HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces

In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which are continuously differentiable in the space variable. We also provide an application of our results to an exit-time optimal control problem, and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.


Introduction
A typical and nontrivial feature of optimal control problems in real applications (both in the deterministic and in the stochastic case) is the fact that the state variable must satisfy suitable constraints, i.e., to belong to a given subset D of its state space H.In some settings, as in the case of state constrained problems (see, e.g, Bardi and Capuzzo-Dolcetta [1, Chapter IV]), one restricts the set of admissible control strategies and considers only those that keep the state inside the given subset D. In some other settings, as in exit time optimal control problems (see, e.g., Cannarsa and Sinestrari [6,Chapter 8]), one does not operate such restriction but terminates the control actions (and, thus, stops the optimization problem) once the state goes out of D.
In both cases, standard arguments involving the dynamic programming principle provide a Hamilton-Jacobi-Bellman (HJB) equation associated to the optimal control problem, which is a Partial Differential Equation (PDE) in the domain D, satisfying suitable boundary conditions.Such type of HJB equations have been studied in many papers in the finite dimensional case, finding theorems on existence and uniqueness of viscosity solutions (see, e.g., Soner [21,22]) and, in some cases, results on their regularity (see, e.g., Crandall et al. [7,Section 9]).In the infinite dimensional case, the theory of existence and uniqueness of viscosity solutions of HJB equations in domains has been studied as well, both in the deterministic (see, e.g., Li and Yong [15,Chapter 4]) and in the stochastic case (see, e.g., Fabbri et al. [13,Chapter 3]).
However, in general, solutions to HJB equations in the viscosity sense are not regular enough to find the optimal strategies of a control problem.Said otherwise, if one wants to find optimal strategies it is fundamental to establish results on the regularity of solutions to such HJB equations, showing that they are, at least, continuously differentiable in the state variable.To the best of our knowledge, these results seem to be completely missing in the literature in a stochastic framework, except for some cases where explicit solutions can be found (see e.g.Fabbri et al. [13,Section 4.10], Da Prato and Zabczyk [8], Biffis et al. [2], Gozzi and Leocata [14]).
The aim of this paper is twofold: we want to establish, first, a result on the existence and the uniqueness of regular solutions (i.e., continuously differentiable in the state variable) of second-order infinite dimensional semilinear HJB equations in domains; then, we want to prove a verification theorem for the associated exit time stochastic optimal control problem, using the aforementioned existence and uniqueness result.To do so, we extend to the case of domains the theory of mild solutions for second-order semilinear HJB equations in Hilbert spaces (for a summary on this theory see, e.g., Fabbri et al. [13,Chapter 4]).As a starting point for future research, in this paper we consider the domain D to be a half-space.
1.1.Methodology and main results.The starting point of our paper are global gradient estimates up to the boundary for solutions to second-order linear PDEs in special half-spaces of Hilbert spaces, see Priola [18] and Chapter 5 in Priola [16].We also mention Priola [20] for related finite-dimensional Dirichlet problems.We point out that in general domains of Hilbert spaces local regularity results are obtained in [11] and [23] (see also Da Prato and Zabczyk [9,Chapter 8]).On the other hand, global gradient estimates for Ornstein-Uhlenbeck Dirichlet semigroups can fail even in a half space of R 2 (see an example in [24]).Using suitable extension operators from the half-space to the whole Hilbert space, we define a family of operators on the set of bounded measurable functions on the half-space.We show that this family is a semigroup of contractions, which coincides with the semigroup defined in Da Prato and Zabczyk [9,Chapter 8], providing the so-called generalized solution to these second-order linear PDEs.We also prove some regularizing properties of this semigroup.
Next, we turn our attention to establishing existence and uniqueness of mild solutions (i.e., solutions in integral form) of semilinear HJB equations in the half-space.This result, given in Theorem 3.3, extends analogous ones provided in Fabbri et al. [13,Chapter 4], which were proved in the whole Hilbert space case.
As stated previously, we need more regular solutions to be able to find optimal strategies.Thus, in Theorem 4.4 we show that mild solutions are indeed strong solutions, in the sense they can be approximated by classical solutions.More precisely, we rely on the concept of K-convergence, see Definition 4.1.
These results can be profitably applied to study a family of stochastic optimal control problems with exit time and they permit us to prove a verification theorem (see Theorem 5.9).This is a sufficient condition of optimality which allows to write the optimal control in feedback form, when a solution to the so-called closed-loop equation (5.12) can be found.
1.2.Plan of the paper.The plan of the paper is the following.
• Section 2 contains some important preliminary material on second-order linear PDEs, which are needed to prove our main results.It is divided in four subsections: -Subsection 2.1, where we recall some basic notation used in this paper; -Subsection 2.2, where we present some results on second-order linear PDEs in half-spaces; -Subsection 2.3, where we define some function spaces needed in the main results; -Subsection 2.4, where we prove a regularization result for the semigroup associated to linear PDEs in half-spaces.• In Section 3 we establish our first main result, namely, Theorem 3.3 on the existence and uniqueness of mild solution for HJB equation (3.1).• In Section 4 we prove our second main result, i.e., Theorem 4.4, which shows that the mild solution of HJB equation (3.1) is also a K-strong solution, that is, it can be approximated by classical solutions.• In Section 5 we establish our third main result, namely, the Verification Theorem 5.9, providing a sufficient condition of optimality for the optimal control problem (5.4).In what follows, we will just write measurable instead of Borel measurable.We denote by

Preliminaries
denotes the set of all real-valued functions on X, that are uniformly continuous and bounded on X together with their first-order Fréchet derivatives.
Throughout the paper, H is a real separable Hilbert space, with inner product •, • and associated norm |•|.The symbol L(H) denotes the Banach space of all linear and bounded operators from H into itself, endowed with the supremum norm • L(H) , and L + (H) indicates the subset of L(H) consisting of all positive and self-adjoint operators.L 1 (H) indicates the set of all trace class (or nuclear ) operators from H into itself and where {e k } k∈N is a complete orthonormal system for H.The symbol N (x, B) denotes the Gaussian measure on H with mean x ∈ H and covariance operator B ∈ L + 1 (H).The Gaussian measure on R with mean µ ∈ R and variance σ 2 ≥ 0, is indicated by N 1 (µ, σ 2 ).
If f : [0, +∞) × H → R is a differentiable function, f t denotes the derivative of f (t, x) with respect to t and Df , D 2 f denote the first-and second-order Fréchet derivatives of f (t, x) with respect to x, respectively.
From this point onward, let {ȳ, e k } k≥2 be a fixed orthonormal basis of H.We consider the open half-space of H generated by ȳ, namely, and we also define the sets We will always identify any element x ∈ H with the sequence of Fourier coefficients (x k ) k∈N with respect to {ȳ, e k } k≥2 , where x 1 := x, ȳ and x k := x, e k , for k ≥ 2. It is important to recall that this identification defines an isometry between H and ℓ 2 , the Hilbert space of real-valued, square-summable sequences.
We will also consider the sub-space H ′ of H generated by the system {e k } k≥2 .As above, we identify any element x ′ ∈ H ′ with the sequence of Fourier coefficients (x ′ k ) k≥2 , which will be still denoted by x ′ .To be precise, we should identify x ′ with the sequence (0, x ′ 2 , x ′ 3 , . ..) but, for the sake of brevity, we will omit the leading zero.This notation allows to identify any element x ∈ H with the pair (x 1 , x ′ ) ∈ R × H ′ , and hence we will write (x 1 , x ′ ) in place of x whenever necessary.Notice, also, that we have an isometry between ∂H + and H ′ .
Finally, we introduce the function spaces B 0 (H + ), C 0 (H + ), UC 0 (H + ) indicate, respectively, the sets of bounded measurable, bounded continuous, bounded uniformly continuous functions of H + that vanish on ∂H + .It is easy to prove that all these three spaces, endowed with the supremum norm, are Banach spaces.Other spaces, such as B b (H + ), can be defined similarly.We also recall that UC b (H + ) = UC b (H + ).

2.2.
The linear problem on the half space.In this section we will study the following equation on the closed half-space where Q ∈ L + 1 (H) is a positive, self-adjoint, trace class operator in H, A is a linear operator on H (whose adjoint is denoted by A * ) and φ ∈ B b (H + ) is a given function.We will work under the following hypothesis, that will stand from now on.We will need also the following hypothesis.
(2.6) where τ x := inf{t > 0 : X(t; x) ∈ H − }.The purpose of this section is to show that it is possible to define a semigroup {P t } t≥0 that allows to provide the mild solution to (2.1).We will also show that {P t } t≥0 coincides with {M t } t≥0 (called restricted semigroup in [9,Chapter 8]) on a suitable space.Moreover, we will prove some regularizing properties.The idea (cf.[17,18,19,20]) is to define suitable extension operators, so that it is possible to exploit the semigroup {T t } t≥0 , given in (2.5).
For any η ∈ B b (H + ) and x ∈ H (identified with By abuse of notation, we can adopt the same symbol to denote the following extension operator, defined for any η ∈ B b (H + ) and x ∈ H, Clearly, Eη ∈ B b (H), in both cases.
We need to define a suitable counterpart in H ′ of the semigroup {e tA } t≥0 .This can be easily done thanks to the following lemma.Lemma 2.4.For each t ≥ 0, the operator e tA leaves ∂H + invariant.
Using the isometry between ∂H + and H ′ , here denoted by γ : ∂H + → H ′ , we can define the family of operators S(t) : H ′ → H ′ , given by It is immediate to prove that each of the operators S(t), t ≥ 0, is linear and bounded.We also have the following result, whose proof is omitted because it is a standard consequence of the invariance property proved in Lemma 2.4.
Lemma 2.5.The family of operators { S(t)} t≥0 is a semigroup on H ′ .Thanks to Hypothesis 2.1-(iii), also the operator Q leaves ∂H + invariant.Therefore, we can define the operator Q := H ′ → H ′ as:

Finally, we define
Thanks to the isometry γ and recalling that Q ∈ L + 1 (H), it is easy to show that also Q ∈ L + 1 (H).Moreover, considering in addition Hypothesis 2.1-(vi), we have that Qt ∈ L + 1 (H), for all t > 0.
Remark 2.6.By Hypothesis 2.1-(ii)-(iii), we get We deduce that, for any x ∈ H, x = (x 1 , x ′ ), the Gaussian measure N (e tA x, Q t ) can be split as follows where the Gaussian measure on H ′ is still denoted by N .
The following lemma provides some useful formulas (cf.[20] for the finite-dimensional case).
Lemma 2.7.Let us define, for all t ≥ 0, θ ∈ R, and Then, for any η ∈ B b (H + ) and any x ∈ H, (2.10) We introduce, next, a family of operators on B b (H + ).
Definition 2.8.For any η ∈ B b (H + ) and any x ∈ H + , we define the family of operators {P t } t≥0 as where Rf is the restriction of f ∈ B b (H) to H + .
Remark 2.9.Using the extension defined in (2.8), we immediately see that the family {P t } t≥0 is uniquely defined on functions η ∈ B b (H + ).To see this, it is enough to recall that Note that for this reason, while in (2.8) we choose to extend η to be null on ∂H + , one can opt for an arbitrarily different measurable extension on the boundary of H + and still obtain a well-defined family {P t } t≥0 on B b (H + ).It is also worth remembering that, in any case, P t η is a function defined on H + , for all t ≥ 0.
Proposition 2.10.The family of operators {P t } t≥0 is a semigroup of contractions on , for all t > 0.
(ii) For every f ∈ C 0 (H + ) and x ∈ H + , the map t → P t f (x), defined on [0, +∞) with real values, is continuous.
Proof.We prove, first, the semigroup property for {P t } t≥0 .Applying the definition of the restriction R, the fact that G, defined in (2.9), is an odd function in the second argument, and using (2.10), we get that, for all η ∈ B b (H + ), From this equality, we obtain that, for all t, s ≥ 0, i.e., the semigroup property.Now we can prove the others statements.
We are now ready to show the following important result on semigroups {P t } t≥0 and {M t } t≥0 .Proposition 2.11.On B b (H + ) the identity P t = M t holds true for any t ≥ 0.
Proof.First, we set a useful notation.For any x = (x 1 , x ′ ) ∈ H + , where x 1 ∈ R + and x ′ ∈ H ′ , we can write the solution X(t, x) to SDE (2.4) as where X 1 (t, x 1 ) is a stochastic process with values in R + while X ′ (t, x ′ ) is a stochastic process with values in H ′ .Therefore Let f ∈ B b (H + ).Then, using the tower property, we get Noting that X 1 (t, x 1 ) and X ′ (t, x ′ ) are independent (see [10, Proposition 2.12]), we have 2.3.Some useful function spaces.In this section we introduce some further function spaces, that will be needed in the sequel.Let T > 0 and set These are Banach spaces when endowed with the supremum norm and, clearly, the latter is a subspace of the former.
Next, for δ ∈ (0, 1) as in Hypothesis 2.3, X denoting H, H + , or H + , and Y indicating either H or R, define The latter is a subspace of the former space.If Y = R we will simply denote them by B b,δ ((0, T ] × X ) and C b,δ ((0, T ] × X ).We endow them with the following norm, making them Banach spaces: Also in this case, the latter is a subspace of the former space.We endow them with the norm: 2.4.Regularization property of P t .In this section we show the regularization property of the semigroup {P t } t≥0 , introduced in Definition 2.8, and we provide some joint time-space regularity properties of this semigroup that will be particularly useful in the next sections.We recall that, under Hypotheses 2.1 and 2.3, the semigroup {T t } t≥0 satisfies for some positive constant C = C T independent of f (see, e.g., [9,Chapter 6]).
Proof.We recall, first, the definition of semigroup {P t } t≥0 .For any f ∈ B b (H + ), where is the extension operator defined in (2.7), R is the restriction to the half-plane H + and T t is the semigroup defined in (2.5).Recall, also, that P t f ∈ UC 0 (H + ), for all t > 0, by Proposition 2.10-(i).

Mild solutions of HJB equations
The purpose of this section is to establish the existence and the uniqueness of the mild solution (see Definition 3.2 below) to the following HJB equation where T > 0 is a given time horizon (which will be fixed from now on), F and φ are given measurable functions.We introduce the following assumption.
Hypothesis 3.1.The measurable functions (ii) For any t ∈ [0, T ], x ∈ H + , y ∈ R, and z ∈ H, we have Definition 3.2.A function u : [0, T ]×H + → R is a mild solution to the HJB equation (3.1) if (i) There exists η ∈ (0, 1) such that u ∈ B 0,1 b,η ([0, T ] × H + ); (ii) for all t ∈ [0, T ] and all x ∈ H + , the following equality holds The following theorem establishes existence and uniqueness of the mild solution to (3.1), in the sense of Definition 3.2.Theorem 3.3.Let δ ∈ (0, 1) be such that Hypotheses 2.1, 2.3 and 3.1 are satisfied.Then, Proof.We use the contraction mapping principle on a suitable space to establish the claim.We consider the space B := B 0 ([0, T ] × H + ) × B b,δ ((0, T ] × H + ; H) endowed with the product norm given by the sum of the norms of the factor spaces.To ease notations, we will denote the norm of B 0 ([0, T ] × H + ) (which is the sup-norm) by • B 0 .
Let us define the operator Υ = (Υ 1 , Υ 2 ) as To begin with, we need to ensure that the Υ is well defined as a map from B into itself.Let (u, v) ∈ B and consider, first, the function Υ 1 [u, v].This is the sum of two functions belonging to B 0 ([0, T ] × H + ).Indeed, on the one hand, by Proposition 2.13-(i) and Remark 2.15, P t φ(x) ∈ B 0 ([0, T ] × H + ).On the other hand, we readily see that the function ψ(s, x) := F (s, x, u(s, x), v(s, x)), (s, x) ∈ (0, T ] × H + , (3.3) is Borel measurable on (0, T ] × H + , that (s, x) → s δ ψ(s, x) is bounded on (0, T ] × H + , and that, for all (s, x) This implies that ψ ∈ B b,δ ((0, T ] × H + ), and hence, by Proposition 2.13-(ii), the map Next, considering the function Υ 2 [u, v] we see that the first term belongs to B b,δ ((0, T ] × H + ; H), thanks to Proposition 2.12, and that the second term is measurable, as a consequence of Proposition 2.13-(iv) (use the same ψ above to apply this result).We are left to prove that this latter term belongs to B b,δ ((0, T ] × H + ; H), a fact that is justified by the following estimate, holding for all (t, x) ∈ (0, T ]×H + , where we use once more Proposition 2.12 and (3.4): where Γ is the gamma function.
We proceed, next, to show that Υ is a contraction on B. To this end, it is convenient to use an equivalent norm on B, given by the sum of the equivalent norms • β,B b and • β,B b,δ on B 0 ([0, T ] × H + ) and on B b,δ ((0, T ] × H + ; H), respectively, defined by: where β > 0 is a constant to be fixed later in the proof.We want, now, to find a suitable β > 0 such that the map We start with an estimate on Υ 1 .Taking any (u 1 , v 1 ), (u 2 , v 2 ) ∈ B and using that {P t } t≥0 is a semigroup of contractions (cf.Proposition 2.10) and Hypothesis 3.1-(i), we have that, for all (t, x) Now, multiplying and dividing by e βs in the integrals appearing in the last line, we get where An immediate application of [13, Proposition 4.21-(iv)] entails that C 1 (β) → 0, as β → +∞.We provide, next, an estimate on Υ 2 , Considering any (u 1 , v 1 ), (u 2 , v 2 ) ∈ B and using Proposition 2.12 and Hypothesis 3.1-(i), we have that, for all (t, x) Now, multiplying and dividing by e βs in the integrals appearing in the last line, we get Applying [13, Proposition 4.21-(iv) and (v)], we get that C 2 (β) → 0, as β → +∞.By the reasoning above, there exists β 0 > 0 such that for β > β 0 we have which entails that Υ is a contraction and, therefore, that it has a unique fixed point.
Furthermore, uniqueness easily follows noticing that any other solution u * ∈ B 0,1 b,δ ([0, T ]× H + ) must be equal to the first component of the fixed point of Υ in B, i.e., it must hold that u * = ū.
The continuity of the solution and of its derivative follows exactly with the same argument of [13, Theorem 4.149-(ii)], exploiting the definition of semigroup {P t } t≥0 and that semigroup {T t } t≥0 has a regularizing effect for t > 0.
To deduce the last assertion we proceed as follows.Let (t, x), (t 0 , x 0 ) ∈ [0, T ] × H + and assume, without loss of generality, that t 0 < t.Then, recalling (3.3), we have that The result follows from the continuity of φ and of semigroup {P t } t≥0 with respect to t, from Proposition 2.10, and from an application of the dominated convergence theorem.

Strong solutions of HJB equations
In applications to optimal control it is useful to know that mild solutions to an HJB equation can be approximated by regular solutions, where by regular we mean smooth enough to apply the Itô or the Dynkin formulas.Solutions constructed by this approximating procedure are called strong solution (see Definition 4.3 below for a precise statement).
The idea is to approximate mild solutions to (3.1) with classical solutions in UC 2 b (H).However, it is well-known that UC 2 b (H) is not dense in UC b (H), when dim(H) = +∞ (since unit balls are not compact in this case).As a consequence, we cannot hope for uniform convergence and we need to resort to a different concept of convergence.
We follow the approach of [13, Section 4.5] and introduce K-convergence (cf. Finally, for δ ∈ (0, 1), we say that a sequence To give the definition of classical solution, we need to introduce the space In this space we introduce the norm if: We are now ready to introduce the concept of strong solution mentioned at the beginning of this section.Recall that the aim is to show that mild solutions to (3.1) can be approximated, in the sense of K-convergence, by classical solutions.Solutions to (3.1) constructed by this approximating procedure are called K-strong solutions.(i) There exists η ∈ (0, 1) such that u ∈ B 0,1 b,η ([0, T ] × H + ); (ii) There exist three sequences Then, the function u is a K-strong solution to (3.1), which is unique among all solutions in B 0,1 b,δ ([0, T ] × H + ).
Proof.We note, first, that u is continuous both in [0, T ] × H + and in (0, T ] × H + , thanks to Theorem 3.3, and that f is bounded, thanks to Hypothesis 3.1-(ii).
To prove the theorem, we need to provide the sequence {u n } n∈N of classical solutions to (4.2) that approximate the mild solution u to (3.1).To do so, we construct approximating sequences {f n } n∈N and {φ n } n∈N for f and φ, respectively.
Let us fix an orthonormal basis E ⊂ D(A * ) of H (this can always be done, as D(A * ) is dense).Since ȳ ∈ D(A * ), we can choose E so that ȳ ∈ E. We denote by w j , j ≥ 2, the other elements of E. Recall that we can identify any element x ∈ H with the sequence of Fourier coefficients (x k ) k∈N with respect to the orthonormal basis E. Let us denote, for each n ∈ N, the orthogonal projection P n onto Span{ȳ, w 2 , . . ., w n } (clearly, P 1 denotes the projection onto Span{ȳ}) and let us define the maps Π n : H → R n and Q n : R n → H as x j w j .
Note that with support in the ball centered at the origin with radius We approximate φ, first.Since the behaviour of φ on ∂H + is not known we do not directly define regularising convolutions for φ, but rather for some approximations of this function.First, we extend φ to be equal to 0 for all x ∈ ∂H + .Then we define, for each h ∈ N, the functions where, for any ε > 0, We define, for all x ∈ H and for each h ∈ N, the following regularizing convolutions of the functions E φ h , where E is the extension operator introduced in (2.7), We immediately obtain that, for all k, h, n ∈ N, Moreover, we can easily show that ψ n k,h vanishes on ∂H + .Indeed, for all x ∈ ∂H + and recalling that we chose the mollifiers to be symmetric, we have that Let us define, for all h, n ∈ N, the functions ξ h,n (x) := E φ h (P n x).Clearly, ψ n k,h and ξ h,n can be also seen as functions depending on n real variables.Therefore, by standard facts on convolutions (see, e.g., [12]), we have that, for all k, h, n ∈ N, ψ n k,h ∈ C ∞ (R n ) and that, for any h, n ∈ N, the sequence {ψ n k,h } k∈N converges to ξ h,n uniformly on compact subsets of R n .Next, for each h, n ∈ N, we take k(h, n) ∈ N such that and we set ψ h,n := ψ n k(h,n),h .We have that, for any compact set K ⊂ H + , any h ∈ N, and any n ≥ sup x∈K |x|, where we used the fact that E φ h (x) = φ h (x), for all x ∈ H + .From this estimate, observing that the set {P n x : x ∈ K, n ∈ N} ⊂ H is relatively compact (cf.[13, Lemma B.77]) and using the continuity of φ h and (4.4), we get that, for each h ∈ N, Finally, taking the diagonal sequence and, applying (4.7), we get that φ n ∈ C 0 (H + ), for all n ∈ N.
We now turn our attention to the approximation of f , which may not belong to C 0 ((0, T ]× H + ), due to the singularity at t = 0 and the fact that the behaviour of f on ∂H + is not known.For this reason, also in this case we define, first, regularising convolutions for some approximations of f .First, we extend f to be equal to 0 for all (t, x) ∈ (0, T ] × ∂H + .We start approximating f in space, by defining, for each h ∈ N, the functions Next, we approximate in time by defining, first, the following extensions, for each h ∈ N and x ∈ H + , and then by introducing, for each h ∈ N, the approximations Note that, for each h ∈ N, f h is continuous on (0, T ] × H + and vanishes on ∂H + , and hence Such a sequence can be constructed proceeding in a similar way as in the first part of this proof, by choosing for each n a suitable family of C ∞ mollifiers (see also the proof of [13,Lemma B.78]).In particular, this sequence can be chosen so that We consider, next, the diagonal sequence {f n,n } n∈N .Using again the fact Eφ = φ on H + , we have that, for any compact subsets I 0 ⊂ (0, T ], K ⊂ H + , and any n ≥ sup (t,x)∈I 0 ×K {t+|x|}, Observe that, by construction, the sequence Indeed, thanks to the definition of f h , f h converges to f uniformly on compact subsets of (0, T ] × H + , as h → +∞.Therefore, the last term in the second line of (4.9) converges to 0.
To deal with the first term, instead, we note that, if n is large enough, then f n (t, x) = f (t, x), for all (t, x) ∈ I 0 × K. Therefore, applying also (4.Next, following the same reasoning of Step 2 of the proof of [13,Theorem 4.135], we get that also points (i) and (v) of Definition 4.2 are verified, by choosing g n (t, x) := F (t, x, u n (t, x), Du n (t, x)) − f n (t, x).Hence, u n is a classical solution to (4.2), for all n ∈ N.
We are left to check the convergences of the three sequences {u n } n∈N , {φ n } n∈N , {g n } n∈N and that u is the unique K-strong solution to (3.1).This can be done exactly in the same way as in Step 3 of the proof of [13,Theorem 4.135].

Application to a control problem with exit time
In this section we discuss an exit-time optimal control problem.In particular, we study the associated HJB equation and we provide a verification theorem and a result concerning optimal feedback controls.
Fix T > 0, a real separable Hilbert space Ξ, and a complete and separable metric space U , representing the control space.
We consider a fixed complete filtered probability space (Ω, F, F := (F s ) s∈[0,T ] , P), where filtration F satisfies the usual assumptions, supporting a cylindrical Wiener process W = (W (s)) s∈[0,T ] on Ξ.We introduce also the following set of admissible controls For any fixed t ∈ [0, T ) and any u ∈ U ad we consider the SDE (5.1) The following assumption will stand from now on.
(i) A and Q satisfy Hypotheses 2.1 and 2.3, for some δ ∈ (0, 1); x) uniformly with respect to u, and satisfies Under these assumptions (see, e.g., [13,Theorem 1.152]), for any t ∈ [0, T ), x ∈ H, and u ∈ U ad , SDE (5.1) admits a unique mild solution X = (X(s; t, x, u)) s∈[t,T ] (in the sense of [13, Moreover, X has continuous trajectories and verifies, for some constant C > 0, In what follows, we will denote this solution by (X(s)) s∈[t,T ] , when no confusion can arise.
The aim of the optimal control problem is to drive the dynamics of the state process X, by choosing a control u ∈ U ad to minimize the cost functional J : [0, T ] × H + × U ad → R defined, for all t ∈ [0, T ], x ∈ H + , u ∈ U ad , as where τ := inf{s ≥ t : X(s; t, x, u) ∈ H − }.
The following assumption will be in force in the remainder of the paper.
The value function of the optimal control problem is (5.4) Remark 5.3.Clearly, τ is an F-stopping time and depends on t, x, and the control u.However, we will omit this dependence to ease notations.
We can associate to the exit-time optimal control problem introduced above an HJB equation, that the value function V is expected to satisfy.
To start, let us introduce the current value Hamiltonian and the Hamiltonian The HJB equation associated to it is (5.7) Remark 5.4.Note that, while Equation (3.1) corresponds to an initial value problem, here Equation (5.7) is associated to a terminal value problem, which is a more natural formulation for an optimal control problem.It is clear that all the results in Sections 3 and 4 can still be applied, simply by reversing time.To be more precise, we give the following definition.We want now to verify that the HJB equation (5.7) has a unique mild and K-strong solution.To do so, we need to check that the Hamiltonian F verifies appropriate assumptions.
Finally, to check Hypothesis 3.1-(ii), let (t, x) ∈ [0, T ] × H + and p ∈ H.Then, To establish the continuity of F consider (t, x, p) ∈ [0, T ] × H + × H and a sequence (t n , x n , p n ) → (t, x, p).Then, and hence the left hand side tends to 0, since b and ℓ are continuous in (t, x), uniformly with respect to u ∈ U .Our next aim is to establish a verification theorem.To do so, we need to show that the following fundamental identity holds.Lemma 5.8.Suppose that Hypotheses 5.1 and 5.2 hold.Let v be the unique mild and K-strong solution to (5.7).Then, for every t ∈ [0, T ], x ∈ H + , and u ∈ U ad , it holds that s), Dv(s, X(s), u(s))) − F (s, X(s), Dv(s, X(s)))] ds.(5.8)where X = (X(s; t, x, u)) s∈[t,T ] is the mild solution to (5.1).
We show, first, that (5.8) holds for the functions v n (t, x) := z n (T − t, x), (t, x) ∈ [0, T ] × H + .It is clear that, for all n ∈ N, v n and z n enjoy the same regularity properties, as listed in Definition 4.2, and that where ∂ t denotes the time-derivative.It is also clear that Thanks to these facts and to Hypothesis 5.1, we can apply the Dynkin formula of [13, Equation (1.109)] to the process (v n (s, X(s))) s∈[t,T ] .For ε > 0, let us define the stopping time τ s, X(s), Dv n (s, X(s))) + g n (s, X(s)) ds. (5.9) Clearly, on {T < τ ε } we have that T ∧ τ ε = T , whence X(T ∧ τ ε ) = X(T ) ∈ H + , and Therefore we have, P-a.s., = E[1 T <τε φ n (X(T ))] + E[1 T ≥τε v n (τ ε , X(τ ε ))].Taking into account this equality, adding J(t, x, u) on both sides of (5.9), and rearranging the terms, we get Using the K-convergences previously recalled and the dominated convergence theorem, we can take the limit as n → +∞ in the last equality to get v(t, x) = J(t, x, u) + E[1 T ≥τε v(τ ε , X(τ ε ))].
We conclude our paper with a final result concerning optimal feedback controls, which is a corollary of Theorem 5.9.
Let v be the unique mild and K-strong solution to (5.7) and let us define the set-valued function Φ(t, x) := arg min u∈U F CV (t, x, Dv(t, x), u), (t, x) ∈ (0, T ) × H + . (5.11) Function Φ is the feedback map for our optimal control problem and the associated closed loop equation is the stochastic differential inclusion dX(s) ∈ [AX(s) + b(s, X(s), Φ(s, X(s)))] ds + Q dW (s), s ∈ [t, T ], Here is our final result.Its proof is omitted, since it is a direct consequence of Theorem 5.9 (for more details see, e.g., [13,Corollary 2.38]).
If u φt ∈ U ad , then the pair (u φt , X φt ) is optimal at (t, x) and v(t, x) = V (t, x).Moreover, if Φ(t, x) is always a singleton4 and the mild solution of (5.12) is unique, then u φt is the unique optimal control.Example 5.11.We consider here a simplified version of the spatial economic growth problem of [4] in a stochastic and finite time horizon setting (see also [3], [5], and [14]).We provide this example as a motivation for our paper and for future research in exit time optimal control problems and optimal control problems with state space constraints, as explained in the Introduction.Indeed, as we will see in a moment, our example is related to both kind of dynamic optimization problems and it is formulated in a slightly more general setting than the one of Section 5.This will give us the chance to point out possible future developments of our research.In the following, we denote by S 1 the unit circle and we set H := L 2 (S 1 ), that is, we consider H to be the set of square-integrable functions on S 1 .
We present, first, a finite time horizon version of the model studied in [4], which is formulated in a deterministic setting.In this paper the state variable is the capital stock k(t, ξ) and the control variable, which is assumed to be non-negative, is the consumption flow c(t, ξ); both variables depend on time t ∈ [0, T ], where T > 0 is a fixed time horizon, and on the spatial position ξ ∈ S 1 .Given suitable bounded measurable data A : S 1 → R + , k 0 : S 1 → R + , and t ∈ [0, T ], we consider the state equation    dk(s, ξ) = ∂ 2 ∂ξ 2 k(s, ξ) + A(ξ)k(s, ξ) − c(s, ξ) ds, (s, ξ) ∈ (t, T ] × S 1 , k(t, ξ) = k 0 (ξ), ξ ∈ S 1 . (5.13) We call k t,k 0 ,c the unique solution to such state equation.We require the state to satisfy the constraint k(s, •) ∈ H + , for all s ∈ [t, T ], where

2. 1 .
Notation and basic spaces.In this section we collect the main notations and conventions used in this research article.Throughout the paper the set N denotes the set of natural integers N := {1, 2, . . .}, and the symbol R denotes the set of real numbers, equipped with the usual Euclidean norm |•| R .We set R + := (0, +∞).The symbol 1 A denotes the indicator function of a set A. If X, Y are two Banach spaces, the symbols B b (X; Y ), C b (X; Y ), UC b (X; Y ) indicate the sets of Y -valued bounded Borel measurable, continuous, uniformly continuous functions on X, respectively.

Definition 4 . 3 .
We say that a function u : [0, T ] × H + → R is a K-strong solution to (3.1) if:

Theorem 5 . 7 .
Equation (5.7) has a unique mild and K-strong solution in the sense of Definition 5.5.Proof.First of all, we observe that, thanks to Assumption 5.2-(ii), the terminal cost φ satisfies 3.1-(iii).Moreover, by Proposition 5.6, the Hamiltonian F satisfies 3.1-(i)-(ii).Finally, Q and A, verify Hypotheses 2.1 and 2.3, thanks to Hypothesis 5.1.Therefore, we can apply Theorem 3.3 and conclude that Equation (5.7) has a unique mild solution in the sense of Definition 5.5.Finally, since φ ∈ C b (H + ) by Hypothesis 5.2-(ii) and F is continuous on [0, T ] × H + × H thanks to Proposition 5.6, we can apply Theorem 4.4 to deduce that Equation (5.7) has a unique K-strong solution in the sense of Definition 5.5.
δ | f n (t, P n x) − f (t, x)| and, thus, noting that the set {(t, P n x) :t ∈ I 0 , x ∈ K, n ∈ N} ⊂ R × H is relatively compact (cf.[13,Lemma B.77]) and using the continuity of f , we get that From now on, let us simply denote the diagonal sequence {f n,n } n∈N by {f n } n∈N .Let us define, next, the approximating sequence {u n } n∈N as follows u n (t, x) := P t φ n (x) + We note, first, that u n (0, x) = φ n (x), for all x ∈ H + , and that, as a consequence of Proposition 2.13 (i)-(ii) (see also Remark 2.15), u n ∈ C 0 ([0, T ] × H + ), and hence point (iii) of Definition 4.2 is verified.Arguing as in the proof of [13, Theorem 4.135], we deduce that, for each n ∈ N, φ n ∈ UC 2,A b (H + ) and that, for each t ∈ (0, T ], f n (t, •) ∈ UC 2,A b (H + ).Therefore, by [13, Proposition B.91], we deduce that points (ii) and (iv) of Definition 4.2 are satisfied.
t δ |f n,n (t, x) − f n (t, P n x)| + sup (t,x)∈I 0 ×K t δ | f n (t, P n x) − f n (t, x)| t Definition 1.119]) in the class of processes H 2 (t, T ; H) := {Y : [t, T ] × Ω → H progr. meas., s.t.sup Proposition 5.6.Under Hypothesis 5.2-(i), the Hamiltonian F is a measurable function and satisfies Hypothesis 3.1-(i)-(ii).Moreover, F is continuous on [0, T ] × H + × H.Proof. Since F CV is clearly a measurable function and U is separable, it follows that F is a measurable function, too.