BSDE Representation and Randomized Dynamic Programming Principle for Stochastic Control Problems of Infinite-Dimensional Jump-Diffusions

We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.


Introduction
In the present paper we study a general class of stochastic optimal control problems, where the infinite-dimensional state process, taking values in a real separable Hilbert space H, has a dynamics driven by a cylindrical Brownian motion W and a Poisson random measure π. Moreover, the coefficients are assumed to be path-dependent, in the sense that they depend on the past trajectory of the state process. In addition, the space of control actions Λ can be any Borel space (i.e., any topological space homeomorphic to a Borel subset of a Polish space). More precisely, the controlled state process is a so-called mild solution to the following equation on [0, T ]: γ t (X, α t , z) π(dt dz) − λ π (dz) dt , where A is a linear operator generating a strongly continuous semigroup {e tA , t ≥ 0}, λ π (dz)dt is the compensator of π, while α is an admissible control process, that is a predictable stochastic process taking values in Λ. Given an admissible control α, the corresponding gain functional is given by where the running and terminal reward functionals f and g may also depend on the past trajectory of the state process. The value of the stochastic control problem, starting at t = 0 from x 0 , is defined as V 0 = sup α J(α). (1.1) Stochastic optimal control problems of infinite-dimensional processes have been extensively studied using the theory of Backward Stochastic Differential Equations (BSDEs); we mention in particular the seminal papers [11], [12] and the last chapter of the recent book [9], where a detailed discussion of the literature can be found. Notice however that the current results require a special structure of the controlled state equations, namely that the diffusion coefficient σ = σ(t, x) is uncontrolled and the drift has the following specific form b = b 1 (t, x) + σ(t, x)b 2 (t, x, a). Up to our knowledge, only the recent paper [6], which is devoted to the study of ergodic control problems, applies the BSDEs techniques to a more general class of infinite-dimensional controlled state processes; in [6] the drift has the general form b = b(x, a), however the diffusion coefficient is still uncontrolled and indeed constant, moreover the space of control actions Λ is assumed to be a real separable Hilbert space (or, more generally, according to Remark 2.2 in [6], Λ has to be the image of a continuous surjection ϕ defined on some requires more restrictive assumptions; as a matter of fact, there the authors find the BSDE representation passing through the Hamilton-Jacobi-Bellman equation, and in particular using viscosity solutions techniques; moreover, in order to apply the techniques in [15], one already needs to know that the value function is the unique viscosity solution to the HJB equation. The randomization method developed in the present paper improves considerably the methodology used in [15] and allows to extend the results in [10] and [1] to the infinite dimensional jump-diffusive framework, addressing, in addition, the path-dependent case. We notice that it would be possible to consider a path-dependence, or delay, in the control variable as well; however, in order to make the presentation more understandable and effective, we assume a path-dependence only in the state variable. We underline that our results are also relevant for the finite-dimensional case, as it is the first time the randomization method (understood as a pure probabilistic methodology as explained above) is implemented when a jump component appears in the state process dynamics.
Roughly speaking, the key idea of the randomization method consists in randomizing the control process α, by replacing it with an uncontrolled pure jump process I associated with a Poisson random measure θ, independent of W and π; for the pair of processes (X, I), a new randomized intensity-control problem is then introduced in such a way that the corresponding value coincides with the original one. The idea of this control randomization procedure comes from the well-known methodology implemented in [16] to prove the dynamic programming principle, which is based on the use of piece-wise constant policies. More specifically, in [16] it is shown (under quite general assumptions; the only not usual assumption is the continuity of all coefficients with respect to the control variable) that the supremum over all admissible controls α can be replaced by the supremum over a suitable class of piece-wise constant policies. This allows to prove in a relatively easy but rigorous manner the dynamic programming principle, see Theorem III.1.6 in [16]. Similarly, in the randomization method we prove (Theorem 4.7), under quite general assumptions (the only not usual assumption is still the continuity of all coefficients with respect to the control variable), that we can optimize over a suitable class of piece-wise constant policies, whose dynamics is now described by the Poisson random measure θ. This particular class of policies allows to prove the BSDE representation (Theorem 5.6), as well as the randomized dynamic programming principle. Notice that in the present paper we have made an effort to simplify various arguments in the proof of Theorem 4.7 and streamline the exposition. In the Markovian case (Section 6), namely when the coefficients are non-pathdependent, we consider a family of stochastic control problems, one for each (t, x) ∈ [0, T ] × H, and define the corresponding value function. Then, exploiting the BSDE representation derived in Section 5, we are able to prove the so-called randomized dynamic programming principle (Theorem 6.6), which is as powerful as the classical dynamic programming principle, in the sense that it allows to prove (Proposition 6.12) that the value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation on [0, T ] × H: 2 Tr σ(t, x, a)σ * (t, x, a)D 2 x v + b(t, x, a), D x v + f (t, x, a) + U \{0} (v(t, x + γ(t, x, a, z)) − v(t, x) − D x v(t, x)γ(t, x, a, z))λ π (dz) = 0,

Notations and assumptions
Let H, U and Ξ be two real separable Hilbert spaces equipped with their respective Borel σ-algebrae. We denote by | · | and ·, · (resp. | · | U , | · | Ξ and ·, · Ξ , ·, · U ) the norm and scalar product in H (resp. in U and Ξ). Let (Ω, F, P) be a complete probability space on which are defined a random variable x 0 : Ω → H, a cylindrical Brownian motion W = (W t ) t≥0 with values in Ξ, and a Poisson random measure π(dt dz) on [0, ∞) × U with compensator λ π (dz) dt. We assume that x 0 , W , π are independent. We denote by µ 0 the law of x 0 , which is a probability measure on the Borel subsets of H. We also denote by F x0,W,π = (F x0,W,π t ) t≥0 the P-completion of the filtration generated by x 0 , W , π, which turns out to be also right-continuous, as it follows for instance from Theorem 1 in [13].
So, in particular, F x0,W,π satisfies the usual conditions. When x 0 is deterministic (that is, µ 0 is the Dirac measure δ x0 ) we denote F x0,W,π simply by F W,π . Let L(Ξ; H) be the Banach space of bounded linear operators P : Ξ → H, and let L 2 (Ξ; H) be the Hilbert space of Hilbert-Schmidt operators P : Ξ → H.
Let T > 0 be a finite time horizon. For every t ∈ [0, T ], we consider the Banach space D([0, t]; H) of càdlàg maps x : [0, t] → H endowed with the supremum norm x * t := sup s∈[0,t] |x(s)|; when t = T we also use the notation for all s ∈ [0, t]. We also define its right-continuous version Let Λ be a Borel space, namely a topological space homeomorphic to a Borel subset of a Polish space. We denote by B(Λ) the Borel σ-algebra of Λ. We also denote by d Λ a bounded distance on Λ.
(v) The map g is continuous on D([0, T ]; H) with respect to the supremum norm. For every t ∈ [0, T ], the maps b t (·, ·) and f t (·, ·) are continuous on D(

Stochastic optimal control problem
In the present section we formulate the original stochastic optimal control problem on two different probabilistic settings. More precisely, we begin formulating (see subsection 3.1 below) such a control problem in a standard way, using the probabilistic setting previously introduced. Afterwards, in subsection 3.2 we formulate it on the so-called randomized probabilistic setting (that will be used for the rest of the paper and, in particular, for the formulation of the randomized control problem in Section 4). Finally, we prove that the two formulations have the same value.

Formulation of the control problem
We formulate the stochastic optimal control problem on the probabilistic setting introduced in Section 2. An admissible control process will be any F x0,W,π -predictable process α with values in Λ. The set of all admissible control processes is denoted by A. The controlled state process satisfies the following equation on [0, T ]: We look for a mild solution to the above equation in the sense of the following definition. Definition 3.1. Let α ∈ A. We say that a càdlàg F x0,W,π -adapted stochastic process X = (X t ) t∈[0,T ] taking values in H is a mild solution to equation (3.1) if, P-a.s.,

2)
for some positive constant C p , independent of x 0 and α.
Proof. Under assumption (A), the existence of a unique mild solution X x0,α = (X x0,α t ) t∈[0,T ] to equation (3.1), for every α ∈ A, can be obtained by a fixed point argument proceeding as in Theorem 3.4 in [19], taking into account the fact that the coefficients of equation (3.1) are path-dependent.
We now prove estimate (3.2). In the sequel, we denote by C a positive constant depending only on T and p, independent of x 0 and α, that may vary from line to line. For brevity we will denote X x0,α simply by X. We start by noticing that E sup On the other hand, by the Burkölder-Davis-Gundy inequalities, we have E sup where we have set φ s (z) = e (t−s)A γ s (X, α s , z) and ||φ s || L 2 (U,λπ;H) = ( U \{0} |φ s (z)| 2 λ π (dz)) 1/2 .
Taking the square of both sides and using the Cauchy-Schwarz inequality, we find (we and we conclude by the Gronwall inequality.
The controller aims at maximizing over all α ∈ A the gain functional By assumption (2.1) and estimate (3.2), we notice that J(α) is always finite. Finally, the value of the stochastic control problem is given by

Formulation of the control problem in the randomized setting
We formulate the stochastic optimal control problem on a new probabilistic setting that we now introduce, to which we refer as randomized probabilistic setting. Such a setting will be used for the rest of the paper and, in particular, in Section 4 for the formulation of the randomized stochastic optimal control problem.
We impose the following additional assumptions.
with the same constant C p as in Proposition 3.2, whereÊ denotes the expectation under P.
In the present randomized probabilistic setting the formulations of the control problem reads as follows: the controller aims at maximizing over allα ∈Â the gain functional Proof. The proof is organized as follows: 1) firstly we introduce a new probabilistic setting in product form on which we formulate the control problem (3.10) and denote the new value functionV 0 ; then, we show that 2) we prove that V 0 =V 0 .

Formulation of the randomized control problem
We now formulate the randomized stochastic optimal control problem on the probabilistic setting introduced in subsection 3.2. Our aim is then to prove that the value of such a control problem coincides with V 0 or, equivalently (by Proposition 3.4), withV 0 . Here we simply observe that the randomized problem may depend on λ 0 and a 0 , but its value will be independent of these two objects, as it will coincide with the value V 0 of the original stochastic control problem (which is independent of λ 0 and a 0 ).

4)
with the same constant C p as in Proposition 3.2. In addition, for every t ∈ [0, T ] and any |X s | p ,P-a.s. (4.5) with the same constant C p as in Proposition 3.2.
Proof. Concerning estimate (4.4), the proof can be done proceeding along the same lines as in the proof of Proposition 3.2. Regarding estimate (4.5), we begin noting that, given any two integrableFx 0 ,Ŵ ,π,θ t -measurable random variables η and ξ, then the following property holds: η ≤ ξ,P-a.s., if and only ifÊ[η 1 E ] ≤Ê[ξ 1 E ], for every E ∈Fx 0 ,Ŵ ,π,θ t . So, in particular, estimate (4.5) is true if and only if the following estimate holds: The proof of estimate (4.6) can be done proceeding along the same lines as in the proof of Proposition 3.2, firstly multiplying equation (4.3) by 1 E .
As a consequence, the following generalization of estimate (4.4) holds: for every with the same constant C p as in (4.4), whereÊν denotes the expectation with respect tô Pν.
The controller aims at maximizing over allν ∈V the gain functional By assumption (2.1) and estimate (4.8), it follows thatĴ R (ν) is always finite. Finally, the value function of the randomized control problem is given bŷ In the sequel, we denote the probabilistic setting we have adopted for the randomized control problem shortly by the tuple (Ω,F,P;x 0 ,Ŵ ,π,θ;Î,X;V).
Our aim is now to prove thatV R 0 coincides with the value V 0 of the original control problem. Firstly, we state three auxiliary results: 1) the first result (Lemma 4.2) shows that the valueV R 0 of the randomized control problem is independent of the probabilistic setting on which the problem is formulated; 2) in Lemma 4.3 we prove that there exists a probabilistic setting for the randomized control problem whereĴ R can be expressed in terms of the gain functionalĴ in (3.9); as noticed in Remark 4.5, this result allows to formulate the randomized control problem in "strong" form, rather than as a supremum over a family of probability measures; 3) finally, in Lemma 4.6 we prove, roughly speaking, that given any α ∈ A and ε > 0 there exist a probabilistic setting for the randomized control and a suitableν such that the "distance" underPν between the pure jump processÎ and α is less than ε.
In order to do it, we need to introduce the following distance onÂ (see Definition 3.2.3 in [16]), for every fixedν ∈V: for allα,β ∈Â.
Proof of Lemma 4.3. Let (Ω, F, P; x 0 , W, π; X; A) be the setting of the original stochastic control problem in Section 3.1.
Finally, concerning the existence of a sequence (Tν n ,ην n ) n≥1 satisfying (i)-(ii)-(iii)-(iv), we do not report the proof of this result as it can be done proceeding along the same lines as in the proof of Lemma 4.3 in [1], the only difference being that the filtration F W in [1] (notice that in [1] W denotes a finite dimensional Brownian motion) is now replaced by F x0,W,π : this does not affect the proof of Lemma 4.3 in [1].
Then, from item (b) above we immediately deduce item (ii).
It remains to prove (4.12). By item (i) above we notice thatJ R,α,k (ν α,k ) can be equivalently written in terms ofĒQ: On the other hand, by item (iii) above, J(α) is also given by Hence, (4.12) can be equivalently rewritten as follows: .  It is then easy to see that, from the continuity and polynomial growth assumptions on f and g in (A)-(v) and (A)-(vi), convergence (4.13) follows directly from (4.11) and (4.14). This concludes the proof of the inequality V 0 ≤V R
for all 0 ≤ t ≤ T ,P-a.s.. Observe that on the left-hand side of (5.3) there is a predictable process, which has therefore no totally inaccessible jumps, while on the right-hand side in (5.3) there is a pure jump process which has only totally inaccessible jumps. We deduce that both sides must be equal to zero. Therefore, we obtain the two following equalities: for all 0 ≤ t ≤ T ,P-a.s., Concerning the first equation, the left-hand side is a finite variation process, while the process on the right-hand side has not finite variation, unless Z =Z and K −K + · 0 U (L s (z) −L s (z))λ π (dz)ds = 0. On the other hand, sinceπ andθ are independent, they have disjoint jump times, therefore from the second equation above we find L =L and R =R, from which we also obtain K =K.
We now prove that focus on the existence of a minimal solution to (5.1)-(5.2). To this end, we introduce, for every integer n ≥ 1, the following penalized backward stochastic differential equation: s.
with f + = max(f, 0) denoting the positive part of the function f .

Proposition 5.4. Under assumptions (A)-(A R )
, for every integer n ≥ 1 there exists a unique solution (Y n , Z n , L n , R n ) ∈ S 2 × L 2 (Ŵ ) × L 2 (π) × L 2 (θ) to equation (5.4). In addition, the following estimate holds: for some constantĈ ≥ 0, depending only on T and on the constant L in assumption (A)-(vi), independent of n.
Proof. The existence and uniqueness result can be proved as in the finite-dimensional case dim Ξ < ∞, see Lemma 2.4 in [21]. We simply recall that, as usual, it is based on a fixed point argument and on the martingale representation (concerning this latter result, since we did not find a reference for it suitable for our setting, we proved it in Lemma 5.3). Similarly, estimate (5.6) can be proved proceeding along the same lines as in the finite-dimensional case dim Ξ < ∞, for which we refer to Lemma 2.3 in [15]; we just recall that its proof is based on the application of Itô's formula to |Y n | 2 , as well as on Gronwall's lemma and the Burkholder-Davis-Gundy inequality.
For every integer n ≥ 1, we provide the following representation of Y n in terms of a suitable penalized randomized control problem. To this end, we defineV n as the subset ofV of all mapsν bounded from above by n. We recall that, for everyν ∈V,Êν denotes the expectation with respect to the probability measure on (Ω,Fx 0,Ŵ ,π,θ T ) given by dPν =κν T dP, where (κν t ) t∈[0,T ] denotes the Doléans-Dade exponential defined in (4.7). Proof. Proof of formulae (5.7) and (5.8). We report the proof of formula (5.7), as (5.8) can be proved proceeding along the same lines (simply replacing all thePν -conditional expectations with normalPν -expectations, and also noting thatPν coincides withP on Fx 0 ,Ŵ ,π,θ 0 , which is theP-completion of the σ-algebra generated byx 0 ). Fix an integer n ≥ 1 and let (Y n , Z n , L n , R n ) be the solution to (5.4), whose existence follows from Proposition 5.4. As consequence of the Girsanov Theorem, the two following processes arePν -martingales (see e.g. Theorem 15.3.10 in [5] or Theorem 12.31 in [14]). Moreover Therefore, taking thePν -conditional expectation givenFx 0,Ŵ ,π,θ t in (5.4), we obtain Firstly, we notice that nu + − νu ≥ 0 for all u ∈ R, ν ∈ (0, n], so that (5.10) gives Y n t ≥ ess sup ν∈VnÊν g(X T ) + T t f (X s ,Î s ) ds Fx 0,Ŵ ,π,θ t P -a.s., 0 ≤ t ≤ T. On the other hand, since R n ∈ L 2 (θ), by Lebesgue's dominated convergence theorem for conditional expectation, we obtain So, in particular, for every n ≥ 1 there exists a positive integer N n such that Now, let us definê It is easy to see thatν n,ε ∈V n . Moreover, we havê ≤Ê κ where the last equality follows from the fact that, for everyν ∈V, we haveν 2 ∈V, so thatκν 2 is a martingale. Plugging (5.12) and (5.14) into (5.13), we end up witĥ From the arbitrariness of ε, we find the reverse inequality of (5.11), from which (5.7) follows.
Proof of the monotonicity of (Y n ) n . By definitionV n ⊂V n+1 . Then inequality Y n t ≤ Y Finally, by estimate (4.5), together with the fact that Y n is a càdlàg process, we see that (5.9) follows.
We can now prove the main result of this section.
Recalling thatV n ⊂V and Y n ≤ Y , we find Y n t ≤ ess sup Letting n → ∞, we conclude that ,P-a.s.
In order to prove the reverse inequality, take a positive integer m, then, for every n ≥ m, where we have used that Y t ≥ Y n t and Y n τ ≥ Y m τ . From the arbitrariness of n, we end up with Y t ≥Êν[ τ t f s (X,Î s ) ds + Y m τ |Fx 0,Ŵ ,π,θ t ], for anyν ∈V and m ≥ 1. Letting m → ∞ and taking the essential supremum overV, we see that the claim follows.

HJB equation in Hilbert spaces: the Markovian case
In the present section, we replace assumptions (A) by the set of assumptions (A M ) reported below. Before stating (A M ), we notice that in this section, A still denotes a linear operator from D(A) ⊂ H into H, while the coefficients b, σ, γ, f , g are non-path-depedent, In what follows, we shall impose the following assumptions on A, b, σ, γ, f , g.  ( for some constant L ≥ 0 and some modulus of continuity ω, i.e. a continuous, nondecreasing, subadditive map ω : [0, ∞) → [0, ∞) satisfying ω(0) = 0 and ω(r) > 0, for any r > 0.
Stochastic optimal control problem. We now formulate the stochastic optimal control problem in such a setting. Since the formulation can be done proceeding along the same lines as in subsection 3.1, we focus on the main steps. We consider a complete probability space (Ω, F, P) on which are defined a cylindrical Brownian motion W = (W t ) t≥0 , with values in Ξ, and an independent Poisson random measure π(dt dz) on [0, ∞) × U with compensator λ π (dz) dt. For every t ≥ 0, we denote by F t,W,π = (F t,W,π s ) s≥t the P-completion of the filtration generated by (W s − W t ) s≥t and the restriction of π(dt dz) For every t ∈ [0, T ], an admissible control process at time t will be any F t,W,πpredictable process α : [t, T ] × Ω → Λ. For every t ∈ [0, T ], the set of all admissible control processes at time t is denoted by A t . For every (t, x) ∈ [0, T ] × H and any α ∈ A t , the controlled equation has the form We have the following result.
for some positive constant C p , independent of t, x, α.
Proof. The proof can be done proceeding along the same lines as in the proof of Proposition 3.1.
The controller aims at maximizing over all α ∈ A t the gain functional Finally, the value function of the stochastic control problem is given by for all t ∈ [0, T ], x, x ∈ H, α ∈ A t . In particular, Proof. We begin noting that, proceeding along the same lines as in the proof of estimate (3.12) of Theorem 3.4 in [20], we can prove that the following estimate holds: for some constantC ≥ 0, independent of t, x, x , α. Then, (6.4) follows directly from estimate (6.7) and the assumptions on f and g in (A M )-(v). On the other hand, (6.5) follows from estimate (6.2), using again the assumptions on f and g in (A M )-(v).
Randomized setting. We now consider, following Section 4, the randomized setting.
For every t ∈ [0, T ] and a ∈ Λ, we denote byÎ t,a = (Î t,a s ) s∈[t,T ] the stochastic process taking values in Λ defined as (notice that, when Λ is a subset of a vector space, we can write (6.8) also asÎ t,a s = a + where we recall that (T n ,η n ) n≥1 is the marked point process associated with the random measureθ, in particular we haveθ(dt da) = n≥1 δ (Tn,ηn) (dt da). Now, for every (t, x, a) ∈ [0, T ] × H × Λ, we consider the following equation: We have the following result.  for some positive constant C p , independent of t, x, a.
Proof. The proof can be done proceeding along the same lines as in the proof of Proposition 3.2.
BSDE with non-positive jumps. We introduce the following additional notations.
We can now state the two main results of this section: the first result is the probabilistic representation formula (or non-linear Feynman-Kac formula) for the value function v defined in (6.3); the second result is the so-called randomized dynamic programming principle for v. x,a , Z t,x,a , L t,x,a , R t,x,a , K t,x,a ) ∈ S 2 (t, T ) × L 2 (Ŵ ; t, T ) × L 2 (π; t, T ) × L 2 (θ; t, T ) × K 2 (t, T ) to (6.11)-(6.12), satisfying v(s,X t,x,a s ) = Y t,x,a s ,P-a.s., t ≤ s ≤ T 1) For every R > 0, there exists a modulus of continuity ω R such that for all t, t ∈ [0, T ], |x| ≤ R.
2) The randomized dynamic programming principle holds: for every t ∈ [0, T ] and anŷ F t,Ŵ ,π,θ -stopping timeτ taking values ) .  Following the same arguments as in the proof of Theorem 6.6, we see that we can apply Theorem 5.6 to our backward stochastic differential equation ( Now, by (6.20) withτ = t , together with (6.13), we see that (6.18) follows.
Proof of 1). We proceed as in the proof of Lemma 4.3 in [20]. More precisely, fix R > 0, 0 ≤ t < t ≤ T , and |x| ≤ R. Then, by (6.19) we have Now, notice that proceeding along the same lines as in the proof of estimate (3.13) of Theorem 3.4 in [20], we can prove that the following estimate holds: for some modulus ω x . Then, using the assumptions on f in (A M )-(v), estimates (6.10) and (6.22), inequality (6.6), and estimate (D.1) in [9], we obtain from (6.21): for some constantC ≥ 0 and some modulus ω R .

Viscosity property of the value function v
We now exploit the randomized dynamic programming principle (6.18) in order to prove that the value function v in (6.3) is a viscosity solution to the following Hamilton-Jacobi-Bellman equation on [0, T ] × H: x, a, z))λ π (dz) = 0, v(T, x) = g(x). (6.24) We adopt the definition of viscosity solution given in [20], Definition 5.2, which requires the following notions. (i) ϕ t , D x ϕ, D 2 x ϕ, A * D x ϕ, δ t , D x δ, D 2 x δ, A * D x δ are uniformly continuous on (ε, T − ε) × H, for every ε > 0; in addition, ϕ is B-lower semicontinuous; finally, δ ≥ 0, bounded, and B-continuous.
(iii) A function u : (0, T ) × H → R is a viscosity solution of (6.24) if it is both a viscosity subsolution and a viscosity supersolution of (6.24).
In order to prove that v is a viscosity solution to equation (6.24) we will need the following technical result.
Lemma 6.11. Let assumption (A M ) hold. Let ψ = ϕ + δh(| · |) be a test function. Fix t, t ∈ (0, T ), with t < t , and letτ be aF t,Ŵ ,π,θ -stopping time taking values in [t, t ]. Then, for any (x, a) ∈ H × Λ,ν ∈V t , E t,ν ψ(τ ,X t,x,â τ ) ≥ ψ(t, x) +Ê t,ν ψ r,X t,x,a r + γ(r,X t,x,a r ,Î t,a r , z) − ψ(r,X t,x,a r ) − D x ψ(r,X t,x,a r )γ(r,X t,x,a r ,Î t,a r , z) λ π (dz)dr . Proof. The proof can be done proceeding along the same lines as in the proof of Lemma 5.3 in [20], the only difference being the presence of the pure jump processÎ t,a . For this reason, here we just give an outline. The proof consists in approximating the procesŝ X t,x,a by means of a sequence of more regular processesX n,t,x,a , which are obtained replacing the operator A in equation (6.9) by its Yosida approximations (A n ) n . It is wellknown, see e.g. Theorem 27.2 in [17], that ψ(·,X n,t,x,a · ) satisfies an Itô formula. Then, using convergence results ofX n,t,x,a towardsX t,x,a , which can be found for instance in Proposition 1.115 of [9], and taking the expectation underP t,ν , we deduce (6.25) using that −AX t,x,a r , δ(r,X t,x,a r ) h (|X t,x,a r |) |X t,x,a r |X t,x,a r ≥ 0. Proof. We split the proof into two steps.