T ]. Using Lemma 4.3 with E = S d , f = σσ , δ = 2, η = η 2 1 and β = p/2 > 1, we obtain σn σn, σσ , weakly in L 1 ([0, T ] × Ω; S d ).
Let t ∈ [0, T ] be fixed. By (33) and the Burkholder-Davis-Gundy inequality we have M n t → M t in L 2 ( Ω). Then, for any ε > 0 there exists an integer m = m

(ε) ≥ 1 such that Ẽ M n t -M t | < ε, ∀n ≥ m.
As in the proof of Step 5, there also exists an integer N ≥ 1 and {α 1 , . . . , α N } with α i ≥ 0, for each t ∈ [0, T ]. By Lemma 2.1 this is equivalent to X being solution to stochastic Volterra equation controlled by q. In other words, π = Ω, F, P, F, W , X, q is a weak admissible relaxed control. By the Fiber Product Lemma B.2 we have δ Xn ⊗ λ n → δ X ⊗ λ, stably in Y(0, T ; R × M ), P -a.s.

Since R × M is also a metrisable Suslin space, using Lemma B.1 and Fatou's Lemma we get Ẽ J(π), that is, π is a weak optimal relaxed control for (RCP), and this concludes the proof of Theorem 3.1.

Proof of Theorem 3.2

Let π = (Ω, F, P, F, X, W, q) ∈ Ū(x 0 , T ). Define κ : [0, T ] × Ω → R d×d × R d × R as κ(t, ω) := (σσ , b, h)(t, X t (ω), q t (ω)).

The notion of tightness for Young measures that we use was introduced by [START_REF] Valadier | Young measures, methods of nonconvex analysis[END_REF]. See also the book by [START_REF] Crauel | Random probability measures on Polish spaces[END_REF]. Recall that a set-valued function [0, T ] t → K t ⊂ M is said to be measurable if and only if for every open set U ⊂ M , {t ∈ [0, T ] : K t ∩ U = ∅} ∈ B([0, T ]).

Definition B.1. We say that a set J ⊂ Y(0, T ; M ) is flexibly tight if, for each ε > 0, there exists a measurable set-valued mapping [0, T ] t → K t ⊂ M such that K t is compact for all t ∈ [0, T ] and Proof. See e.g. (Balder, 1998, Definition 3.3).

Theorem B.2 (Prohorov criterion for relative compactness). Let M be a metrisable Suslin space. Then every flexibly tight subset of Y(0, T ; M ) is sequentially relatively compact in the stable topology.

Proof. See (Castaing et al., 2004, Theorem 4.3.5).

Lemma B.1. Let M be a metrisable Suslin space and g ∈ L 1 (0, T ; R). Let us assume that

h : [0, T ] × M → [-∞, +∞]
is a measurable function such that h(t, •) is lower semi-continuous for every t ∈ [0, T ] and satisfies one of the two following conditions:

1. |h(t, u)| ≤ g(t), a.e. t ∈ [0, T ],

2. h ≥ 0.

If λ n → λ stably in Y(0, T ; M ), then

  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 , T, |x 0 | C(0,T ;R d ) and L m -continuously, on K| [0,T ] . 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

  0, T ; M ) with Σ t as in (18) with f = b. Using Lemma 4.2 with η = η 1 , it follows that Γ t is measurable. Hence by (25) and the definition of ζ n , for each t ∈ [0, T ] and n ∈ N we have ζn Xn s , u) λn (du, ds)

  -s) ζs + Ms ds.

  [0,T ] . Using Lemma 4.3 with E = R d , f = b, β = p > 1, δ = 1, and η = η 1 we obtain bn b, weakly in L 1 ([0, T ] × Ω; R d ) (34)with bt = b(t, Xt , qt ), t ∈ [0, T ]. We claim that the process M satisfies dτ + Ms ds.

  ) we have b ∈ { b m, b m+1 , . . .} w ⊂ co{ b m, b m+1 , . . .} w where co(•) and • w denote the convex hull and weak-closure in L 1 ([0, T ] × Ω; R d ) respectively. By Mazur's, see for example Theorem 2.5.16 in Megginson (2012) co{ b m, b m+1 , . . .} w = co{ b m, b m+1 , . . .}.

  ds < (1 + T )ε.Since ε > 0 and t ∈ [0, T ] are arbitrary, it follows M t = t 0 σs σ s ds P-a.s. for all t ∈ [0, T ]. By the martingale representation theorem (see e.g. Theorem 4.2 in Chapter 3.4[START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]) there exist an extension of the probability space ( Ω, F, P), which we also denote ( Ω, F, P), and a d -dimensional Brownian motion ( Wt ) t≥0 defined on ( Ω, F, Xs , qs ) d Ws , P -a.s., t ∈ [0, Ms ds, P -a.s.

  Xt , u) λ(du, dt) + Ẽg( XT ) n t , u) λ n (du, dt) + lim inf n→∞ E n g(X n T ) n t , u) λ n (du, dt) + E n g(X n T ) = inf π∈U (x 0 )

  λ(du, dt) < ε. Theorem B.1 (Equivalence theorem for flexible tightness). For any J ⊂ Y(0, T ; M ) the two following conditions are equivalent: 1. J is flexibly tight 2. There exists η ∈ IC([0, T ], M ) such that sup u) λ(du, dt) < +∞.

  u) λ n (du, dt).

CB is the constant in the Burkholder-Davis-Gundy inequality, see e.g. Section 4, Chapter IV in[START_REF] Protter | Stochastic differential equations[END_REF].

* Andrés Cárdenas and Rafael Serrano thank Alianza EFI-Colombia Cientifica grant, codes 60185 and FP44842-220-2018 for financial support. The research of the Sergio Pulido benefited from the financial support of the chairs "Deep finance & Statistics" and "Machine Learning & systematic methods in finance" of École Polytechnique. Sergio Pulido acknowledges support by the Europlace Institute of Finance (EIF) and the Labex Louis Bachelier, research project: "The impact of information on financial markets".

By Assumption IV, we have that κ(t, ω) ∈ Γ(t, X t (ω)), as defined in ( 14), for all (t, ω) ∈ [0, T ] × Ω. We also define c 1 (t, ω) := (σσ , b)(t, X t (ω), q t (ω)), c 2 (t, ω) := h(t, X t (ω), q t (ω)).

By Lemma 4.1, c 1 and c 2 are measurable with respect to the predictable σ-algebra G on Y = [0, T ] × Ω. Using Theorem A.1 we conclude the existence of a function u :

and the desired result follows.

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

Theorem A.1. If (c 1 (y), c 2 (y)) ∈ Γ(y, M ) for all y ∈ Y , then there exists a measurable function u : Y → M such that c 1 (y) = φ(y, u(y)) and c 2 i (y) ≥ ψ i (y, u(y)), i = 1, . . . , m.

Proof. See Theorem A.9 of [START_REF] Haussmann | On the existence of optimal controls[END_REF].

B Relative compactness and limit theorems for Young measures

Young measures on metrisable Suslin control sets have been studied by [START_REF] Balder | On ws-convergence of product measures[END_REF] and de Fitte (2003). We refer to the book by Castaing et al. (2004) for more details.

Proposition B.1. Let M be metrisable (resp. metrisable Suslin). Then the space Y(0, T ; M ) endowed with the stable topology is also metrisable (resp. metrisable Suslin).

Proof. For the metrisability part, see Proposition 2.3.1 in Castaing et al. (2004). For the Suslin part, see Proposition 2.3.3 in Castaing et al. (2004).

Proof. If (1) holds, the result follows from Theorem 2.1.3-Part G in Castaing et al. (2004). If (2) holds, the result follows from Proposition 2.1.12-Part (d) in Castaing et al. (2004).

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 .

We will need the following version of the so-called Fiber Product Lemma. For a measurable map y : [0, T ] → M , we denote by δ y(•) (•) the degenerate Young measure defined as δ y(•) (du, dt) = δ y(t) (du) dt.

Lemma B.2 (Fiber Product Lemma). Let S and M be separable metric spaces and let y n : [0, T ] → S be a sequence of measurable mappings which converge pointwise to a mapping y : [0, T ] → S. Let λ n → λ stably in Y(0, T ; M ) and consider the following sequence of Young measures on S × M :

and

(δ y ⊗ λ)(dx, du, dt) = δ y(t) (dx) λ(du, dt).

Then δ yn ⊗ λ n → δ y ⊗ λ stably in Y(0, T ; S × M ).

Proof. Proposition 1 in [START_REF] Valadier | A course on Young measures[END_REF] 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 Castaing and de Fitte (2004).