Optimal control of path-dependent McKean-Vlasov SDEs in infinite dimension
Andrea Cosso, Fausto Gozzi, Idris Kharroubi, Huyên Pham, Mauro Rosestolato
aa r X i v : . [ m a t h . O C ] D ec Optimal control of path-dependentMcKean-Vlasov SDEs in infinite dimension
Andrea COSSO ∗ Fausto GOZZI † Idris KHARROUBI ‡ Huyên PHAM § Mauro ROSESTOLATO ¶ January 1, 2021
Abstract
We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbertspaces motivated by non Markovian mean-field models driven by stochastic PDEs. We firstestablish the well-posedness of the state equation, and then we prove the dynamic programmingprinciple (DPP) in such a general framework. The crucial law invariance property of the valuefunction V is rigorously obtained, which means that V can be viewed as a function on theWasserstein space of probability measures on the set of continuous functions valued in Hilbertspace. We then define a notion of pathwise measure derivative, which extends the Wassersteinderivative due to Lions [41], and prove a related functional Itô formula in the spirit of Dupire[24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by meansof a suitable notion of viscosity solution. We provide different formulations and simplificationsof such a Bellman equation notably in the special case when there is no dependence on the lawof the control. Mathematics Subject Classification (2010):
Keywords:
Path-dependent McKean-Vlasov SDEs in Hilbert space, dynamic programming prin-ciple, pathwise measure derivative, functional Itô calculus, Master Bellman equation, viscosity so-lutions. ∗ University of Bologna, Italy; [email protected] † Luiss University, Roma, Italy; [email protected] ‡ LPSM, UMR CNRS 8001, Sorbonne University and Université de Paris; [email protected] § LPSM, UMR CNRS 8001, Sorbonne University and Université de Paris; [email protected] ¶ Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100 Lecce, Italy;[email protected]). ontents V . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Law invariance property of the lifted value function V . . . . . . . . . . . . . . . . . 15 ˆ ϕ : ˆ H → R and Itô’s formula . . . . . . . . . . . . . . 204.3 Pathwise derivatives for a map ϕ : H → R and Itô’s formula . . . . . . . . . . . . . . 23 A State equation: proofs 32B Law invariance property of V : technical results 38C Pathwise measure derivative: law invariance property 40 C.1 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40C.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
D Proof of Itô’s formula 43E Consistency property of pathwise derivatives 47F Hamilton-Jacobi-Bellman equation: technical results 49
Given two real separable Hilbert spaces H and K , let us consider the nonlinear stochastic differentialequation (SDE) on H in the form: dX t = AX t dt + b t ( X, P X ·∧ t , α t , P α t ) dt + σ t ( X, P X ·∧ t , α t , P α t ) dB t , (1.1)over a finite interval [0 , T ] . Here, A : D ( A ) ⊂ H → H is the generator of a C -semigroup ofcontractions in H , and B = ( B t ) t ≥ is a K -valued cylindrical Brownian motion on a completeprobability space (Ω , F , P ) with F B its completed natural filtration. The coefficients b and σ ,valued respectively in H and L ( K ; H ) (the space of Hilbert-Schmidt operators from K to H ),2epend on time, on the whole path of the state process X , on an input control process α , that isan F B -adapted process valued in some Borel space U , and furthermore on the distribution of thestate/control process.Equation (1.1) is referred to as controlled McKean-Vlasov SDE in Hilbert spaces, and we areinterested in the optimal control for (1.1) by minimizing, over control processes α , a functional inthe form J ( X , α ) = E h Z T f t ( X, P X ·∧ t , α t , P α t ) dt + g ( X, P X ) i , given running cost and terminal cost functions f and g .When the coefficients b, σ, f, g do not depend on the law of the state process, the control ofequation (1.1) is motivated by various kinds of stochastic partial differential equations (SPDEs)like stochastic heat equations (see e.g. [12, 31, 32, 42, 29]) stochastic reaction-diffusion equations(see e.g.[16, 17]), stochastic porous media equations (see e.g. [4]), singular stochastic dissipativeequations (see e.g. [46]), stochastic Burgers and Navier-Stokes equations (see e.g. [19, 45, 35]),Zakai equation in filtering (see e.g. [36]), stochastic first-order equations (see e.g. [30]), stochasticdelay equations (see e.g. [28][47], [33, 34] [9]). We refer also to the lecture notes and monographs[20], [3], [25], for an account on this topic.The main novelty of this paper is to consider a mean-field dependence on the coefficients of theinfinite-dimensional stochastic differential equation (1.1), and to study the corresponding controlproblem. Mean-field diffusion processes, also called McKean-Vlasov equations, in finite dimensionhave a long history with the pioneering works [38], [43], and later on with the seminal paper[48] in the framework of propagation of chaos. The control of such equations has attracted anincreasing interest since the emergence of mean-field game theory initiated independently in [40]and [37], aiming at describing control of large systems or population of interacting particles, and hasgenerated over the last few years numerous contributions, see e.g. [44], [18], [23], and the referencemonographs [7] and [15].In addition to the infinite-dimensional feature of the McKean-Vlasov equation (1.1), we empha-size the path-dependency (in a nonanticipative way) of the coefficients b, σ, f, g , on the state processas well as on its distribution. This general setting, which is motivated by various applications, seee.g. Example 2.12, seems to be considered for the first time in the present paper.Our basic objective here is to extend to our infinite-dimensional path-dependent setting thetools required in the dynamic programming approach for McKean-Vlasov control problems.In the finite-dimensional case, i.e. H = R d , and in the Markovian case, i.e. without path-dependency of the coefficients, the Wasserstein derivative in the lifted sense of Lions [41], turns outto be a convenient notion of measure derivative when combined with Itô’s formula along the flow ofprobability measures (see [10]) in order to define the Master equation in mean-field game/control.These concepts have been recently extended to the path-dependent case in [51] with a functionalItô formula in the McKean-Vlasov setting. Our contributions.
Our first main result is to prove the crucial law invariance property of thevalue function to the control problem (see Theorem 3.5), which implies that the value function canbe considered as a function on the Wasserstein space of probability measures on C ([0 , T ]; H ) . Wealso state and provide a direct proof of the dynamic programming principle in this context (seeTheorem 3.4 and Corollary 3.8). Next, we introduce a notion of pathwise derivative in Wassersteinspace and a related functional Itô formula in our infinite-dimensional McKean-Vlasov context thatextend the concepts in [51]. Equipped with these tools, we can then derive from the dynamic3rogramming principle the associated Master HJB equation, which is a PDE where the state variableis a probability measure on C ([0 , T ]; H ) . For such PDE, we provide equivalent formulations andsimplifications, notably in the special case when there is no dependence of the coefficients on thelaw of the control. We define an intrinsic notion of viscosity solution in P ( C ([0 , T ]; H )) , the spaceof square-integrable probability measures on C ([0 , T ]; H ) , together with the viscosity property ofthe value function. Comparison principle for Master Bellman equation is postponed to furtherinvestigation, as it is already a challenging issue in the finite-dimensional case where only partialresults exist in the literature, see [51] and [11].We also point out that our results clarify and improve in particular some statements fromthe finite-dimensional case, like the law invariance property (see Remark 3.7) and the dynamicprogramming principle.The outline of the paper is organized as follows. In Section 2, we present the notations andformulate the McKean-Vlasov state equation valued in Hilbert space: due to the generality of thesetting basic results on well-posedness and approximation of this equation are not known and arecarefully proved. Section 3 is devoted to the formulation of the optimal control problem, the dynamicprogramming principle and law invariance property of the associated value function. We introducein Section 4 the notion of pathwise derivative in Wasserstein space and the related functional Itôformula. Section 5 is concerned with the derivation of the Master Bellman equation and the viscosityproperty of the value function. Finally, Appendices A, B, C, E, F collect some technical resultsused throughout the paper. State space and functional analytic setting.
We fix two real separable Hilbert spaces H and K , with inner products h· , ·i H , h· , ·i K and induced norms | · | H , | · | K , respectively, omitting thesubscripts H or K when clear from the context. We denote by L ( K ; H ) (resp. L ( H ) ) the space ofbounded linear operators from K to H (resp. H to H ). We endow L ( K ; H ) with the operator norm k · k L ( K ; H ) defined by k F k L ( K ; H ) = sup k ∈ K, k =0 | F k | H | k | K , F ∈ L ( K ; H ) . Similarly, we endow L ( H ) with the corresponding norm k · k L ( H ) . We also denote by L ( K ; H ) thespace of Hilbert-Schmidt operators from K to H , that is the set of all F ∈ L ( K ; H ) such that X n ∈ N | F e n | H < + ∞ for some orthonormal basis { e n } n ∈ N of K . We endow L ( K ; H ) with the norm k · k L ( K ; H ) definedby k F k L ( K ; H ) = sX n ∈ N | F e n | H , F ∈ L ( K ; H ) .
4e recall that the definitions of L ( K ; H ) and k · k L ( K ; H ) do not depend on the choice of theorthonormal basis { e n } n ∈ N of K .We now fix a finite time horizon T > and consider the state space of our optimal controlproblem which is given by the set C ([0 , T ]; H ) of continuous H -valued functions on [0 , T ] . Given x ∈ C ([0 , T ]; H ) and t ∈ [0 , T ] , we denote by x t the value of x at time t and we set x ·∧ t :=( x s ∧ t ) s ∈ [0 ,T ] . Notice that x t ∈ H , while x ·∧ t ∈ C ([0 , T ]; H ) . We endow C ([0 , T ]; H ) with theuniform norm k · k T defined as k x k T = sup s ∈ [0 ,T ] | x s | H , x ∈ C ([0 , T ]; H ) . Notice that ( C ([0 , T ]; H ) , k · k T ) is a Banach space. We denote by B the Borel σ -algebra of C ([0 , T ]; H ) . Finally, for every t ∈ [0 , T ] we introduce the seminorm k · k t defined as k x k t = k x ·∧ t k T , x ∈ C ([0 , T ]; H ) . Spaces of probability measures and Wasserstein distance.
Given a metric space M , if M denotes its Borel σ -algebra, we denote by P ( M ) the set of all probability measures on ( M, M ) . Weendow P ( M ) with the topology of weak convergence. When M is a Polish space S , with metric d S ,we also define, for q ≥ , P q (S) := (cid:26) µ ∈ P (S) : Z S d S ( x , x ) q µ ( dx ) < + ∞ (cid:27) , where x ∈ S is arbitrary. This set is endowed with the q -Wasserstein distance defined as W q ( µ, µ ′ ) := inf (cid:26) Z S × S d S ( x, y ) q π ( dx, dy ) : π ∈ P (S × S) such that π ( · × S) = µ and π (S × · ) = µ ′ (cid:27) q , q ≥ , for every µ, µ ′ ∈ P q (S) . The space (cid:0) P q (S) , W q ) turns out to be a Polish space (see for instance [49,Theorem 6.18]). Probabilistic setting.
We fix a complete probability space (Ω , F , P ) on which a K -valued cylin-drical Brownian motion B = ( B t ) t ≥ is defined (see e.g. [20, Section 4.1] and [25, Remark 1.89] onthe definition of cylindrical Brownian motion). We denote by F B = ( F Bt ) t ≥ the P -completion ofthe filtration generated by B . Notice that F B is also right-continuous (see [25, Lemma 1.94]), so,in particular, it satisfies the usual conditions. We assume that there exists a sub- σ -algebra G of F satisfying the following standing assumptions. Standing Assumption (A G ). i) G and F B ∞ are independent; Notice that it may be not obvious to define the natural filtration for cylindrical Brownian motion as, in principle,it may depend on the choice of the reference system where such process is considered, which, in general, is not unique.However, as noted in [25, Remark 1.89]) this will not affect the class of integrable processes and, consequently, thefiltration. G is “rich enough” in the sense that the following property holds: P (cid:0) C ([0 , T ]; H ) (cid:1) = (cid:8) P ξ with ξ : [0 , T ] × Ω → H continuous and B ([0 , T ]) ⊗ G -measurable process satisfying E (cid:2) k ξ k T (cid:3) < ∞ (cid:9) , i.e. for every µ ∈ P ( C ([0 , T ]; H )) there exists a continuous and B ([0 , T ]) ⊗ G -measurableprocess ξ : [0 , T ] × Ω → H , satisfying E k ξ k T < ∞ , such that ξ has law equal to µ . As stated in the following lemma (take H = G in Lemma 2.1), property (A G ) -ii) holds if and only ifthere exists a G -measurable random variable U G : Ω → R having uniform distribution on [0 , (seealso Remark 2.2). Lemma 2.1.
On the probability space (Ω , F , P ) consider a sub- σ -algebra H ⊂ F . The followingstatements are equivalent. There exists a H -measurable random variable U H : Ω → R having uniform distribution on [0 , . H is “rich enough” in the sense that the following property holds: P (cid:0) C ([0 , T ]; H ) (cid:1) = (cid:8) P ξ with ξ : [0 , T ] × Ω → H continuous and B ([0 , T ]) ⊗ H -measurable process satisfying E (cid:2) k ξ k T (cid:3) < ∞ (cid:9) . Remark 2.2.
Using the same notations as in Lemma 2.1, if the probability space (Ω , H , P ) is atomless (namely, for any E ∈ H such that P ( E ) > there exists F ∈ H , F ⊂ E , such that < P ( F ) < P ( E ) ) then property 1), or equivalently 2), of Lemma 2.1 holds (see for instance [15,vol. I, p. 352]). Remark 2.3.
The additional randomness (other than B ) coming from the σ -algebra G is used forthe initial condition ξ of the state equation (2.2) (notice that it is necessary to consider random initial conditions in order to state and prove the dynamic programming principle, Theorem 3.4,where for instance the initial condition at time s is given by the random variable X t,ξ,α ). However,we remark that whenever t > the σ -algebra G can be replaced by F Bt ; in other words, the initialcondition ξ can be taken only F Bt -measurable. As a matter of fact, for every t > , it holds that:i) F Bt and σ ( B s − B t , s ≥ t ) are independent (in item i ) of (A G ) we have imposed the strongercondition that G and F B ∞ have to be independent; however, if we consider only the controlproblem with initial time t , then the assumption imposed here is enough.ii) F Bt satisfies the property of being “rich enough” or, equivalently, it satisfies property 1) ofLemma 2.1.Therefore, the σ -algebra G is really necessary only for the control problem with initial time t =0 . In fact, this allows to define the value function v (see (3.10) ) for every pair ( t, µ ) in [0 , T ] ×P ( C ([0 , T ]; H )) , with µ being the law of ξ . On the other hand, if we do not use G , v is defined on (0 , T ] × P ( C ([0 , T ]; H )) and for t = 0 is defined only at the Dirac measures. Proof of Lemma 2.1. = ⇒ µ ∈ P ( C ([0 , T ]; H )) . Our aim is to find a process ξ : [0 , T ] × Ω → H continuous and B ([0 , T ]) ⊗ H -measurable with law equal to µ . To this end,consider the probability space ([0 , , B ([0 , , λ ) , where λ denotes the Lebesgue measure on the6nit interval. Given such a µ , it follows from Theorem 3.19 in [39] that there exists a measurablefunction Ξ : [0 , → C ([0 , T ]; H ) such that the image (or push forward) measure of λ by Ξ is equalto µ . Now, denote ξ t ( ω ) := Ξ( U H ( ω )) t , ∀ ( t, ω ) ∈ [0 , T ] × Ω , where the subscript in Ξ( U H ( ω )) t denotes the valuation at time t of the continuous function Ξ( U H ( ω )) .Notice that ξ is a continuous process with law equal to µ . Moreover, for every fixed t ∈ [0 , T ] , ξ t is H -measurable. Since ξ has continuous paths, it follows that ξ is also B ([0 , T ]) ⊗ H -measurable(see for instance [21], Chapter IV, Theorem 15). This concludes the proof of the implication 1) = ⇒ = ⇒ ([0 , , B ([0 , , λ ) . Fix an orthonormal basis { e n } n ∈ N of H . Let I : [0 , → C ([0 , T ]; H ) be the map defined as I ( r ) := re , ∀ r ∈ [0 , , where re is the constant path of C ([0 , T ]; H ) identically equal to re . Clearly I is continuous, hencemeasurable. Let ˜ µ denote the image measure of λ by I . Notice that ˜ µ ∈ P ( C ([0 , T ]; H )) . From2) it follows that there exists a continuous and B ([0 , T ]) ⊗ H -measurable process ˜ ξ : [0 , T ] × Ω → H with law equal to ˜ µ .Now, define the map Π : C ([0 , T ]; H ) → R as Π( x ) := h x , e i H , ∀ x ∈ C ([0 , T ]; H ) , where we recall that x is the value of the path x at time t = 0 . Consider the random variable U H : Ω → R given by U H := Π( ˜ ξ ) . Notice that U H is H -measurable. Moreover, U H has uniformdistribution on [0 , . As a matter of fact, it holds that P (cid:0) U H ∈ B (cid:1) = P (cid:0) Π( ˜ ξ ) ∈ B (cid:1) = P (cid:0) h ˜ ξ , e i H ∈ B (cid:1) = ˜ µ (cid:0)(cid:8) x ∈ C ([0 , T ]; H ) : h x , e i H ∈ B (cid:9)(cid:1) = λ (cid:0)(cid:8) r ∈ [0 ,
1] : h re , e i H ∈ B (cid:9)(cid:1) = λ ( B ∩ [0 , , for every Borel subset B of R .We denote by F = ( F t ) t ≥ the filtration defined as F t = G ∨ F Bt , t ≥ . Notice that F satisfies the usual conditions of completeness and right-continuity. We then denote by S ( F ) (resp. S ( G ) ) the set of H -valued continuous F -progressively measurable (resp. B ([0 , T ]) ⊗ G -measurable) processes ξ such that k ξ k S := E (cid:2) k ξ k T (cid:3) < ∞ . Control processes.
The space of control actions , denoted by U , satisfies the following standingassumption. Standing Assumption (A U ). U is a Borel space (see for instance Definition 7.7 in [8]), namelya Borel subset of some Polish space E . U denotes its Borel σ -algebra. emark 2.4. Our Assumption (A U ) is quite general. Indeed in most applications it is enough that U is a Polish space or even a Hilbert space, as in the examples of Example 2.12. Finally, we denote by U the space of control processes , namely the family of all F -progressivelymeasurable processes α : [0 , T ] × Ω → U . Assumptions on the coefficients of the state equation.
We consider a linear, possibly un-bounded, operator A : D ( A ) ⊂ H → H and two functions b, σ : [0 , T ] × C ([0 , T ]; H ) × P (cid:0) C ([0 , T ]; H ) (cid:1) × U × P (U) −→ H, L ( K ; H ) , where we recall that P (U) is endowed with the topology of weak convergence. We impose thefollowing assumptions on A , b , σ . Assumption (A
A,b,σ ). (i) A generates a C -semigroup of pseudo-contractions { e tA , t ≥ } in H . Hence, there existsand η ∈ R such that k e tA k L ( H ) ≤ e ηt . (2.1) (ii) The functions b and σ are measurable.(iii) There exists a constant L such that | b t ( x, µ, u, ν ) − b t ( x ′ , µ ′ , u, ν ) | H ≤ L (cid:0) k x − x ′ k t + W ( µ, µ ′ ) (cid:1) , k σ t ( x, µ, u, ν ) − σ t ( x ′ , µ ′ , u, ν ) k L ( K ; H ) ≤ L (cid:0) k x − x ′ k t + W ( µ, µ ′ ) (cid:1) , | b t (0 , δ , u, ν ) | H + k σ t (0 , δ , u, ν ) k L ( K ; H ) ≤ L, for all ( t, u, ν ) ∈ [0 , T ] × U × P (U) , ( x, µ ) , ( x ′ , µ ′ ) ∈ C ([0 , T ]; H ) × P ( C ([0 , T ]; H )) , with δ being the Dirac measure at , namely the probability measure on C ([0 , T ]; H ) putting massequal to to the constant path . Remark 2.5.
Notice that, from the Lipschitz property of b and σ with respect to the variable x ∈ C ([0 , T ]; H ) , it follows that b and σ satisfies the following non-anticipativity property: b t ( x, µ, u, ν ) = b t ( x ·∧ t , µ, u, ν ) , σ t ( x, µ, u, ν ) = σ t ( x ·∧ t , µ, u, ν ) , for every ( t, x, µ, u, ν ) ∈ [0 , T ] × C ([0 , T ]; H ) × P ( C ([0 , T ]; H )) × U × P (U) . Remark 2.6.
The optimal control problem of McKean-Vlasov SDEs is sometimes called extended or generalized (see [1, 18]) when the coefficients also depend on the law of the control process P α s (asin the present framework, see equation (2.2) below) or, more generally, on the joint law P ( X ·∧ s ,α s ) .Notice however that, under “standard” assumptions, the latter case is only apparently more generalthan our framework. As a matter of fact, consider for simplicity only the drift coefficient b andsuppose that it is replaced by a function ¯ b from [0 , T ] × C ([0 , T ]; H ) × U × P ( C ([0 , T ]; H ) × U) to H .Recall from (A A,b,σ ) that the Lipschitz continuity of b with respect to the law of the path reads as | b t ( x, µ, u, ν ) − b t ( x, µ ′ , u, ν ) | H ≤ L W ( µ, µ ′ ) , or all ( t, x, u, ν ) ∈ [0 , T ] × C ([0 , T ]; H ) × U × P (U) , µ, µ ′ ∈ P ( C ([0 , T ]; H )) . Then, it is reasonableto impose on ¯ b the following Lipschitz continuity with respect to the law of the path: | ¯ b t ( x, u, π ) − ¯ b t ( x, u, π ′ ) | H ≤ L W (cid:0) π ( · × U) , π ′ ( · × U) (cid:1) , for every ( t, x, u ) ∈ [0 , T ] × C ([0 , T ]; H ) × U , π, π ′ ∈ P ( C ([0 , T ]; H ) × U) with π ( · × U) , π ′ ( · × U) ∈ P ( C ([0 , T ]; H )) and π ( C ([0 , T ]; H ) × · ) = π ′ ( C ([0 , T ]; H ) × · ) . It follows directly from thisassumption that ¯ b t ( x, u, π ) = ¯ b t ( x, u, π ′ ) whenever the marginals of π and π ′ coincide: π ( · × U) = π ′ ( · × U) and π ( C ([0 , T ]; H ) × · ) = π ′ ( C ([0 , T ]; H ) × · ) . This shows that ¯ b depends on π only throughits marginals. Given an initial time t ∈ [0 , T ] , an initial path ξ ∈ S ( F ) , a control process α ∈ U , the state processevolves according to the following controlled path-dependent McKean-Vlasov stochastic differentialequation: ( dX s = AX s + b s (cid:0) X, P X , α s , P α s (cid:1) ds + σ s (cid:0) X, P X , α s , P α s (cid:1) dB s s > tX s = ξ s s ≤ t. (2.2) Definition 2.7.
Fix t ∈ [0 , T ] , ξ ∈ S ( F ) , α ∈ U . A mild solution of (2.2) is a process X =( X s ) s ∈ [0 ,T ] in S ( F ) satisfying X s = e (( s − t ) ∨ A ξ s ∧ t + Z s ∨ tt e ( s − r ) A b r (cid:0) X, P X ·∧ r , α r , P α r (cid:1) dr + Z s ∨ tt e ( s − r ) A σ r (cid:0) X, P X ·∧ r , α r , P α r (cid:1) dB r ∀ s ∈ [0 , T ] , P -a.s. Proposition 2.8.
Fix t ∈ [0 , T ] , ξ ∈ S ( F ) , α ∈ U . Under (A A,b,σ ) , equation (2.2) admits a uniquemild solution X t,ξ,α ∈ S ( F ) . The map [0 , T ] × S ( F ) → S ( F ) , ( t, ξ ) X t,ξ,α is jointly continuous in ( t, ξ ) , uniformly with respect to α ∈ U , and Lipschitz continuous in ξ ,uniformly in t and α . Moreover, X t,ξ,α = X t,ξ ·∧ t ,α and there exists a constant C , independent of t, ξ, α , such that (cid:13)(cid:13) X t,ξ,α (cid:13)(cid:13) S ≤ C (cid:0) k ξ ·∧ t k S (cid:1) . (2.3) Proof.
See Appendix A.
Remark 2.9.
From Proposition 2.8 we have, in particular, for r, t ∈ [0 , T ] , ξ ∈ S ( F ) , lim r → t sup α ∈U k X α,t,ξr ∧· − ξ t ∧· k S ≤ lim r → t (cid:18) sup α ∈U k X α,t,ξr ∧· − X α,r,ξr ∧· k S + k ξ r ∧· − ξ t ∧· k S (cid:19) ≤ lim r → t (cid:18) sup α ∈U k X α,t,ξ − X α,r,ξ k S + k ξ r ∧· − ξ t ∧· k S (cid:19) = 0 , (2.4) where we have used the fact that X α,r,ξr ∧· = ξ r ∧· . { A n } n ∈ N be the Yosida approximation of A , i.e. A n = nA ( n − A ) − , for n ∈ N , n > η , with η as in (2.1). Denote by S n the uniformly continuous semigroup generated by A n . Notice that S n is a pseudo-contraction semigroup for all n ∈ N , n > η , and that, for some ˜ η > , k S nt k L ( H ) ≤ e ˜ ηt uniformly for n ∈ N , t ≥ . In particular, we can apply Proposition 2.8 to obtain existence of aunique mild solution X n,t,ξ,α to the following equation: ( dX ns = A n X ns + b s (cid:0) X n , P X n , α s , P α s (cid:1) ds + σ s (cid:0) X n , P X n , α s , P α s (cid:1) dB s , s > t,X ns = ξ s , s ≤ t. (2.5) Proposition 2.10.
There exists a constant
C > such that sup α ∈U n ∈ N t ∈ [0 ,T ] k X n,t,ξ,α − X n,t,ξ ′ ,α k S ≤ C k ξ − ξ ′ k S , ∀ ξ, ξ ′ ∈ S ( F ) . (2.6) Moreover, lim t → t ′ n →∞ k X n,t,ξ,α − X t ′ ,ξ,α k S = 0 , ∀ α ∈ U , ξ ∈ S ( F ) , t ′ ∈ [0 , T ] . (2.7) Proof.
See Appendix A.
Remark 2.11.
In the third inequality of Assumption (A A,b,σ ) -(iv) we assume, for simplicity, theboundedness of b and σ with respect to the controls. In many applications (e.g. in linear quadraticcontrol cases see Example 2.12 below) the state equation contains unbounded control terms. It istherefore interesting to understand what happens in such cases. Assume that U is a closed subsetof a Hilbert space and that the right-hand side of the third inequality of Assumption (A A,b,σ ) -(iv) isreplaced by L (1 + | u | U ) . In this case Proposition 2.8 still holds, assuming that α · ∈ L ([0 , T ] × Ω; U) ,with the estimate (2.3) replaced by (cid:13)(cid:13) X t,ξ,α (cid:13)(cid:13) S ≤ C (cid:20) k ξ ·∧ t k S + (cid:18) E Z Tt | α s | ds (cid:19)(cid:21) . (2.8) The only difference is that the joint continuity in ( t, ξ ) is uniform only with respect to α belongingto the bounded sets of L ([0 , T ] × Ω; U) . Consequently also the limit in (2.4) is uniform only in thesame sense. Similarly, also Proposition 2.10 still holds but with the supremum in α belonging to thebounded sets of L ([0 , T ] × Ω; U) . Example 2.12.
Examples of problems where the state equation has the above structure.(i) Lifecycle optimal portfolio problem. In such problems (as it is done in [9, 22]), it is naturalto model the dynamics of the labor income “ y ” (which is one of the state equations of theoptimal portfolio problem) using one-dimensional stochastic delay ODE of McKean-Vlasovtype as follows (here φ ∈ L ( − d, is a given datum and Z is a one-dimensional Brownianmotion). dy ( t ) = (cid:20) b (cid:0) P y ( t ) (cid:1) + Z − d y ( t + ξ ) φ ( ξ ) dξ (cid:21) dt + σ y ( t ) dZ ( t ) . Such equations can be rephrased as SDEs in the Hilbert space R × L ( − d, and the resultingdynamics falls into the class treated in the present section. The equation for y is not controlled, however y is part of the state as it appears in the wealth dynamics, whichis controlled. ii) Optimal investment with vintage capital. In such problems (see e.g. [5, 6, 27, 26]) the statevariable “ x ” is the capital and the control variable is the investment “ u ” both depending on timeand vintage. In the above papers x is required to satisfy a first-order PDE which is rewrittenas an ODE in the space L (0 , ¯ s ) for some given ¯ s > (the maximum vintage). If one wantsto take into account stochastic disturbances, the state equation, written in L (0 , ¯ s ) , becomes(here B is a cylindrical Wiener process): dx ( t ) = [ Ax ( t ) + δx ( t ) + Cu ( t )] dt + σ dB ( t ) , for suitable linear operators A and C and real parameters δ and σ . If one aims to take accountof the behavior of the other investors one has to require that the coefficients of such SPDE (as δ or the ones hidden in A and C here) depend on the distribution of k and u and, possibly, ontheir past values. If the path-dependency arises only for the state variable k , such an equationfalls in the class treated here. A simple way of including the above features would be to requirethe discount factor δ to depend on both the path of k and its law, i.e. δ = δ (cid:0) k ( · ∧ t ) , P k ( ·∧ t ) (cid:1) . We are given two functions f : [0 , T ] × C ([0 , T ]; H ) × P (cid:0) C ([0 , T ]; H ) (cid:1) × U × P (U) −→ R g : C ([0 , T ]; H ) × P (cid:0) C ([0 , T ]; H ) (cid:1) −→ R on which we impose the following assumptions. Assumption (A f,g ). (i) The functions f and g are measurable.(ii) The function f satisfies the non-anticipativity property: f t ( x, µ, u, ν ) = f t ( x ·∧ t , µ, u, ν ) , for all ( t, x, µ, u, ν ) ∈ [0 , T ] × C ([0 , T ]; H ) × P ( C ([0 , T ]; H )) × U × P (U) .(iii) There exists a locally bounded function h : [0 , ∞ ) → [0 , ∞ ) such that | f t ( x, µ, u, ν ) | ≤ h (cid:0) W ( µ, δ ) (cid:1)(cid:0) k x k t (cid:1) , | g ( x, µ ) | ≤ h (cid:0) W ( µ, δ ) (cid:1)(cid:0) k x k T (cid:1) , for all ( t, x, µ, u, ν ) ∈ [0 , T ] × C ([0 , T ]; H ) × P ( C ([0 , T ]; H )) × U × P (U) . We will also need the following continuity assumption on f and g . Assumption (A f,g ) cont . The function f is locally uniformly continuous in ( x, µ ) uniformly withrespect to ( t, u, ν ) . Similarly, g is locally uniformly continuous. More precisely, it holds that: forevery ε > and n ∈ N there exists δ = δ ( ε, n ) > such that, for every ( t, u, ν ) ∈ [0 , T ] × U × P (U) , ( x, µ ) , ( x ′ , µ ′ ) ∈ C ([0 , T ]; H ) ×P ( C ([0 , T ]; H )) , with k x k T + W ( µ, δ ) ≤ n and k x ′ k T + W ( µ ′ , δ ) ≤ n , k x − x ′ k t + W ( µ, µ ′ ) ≤ δ = ⇒ | f ( t, x, µ, u, ν ) − f ( t, x ′ , µ ′ , u, ν ) | ≤ ε and | g ( x, µ ) − g ( x ′ , µ ′ ) | ≤ ε. (A A,b,σ ) and (A f,g ) , from Proposition 2.8 we get that the reward functional J , given by J ( t, ξ, α ) = E (cid:20) Z Tt f s (cid:0) X t,ξ,α , P X t,ξ,α ·∧ s , α s , P α s (cid:1) ds + g (cid:0) X t,ξ,α , P X t,ξ,α (cid:1)(cid:21) , is well-defined for any ( t, ξ, α ) ∈ [0 , T ] × S ( F ) × U . We then consider the function V : [0 , T ] × S ( F ) −→ R , to which we refer as the lifted value function , defined as V ( t, ξ ) = sup α ∈U J ( t, ξ, α ) , ∀ ( t, ξ ) ∈ [0 , T ] × S ( F ) . (3.1) Remark 3.1.
Recall from Proposition 2.8 that X t,ξ,α only involves the values of ξ up to time t ,namely it holds that X t,ξ,α = X t,ξ ·∧ t ,α . As a consequence, both J and V satisfy the non-anticipativityproperty: J ( t, ξ, α ) = J ( t, ξ ·∧ t , α ) , V ( t, ξ ) = V ( t, ξ ·∧ t ) , for every ( t, ξ ) ∈ [0 , T ] × S ( F ) , α ∈ U . Remark 3.2.
In Remark 2.11 we looked at the case when U is a Hilbert space and the coefficientsof the state equation have linear growth in the controls. In such a case, in order to have a well-defined reward functional J , we need some compensating term in the current reward f . A typicalassumption which guarantees that J is well-defined is that the first inequality of Assumption (A f,g ) -(iii) is replaced by f t ( x, µ, u, ν ) ≤ h (cid:0) W ( µ, δ ) (cid:1) (cid:0) k x k t (cid:1) − C | u | θ U , for some θ > and C > . This would include some typical linear-quadratic control cases. Forexample in the case of optimal investment problems mentioned in Example 2.12 a typical form of f would be (recall that here H = L (0 , ¯ s ) ) f t ( x, µ, u, ν ) = f t ( x, u ) = e − rt [ h a , x ( t ) i H − h a , u ( t ) i H − h M u ( t ) , u ( t ) i H ] , (3.2) where r is the interest rate, a , a ∈ L (0 , ¯ s ) , and M is a suitable multiplication operator in L (0 , ¯ s ) . Proposition 3.3.
Suppose that (A A,b,σ ) and (A f,g ) hold. The function V satisfies a quadraticgrowth condition: there exists a constant C such that | V ( t, ξ ) | ≤ C (cid:0) k ξ ·∧ t k S (cid:1) , (3.3) for every ( t, ξ ) ∈ [0 , T ] × S ( F ) . Moreover, if in addition (A f,g ) cont holds, the map V : [0 , T ] × S ( F ) → R is jointly continuous. Proof.
We split the proof into two steps.
Step 1 . Proof of estimate (3.3) . From the definition of J , we have | J ( t, ξ, α ) | ≤ E (cid:20) Z Tt (cid:12)(cid:12) f s (cid:0) X t,ξ,α , P X t,ξ,α ·∧ s , α s , P α s (cid:1)(cid:12)(cid:12) ds (cid:21) + E (cid:2)(cid:12)(cid:12) g (cid:0) X t,ξ,α , P X t,ξ,α (cid:1)(cid:12)(cid:12)(cid:3) . By the quadratic growth of f and g , together with estimate (2.3), we see that there exists a constant C such that | J ( t, ξ, α ) | ≤ C (cid:0) k ξ ·∧ t k S (cid:1) , (3.4)12or every ( t, ξ, α ) ∈ [0 , T ] × S ( F ) × U . Then, estimate (3.3) follows directly from the definition of V and the fact that (3.4) holds uniformly with respect to α ∈ U . Step 2 . Continuity of V . We begin noticing that, for every ( t, ξ ) , ( s, η ) ∈ [0 , T ] × S ( F ) , | V ( t, ξ ) − V ( s, η ) | ≤ sup α ∈U | J ( t, ξ, α ) − J ( s, η, α ) | . Then, the continuity of V follows once we prove that J is continuous in ( t, ξ ) uniformly with respectto α , namely that the following property holds: for every ε > and every ( t, ξ ) ∈ [0 , T ] × S ( F ) ,there exists δ = δ ( ε, t, ξ ) > such that, for every ( s, η, α ) ∈ [0 , T ] × S ( F ) × U , | t − s | ≤ δ and k ξ − η k S ≤ δ = ⇒ | J ( t, ξ, α ) − J ( s, η, α ) | ≤ ε. Such a property is a straightforward consequence of the last statement of Proposition 2.8 and ofassumption (A f,g ) cont . V In this section we prove the dynamic programming principle for the lifted value function V definedin (3.1). Theorem 3.4.
Suppose that (A A,b,σ ) and (A f,g ) hold. The lifted value function V satisfies the dynamic programming principle : for every t, s ∈ [0 , T ] , with t ≤ s , and every ξ ∈ S ( F ) it holdsthat V ( t, ξ ) = sup α ∈U (cid:26) E (cid:20) Z st f r (cid:0) X t,ξ,α , P X t,ξ,α ·∧ r , α r , P α r (cid:1) dr (cid:21) + V (cid:0) s, X t,ξ,α (cid:1)(cid:27) . Proof.
Set Λ( t, ξ ) := sup α ∈U (cid:26) E (cid:20) Z st f r (cid:0) X t,ξ,α , P X t,ξ,α ·∧ r , α r , P α r (cid:1) dr (cid:21) + V (cid:0) s, X t,ξ,α (cid:1)(cid:27) . Step 1 . Proof of the inequality Λ( t, ξ ) ≥ V ( t, ξ ) . For every fixed α ∈ U , the lifted value function at ( s, X t,ξ,α ) is given by V ( s, X t,ξ,α ) = sup β ∈U E (cid:20) Z Ts f r (cid:16) X s,X t,ξ,α ,β , P X s,Xt,ξ,α,β ·∧ r , β r , P β r (cid:17) dr + g (cid:16) X s,X t,ξ,α ,β , P X s,Xt,ξ,α,β (cid:17)(cid:21) . Choosing β = α , we find V ( s, X t,ξ,α ) ≥ E (cid:20) Z Ts f r (cid:16) X s,X t,ξ,α ,α , P X s,Xt,ξ,α,α ·∧ r , α r , P α r (cid:17) dr + g (cid:16) X s,X t,ξ,α ,α , P X s,Xt,ξ,α,α (cid:17)(cid:21) . By the uniqueness property for equation (2.2) stated in Proposition 2.8, we obtain the flow property X t,ξ,α = X s,X t,ξ,α ,α . Hence V ( s, X t,ξ,α ) ≥ E (cid:20) Z Ts f r (cid:0) X t,ξ,α , P X t,ξ,α ·∧ r , α r , P α r (cid:1) dr + g (cid:0) X t,ξ,α , P X t,ξ,α (cid:1)(cid:21) . E R st f r ( X t,ξ,α , P X t,ξ,α ·∧ r , α r , P α r ) dr , we get Λ( t, ξ ) ≥ E (cid:20) Z Tt f r (cid:0) X t,ξ,α , P X t,ξ,α ·∧ r , α r , P α r (cid:1) dr + g (cid:0) X t,ξ,α , P X t,ξ,α (cid:1)(cid:21) . As the latter inequality holds true for every α ∈ U , we conclude that Λ( t, ξ ) ≥ V ( t, ξ ) . Step 2 . Proof of the inequality Λ( t, ξ ) ≤ V ( t, ξ ) . For every ε > , let α ε ∈ U be such that Λ( t, ξ ) ≤ E (cid:20) Z st f r (cid:0) X t,ξ,α ε , P X t,ξ,αε ·∧ r , α εr , P α εr (cid:1) dr (cid:21) + V ( s, X t,ξ,α ε ) + ε. (3.5)From the definition of V ( s, X t,ξ,α ε ) , it follows that there exists β ε ∈ U such that V ( s, X t,ξ,α ε ) ≤ E (cid:20) Z Ts f r (cid:16) X s,X t,ξ,αε ,β ε , P X s,Xt,ξ,αε ,βε ·∧ r , β εr , P β εr (cid:17) dr (3.6) + g (cid:16) X s,X t,ξ,αε ,β ε , P X s,Xt,ξ,αε ,βε (cid:17)(cid:21) + ε. Set γ ε = α ε [0 ,s ] + β ε ( s,T ] . Notice that γ ε ∈ U . Using again the uniqueness property for equation (2.2), we get X s,X t,ξ,αε ,β ε = X t,ξ,γ ε . Hence, (3.6) becomes V ( s, X t,ξ,α ε ) ≤ E (cid:20) Z Ts f r (cid:0) X t,ξ,γ ε , P X t,ξ,γε ·∧ r , γ εr , P γ εr (cid:1) dr + g (cid:0) X t,ξ,γ ε , P X t,ξ,γε (cid:1)(cid:21) + ε. Then, by (3.5) it follows that Λ( t, ξ ) ≤ E (cid:20) Z st f r (cid:0) X t,ξ,α ε , P X t,ξ,αε ·∧ r , α εr , P α εr (cid:1) dr + Z Ts f r (cid:0) X t,ξ,γ ε , P X t,ξ,γε ·∧ r , γ εr , P γ εr (cid:1) dr + g (cid:0) X t,ξ,γ ε , P X t,ξ,γε (cid:1)(cid:21) + 2 ε. (3.7)From the definition of γ ε , we see that X t,ξ,α ε ·∧ s = X t,ξ,γ ε ·∧ s . As a consequence, we can rewrite (3.7) in terms of the only process X t,ξ,γ ε as Λ( t, ξ ) ≤ E (cid:20) Z Tt f r (cid:0) X t,ξ,γ ε , P X t,ξ,γε ·∧ r , γ εr , P γ εr (cid:1) dr + g (cid:0) X t,ξ,γ ε , P X t,ξ,γε (cid:1)(cid:21) + 2 ε ≤ V ( t, ξ ) + 2 ε. The claim follows from the arbitrariness of ε . 14 .3 Law invariance property of the lifted value function V In the present section we introduce the value function of the optimal control problem, which is areal-valued map defined on [0 , T ] × P ( C ([0 , T ]; H )) (see (3.10)). In order to define such a valuefunction, it is necessary to prove that the map V satisfies the following law invariance property: forevery t ∈ [0 , T ] and every ξ, η ∈ S ( F ) it holds that V ( t, ξ ) = V ( t, η ) . This is the subject of the next theorem.
Theorem 3.5.
Suppose that (A A,b,σ ) and (A f,g ) hold. Fix t ∈ [0 , T ] and ξ, η ∈ S ( F ) , with P ξ = P η . Suppose that there exist two random variables U ξ and U η having uniform distribution on [0 , , being F t -measurable and such that ξ and U ξ (resp. η and U η ) are independent. Then, it holdsthat V ( t, ξ ) = V ( t, η ) . If in addition (A f,g ) cont holds, the map V satisfies the law invariance property : for every t ∈ [0 , T ] and every ξ, η ∈ S ( F ) , with P ξ = P η , it holds that V ( t, ξ ) = V ( t, η ) . Proof.
We split the proof into two steps.
Step 1.
Only (A A,b,σ ) and (A f,g ) hold. Fix t ∈ [0 , T ] , ξ, η ∈ S ( F ) , with P ξ = P η , and let U ξ , U η be F t -measurable random variables with uniform distribution on [0 , , such that ξ and U ξ (resp. η and U η ) are independent.By Remark 3.1 we can assume that ξ = ξ ·∧ t and η = η ·∧ t , so, in particular, both ξ and η are B ([0 , T ]) ⊗ F t -measurable (this is needed in order to apply Lemma B.2 of Appendix B). Now, given α ∈ U consider the function a : [0 , T ] × Ω × C ([0 , T ]; H ) × [0 , → U introduced in Lemma B.2.Recall from (B.1) that (cid:16) ( ξ s ) s ∈ [0 ,T ] , (a s ( ξ, U ξ )) s ∈ [ t,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) L = (cid:16) ( ξ s ) s ∈ [0 ,T ] , ( α s ) s ∈ [ t,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) , where L = stands for equality in law (between random objects defined on (Ω , F , P ) ). Then, noticethat (here we use again that ξ and η are B ([0 , T ]) ⊗ F t -measurable, so, in particular, they areindependent of ( B s − B t ) s ∈ [ t,T ] ) (cid:16) ( ξ s ) s ∈ [0 ,T ] , ( α s ) s ∈ [ t,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) L = (cid:16) ( η s ) s ∈ [0 ,T ] , ( β s ) s ∈ [ t,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) , (3.8)where β := (cid:0) a s ( η, U η ) (cid:1) s ∈ [0 ,T ] . Observe that β ∈ U and, by (3.8), (cid:0) ( X t,ξ,αs ) s ∈ [ t,T ] , ( α s ) s ∈ [ t,T ] (cid:1) L = (cid:0) ( X t,η,βs ) s ∈ [ t,T ] , ( β s ) s ∈ [ t,T ] (cid:1) , where the above equality in law can be deduced from (3.8) proceeding along the same lines as inthe proof of Proposition 1.137 in [25]. As a consequence, it holds that J ( t, ξ, α ) = J ( t, η, β ) . J ( t, ξ, α ) ≤ V ( t, η ) . From the arbitrariness of α , we deduce that V ( t, ξ ) ≤ V ( t, η ) . Changingthe roles of ξ and η we get the opposite inequality, from which we deduce that V ( t, ξ ) = V ( t, η ) . Step 2.
Assumptions (A A,b,σ ) , (A f,g ) , (A f,g ) cont hold. Fix t ∈ [0 , T ] and ξ, η ∈ S ( F ) , with P ξ = P η . As in the previous step, we exploit Remark 3.1 and take ξ = ξ ·∧ t , η = η ·∧ t (so, inparticular, both ξ and η are B ([0 , T ]) ⊗ F t -measurable; this is needed in order to apply Lemma B.3). Substep 2.1.
The discrete case.
Suppose that P ξ = m X i =1 p i δ x i , for some { x , . . . , x m } ⊂ C ([0 , T ]; H ) , with x i = x j if i = j , where δ x i is the Dirac measure at x i and p i > , with P mi =1 p i = 1 . Then, by Lemma B.3 there exist two F t -measurable random variables U ξ and U η , with uniform distribution on [0 , , such that ξ and U ξ (resp. η and U η ) are independent.The claim then follows from Step 1 . Substep 2.2.
The general case.
In the general case, we rely on the continuity of the map ξ V ( t, ξ ) , defined from S ( F ) into R , which follows from Proposition 3.3. More precisely, we proceedby approximating ξ and η . For n ∈ N , let { C ni } i ∈ N be a partition of C ([0 , T ]; H ) of Borel sets suchthat diam( C ni ) < − n . For each i ∈ N , choose x ni ∈ C ni . Then, define ˜ ξ n := ∞ X i =1 x ni C ni ( ξ ) , ˜ η n := ∞ X i =0 x ni C ni ( η ) . Notice that ˜ ξ n , ˜ η n ∈ S ( F ) and ˜ ξ n → ξ , ˜ η n → η uniformly with respect to ω ∈ Ω . Moreover, ˜ ξ n and ˜ η n have the same law. By a diagonal argument, we can choose N n ∈ N such that the sequences { ξ n } n ∈ N and { η n } n ∈ N , defined by ξ n := N n X i =1 x ni C ni ( ξ ) , η n := N n X i =1 x ni C ni ( η ) , (3.9)converge respectively to ξ and η , both P -a.s. and in L (Ω; C ([0 , T ]; H )) . From Substep 2.1 wehave V ( t, ξ n ) = V ( t, η n ) , ∀ n ∈ N . Then, using the continuity of V , we can pass to the limit as n → ∞ and conclude that V ( t, ξ ) = V ( t, η ) . Remark 3.6.
Suppose that (A A,b,σ ) , (A f,g ) , (A f,g ) cont hold. Thanks to the law invariance propertystated in Theorem 3.5, in the definition of V we can consider only ξ ∈ S ( G ) rather than ξ ∈ S ( F ) (recall that it was necessary to take ξ ∈ S ( F ) in order to state and prove the dynamic programmingprinciple, Theorem 3.4, where for instance the initial condition at time s is X t,ξ,α and X t,ξ,α ∈ S ( F ) but, in general, X t,ξ,α / ∈ S ( G ) ). Remark 3.7.
In the finite-dimensional and non-path-dependent case, the law invariance propertywas already addressed in [18], Proposition 3.1, under only (A A,b,σ ) and (A f,g ) . Notice however thatthe proof of such a proposition is based on the measurable selection theorem stated in [2], Corollary18.23, which is unfortunately not true. For this reason, Theorem 3.5 is also relevant in the finite-dimensional and non-path-dependent setting. Moreover, we emphasize that assuming only (A A,b,σ ) nd (A f,g ) is not enough for the validity of the law invariance property. To this regard, we give thefollowing example. Example.
Let T = 1 , H = R , K = R , U = [0 , . We consider a non-path-dependent setting. Thecoefficients b, σ, f : [0 , T ] × R × P ( R ) × [0 , × P ([0 , −→ R , R , R and g : R × P ( R ) → R are given by b t ( x, µ, u, ν ) := u , σ t ( x, µ, u, ν ) := , f t ( x, µ, u, ν ) := 0 , g ( x, µ ) := { µ = µ } for every ( t, x, µ, u, ν ) ∈ [0 , T ] × R × P ( R ) × [0 , × P ([0 , , where µ ∈ P ( R ) is the probabilitymeasure defined as µ := Unif (0 , ⊗ Bern (1 / ⊗ N (0 , , with Unif (0 , being the uniform distribution on [0 , , Bern (1 / the Bernoulli distribution withparameter / , N (0 , the standard Gaussian distribution.We now fix the probabilistic setting. Consider the probability spaces (Ω ◦ , F ◦ , P ◦ ) and (Ω , F , P ) where Ω ◦ = [0 , , F ◦ its Borel σ -algebra and P ◦ is the Lebesgue measure on the unit interval, while Ω = { ω ∈ C ([0 , R ) : ω = 0 } , F its Borel σ -algebra and P is the Wiener measure on (Ω , F ) .We then define Ω := Ω ◦ × Ω , F the completion of F ◦ ⊗ F with respect to P ◦ ⊗ P and by P theextension of P ◦ ⊗ P to F . We also denote by G := F ◦ ⊗ {∅ , Ω } the canonical extension of F ◦ tothe product space Ω . Finally, we denote by B = ( B t ) t ∈ [0 , the canonical process B t ( ω ◦ , ω ) := ω t , ∀ t ∈ [0 , . Notice that, under the probability measure P , the process B is a real-valued Brownianmotion. Then, in the present context the lifted value function is given by V ( t, ξ ) = sup α ∈U E h (cid:8) P Xt,ξ,α = µ (cid:9)i , ∀ ( t, ξ ) ∈ [0 , × L (Ω , F t , P ; R ) , where ξ = ξ ξ ξ and X t,ξ,α = ξ ξ + Z t α s dsξ + B − B t . Now, let ξ : Ω → R be given by ξ ( ω ◦ , ω ) := ω ◦ , ∀ ( ω ◦ , ω ) ∈ Ω . Notice that ξ has distribution Unif (0 , . Moreover, ξ is G -measurable and generates the σ -algebra G itself, namely G = σ ( ξ ) . Define η : Ω → R and Z : Ω → [0 , by η := ξ { ξ ≤ / } + (2 ξ − { ξ> / } , Z := { ξ ≤ / } . otice that η and Z are G -measurable and independent. Moreover, the first component of η , namely η , has distribution Unif (0 , , while Z has distribution Bern (1 / .Let us prove that V (0 , ξ ) = V (0 , η ) and, in particular, V (0 , ξ ) = 0 while V (0 , η ) = 1 . If theinitial condition at time t = 0 is η , then taking the control process α ∗ s = Z , ∀ s ∈ [0 , T ] , we get X ,η,α ∗ = η ZB . Notice that P X ,η,α ∗ = µ , which proves that V (0 , η ) = 1 . On the other hand, when the initialcondition is ξ , for every α ∈ U we have X ,ξ,α ( ω ◦ , ω ) = ω ◦ Z α s ( ω ◦ , ω ) dsω , ∀ ( ω ◦ , ω ) ∈ Ω . Observe that the random variable I := R α s ds cannot be independent of ( ξ, B ) , unless it is aconstant. This implies that P X ,ξ,α = µ , ∀ α ∈ U , therefore V (0 , ξ ) = 0 . In conclusion, under assumptions (A A,b,σ ) , (A f,g ) , (A f,g ) cont , we can define the value function v : [0 , T ] × P ( C ([0 , T ]; H )) → R as v ( t, µ ) = V ( t, ξ ) , ∀ ( t, µ ) ∈ [0 , T ] × P ( C ([0 , T ]; H )) , (3.10)for any ξ ∈ S ( F ) with P ξ = µ . By Theorem 3.4 we immediately deduce the dynamic programmingprinciple for v . Corollary 3.8.
Suppose that (A A,b,σ ) , (A f,g ) , (A f,g ) cont hold. The value function v satisfiesthe dynamic programming principle : for every t, s ∈ [0 , T ] , with t ≤ s , and every µ ∈P ( C ([0 , T ]; H )) it holds that v ( t, µ ) = sup α ∈U (cid:26) E (cid:20) Z st f r (cid:0) X t,ξ,α , P X t,ξ,α ·∧ r , α r , P α r (cid:1) dr (cid:21) + v (cid:0) s, P X t,ξ,α (cid:1)(cid:27) , for any ξ ∈ S ( F ) with P ξ = µ . Remark 3.9.
Recall from Remark 3.1 that V is non-anticipative, namely V ( t, ξ ) = V ( t, ξ ·∧ t ) , forevery ( t, ξ ) ∈ [0 , T ] × S ( F ) . As a consequence, the value function v satisfies the following non-anticipativity property: v ( t, µ ) = v ( t, µ [0 ,t ] ) , for every ( t, µ ) ∈ [0 , T ] × C ([0 , T ]; H ) , where we denote by µ [0 ,t ] the measure µ ◦ (cid:0) ( x s ) s ∈ [0 ,T ] ( x s ∧ t ) s ∈ [0 ,T ] (cid:1) − . This section is devoted to the proof of Itô’s formula for a real-valued function ϕ defined on [0 , T ] ×P ( C ([0 , T ]; H )) . Such a formula involves the so-called pathwise derivatives in the Wasserstein spacethat we now define. In the present section we substantially follow [51, Section 2] (see also [50]),extending their framework to our more general setting with H being a real separable Hilbert space(not necessarily a Euclidean space as in [51]). 18 .1 Notations In order to define the pathwise derivatives, we need to extend the canonical space C ([0 , T ]; H ) to thespace of càdlàg paths D ([0 , T ]; H ) , which we endow with the Skorokhod distance (in what follows,we denote paths in D ([0 , T ]; H ) using ˆ · in order to distinguish them from paths in C ([0 , T ]; H ) ; wedo the same with other mathematical objects) d Sk (ˆ x, ˆ x ′ ) := inf λ ∈ Λ sup t ∈ [0 ,T ] (cid:0) | t − λ ( t ) | + | ˆ x t − ˆ x ′ λ ( t ) | H (cid:1) , where Λ denotes the set of strictly increasing and continuous maps λ : [0 , T ] → [0 , T ] , satisfying λ (0) = 0 and λ ( T ) = T . We recall that ( D ([0 , T ]; H ) , d Sk ) is a Polish space. We define the spaces H := [0 , T ] × P (cid:0) C ([0 , T ]; H ) (cid:1) , ˆ H := [0 , T ] × P (cid:0) D ([0 , T ]; H ) (cid:1) . For every ( t, µ ) ∈ H , we denote by µ [0 ,t ] the measure µ ◦ (cid:0) ( x s ) s ∈ [0 ,T ] ( x s ∧ t ) s ∈ [0 ,T ] (cid:1) − . We definesimilarly ˆ µ [0 ,t ] , for every ( t, ˆ µ ) ∈ ˆ H . We then equip H and ˆ H with the following pseudo-distances,respectively: d H (cid:0) ( t, µ ) , ( t ′ , µ ′ ) (cid:1) := (cid:16) | t − t ′ | + W (cid:0) µ [0 ,t ] , µ ′ [0 ,t ′ ] (cid:1) (cid:17) , ( t, µ ) , ( t ′ , µ ′ ) ∈ H ,d ˆ H (cid:0) ( t, ˆ µ ) , ( t ′ , ˆ µ ′ ) (cid:1) := (cid:16) | t − t ′ | + W (cid:0) ˆ µ [0 ,t ] , ˆ µ ′ [0 ,t ′ ] (cid:1) (cid:17) , ( t, ˆ µ ) , ( t ′ , ˆ µ ′ ) ∈ ˆ H . Remark 4.1.
Notice that a function ϕ : H → R (resp. ˆ ϕ : ˆ H → R ) that is measurable with respectto d H (resp. d ˆ H ) satisfies the non-anticipativity property: ϕ ( t, µ ) = ϕ ( t, µ [0 ,t ] ) , ∀ ( t, µ ) ∈ H (cid:0) resp. ˆ ϕ ( t, ˆ µ ) = ˆ ϕ ( t, ˆ µ [0 ,t ] ) , ∀ ( t, ˆ µ ) ∈ ˆ H (cid:1) . We also introduce the lifted spaces H := [0 , T ] × L (cid:0) Ω; C ([0 , T ]; H ) (cid:1) , ˆ H := [0 , T ] × L (cid:0) Ω; D ([0 , T ]; H ) (cid:1) . Remark 4.2.
To alleviate notation, in the present Section 4 we work on the same probabilityspace (Ω , F , P ) adopted in the rest of the paper. Notice however that, for the definition of pathwisederivatives, (Ω , F , P ) can be replaced by any other probability space which supports a random variablehaving uniform distribution on [0 , . See also Remark 2.2. Finally, we introduce the following notation.
Notation (Ntn ˆ P ). For every ξ ∈ L (Ω; C ([0 , T ]; H )) , we denote by ˆ P ξ the law of ξ on D ([0 , T ]; H ) ,while we recall that P ξ denotes the law of ξ on C ([0 , T ]; H ) . So, in particular, ˆ P ξ ∈ P ( D ([0 , T ]; H )) ,while P ξ ∈ P ( C ([0 , T ]; H )) . Clearly, it holds that ˆ P ξ ( B ) = P ξ ( B ) , for every Borel subset B of C ([0 , T ]; H ) . Notice that in [51] the probability ˆ P ξ is denoted simply by P ξ (see the beginning ofSection 2.4 in [51]). .2 Pathwise derivatives for a map ˆ ϕ : ˆ H → R and Itô’s formula We start with the definition of pathwise time derivative for a map ˆ ϕ : ˆ H → R . Definition 4.3.
Let ˆ ϕ : ˆ H → R be a non-anticipative function. We say that ˆ ϕ is pathwise differ-entiable in time at ( t, ˆ µ ) ∈ ˆ H , with t < T , if the following limit exists and is finite: ∂ t ˆ ϕ ( t, ˆ µ ) := lim δ → + ˆ ϕ ( t + δ, ˆ µ [0 ,t ] ) − ˆ ϕ ( t, ˆ µ ) δ . At time t = T , we define ∂ t ˆ ϕ ( t, ˆ µ ) := lim t → T − ∂ t ˆ ϕ ( t, ˆ µ ) , when the limit exists and is finite. We refer to ∂ t ˆ ϕ as the pathwise time derivative (or horizontalderivative ) of ˆ ϕ at ( t, ˆ µ ) . If ∂ t ˆ ϕ exists everywhere as a function ˆ H → R , we refer to it as the pathwise time derivative of ˆ ϕ . Remark 4.4.
Notice that ∂ t ˆ ϕ is a non-anticipative function, namely ∂ t ˆ ϕ ( t, ˆ µ ) = ∂ t ˆ ϕ ( t, ˆ µ [0 ,t ] ) , forevery ( t, ˆ µ ) ∈ ˆ H . In order to define the pathwise measure derivative, we need to consider the lifting of ˆ ϕ . Definition 4.5.
Given ˆ ϕ : ˆ H → R , we say that ˆΦ : ˆ H → R is a lifting of ˆ ϕ if ˆΦ( t, ˆ ξ ) = ˆ ϕ ( t, P ˆ ξ ) , ∀ ( t, ˆ ξ ) ∈ ˆ H , where we recall that P ˆ ξ stands for the law of the random variable ˆ ξ ∈ L (Ω; D ([0 , T ]; H )) . Definition 4.6.
Let ˆΦ : ˆ H → R be a non-anticipative function, namely ˆΦ( t, ˆ ξ ) = ˆΦ( t, ˆ ξ ·∧ t ) , forevery ( t, ˆ ξ ) ∈ ˆ H . We say that ˆΦ is pathwise differentiable in space at ( t, ˆ ξ ) ∈ ˆ H if there exists D ˆΦ( t, ˆ ξ ) ∈ L (Ω; H ) such that lim Y → (cid:12)(cid:12)(cid:12) ˆΦ( t, ˆ ξ + Y [ t,T ] ) − ˆΦ( t, ˆ ξ ) − E (cid:2) h D ˆΦ( t, ˆ ξ ) , Y i H (cid:3)(cid:12)(cid:12)(cid:12) | Y | L (Ω; H ) = 0 . We refer to D ˆΦ( t, ˆ ξ ) as the pathwise space derivative (or vertical derivative ) of ˆΦ at ( t, ˆ ξ ) . If D ˆΦ exists everywhere as a function ˆ H → L (Ω; H ) , we refer to it as the pathwise space derivativeof ˆ ϕ . Remark 4.7.
Notice that, if ˆΦ is pathwise differentiable in space at ( t, ˆ ξ ) , then it is pathwisedifferentiable in space at ( t, ˆ ξ ′ ) for every ˆ ξ ′ ∈ L (Ω; D ([0 , T ]; H )) such that ˆ ξ t ∧· = ˆ ξ ′ t ∧· P -a.s., and,in such a case, D ˆΦ( t, ˆ ξ ) = D ˆΦ( t, ˆ ξ ′ ) P -a.s. We can then give, similarly to [14, Definition 5.22], the following definition.
Definition 4.8.
Let ˆ ϕ : ˆ H → R be a non-anticipative function and ( t, ˆ µ ) ∈ ˆ H . We say that ˆ ϕ is pathwise differentiable in measure at ( t, ˆ µ ) if its lifting ˆΦ is pathwise differentiable in space atsome ( t, ˆ ξ ) ∈ ˆ H such that P ˆ ξ = ˆ µ . oreover, we say that ˆ ϕ admits pathwise measure derivative at ( t, ˆ µ ) if its lifting ˆΦ ispathwise differentiable in space at every ( t, ˆ ξ ) ∈ ˆ H such that P ˆ ξ = ˆ µ and if there exists a masurablefunction ˆ g : D ([0 , T ]; H ) → H such that, for all ( t, ˆ ξ ) ∈ ˆ H with P ˆ ξ = ˆ µ , D ˆΦ( t, ˆ ξ ) = ˆ g ( ˆ ξ ) P -a.s. (4.1) The map ˆ g (which is ˆ µ -a.s. uniquely determined) is called pathwise measure derivative of ˆ ϕ at ( t, ˆ µ ) .Finally, if ˆ ϕ admits pathwise measure derivative at every ( t, ˆ µ ) ∈ ˆ H , then the function ∂ µ ˆ ϕ : ˆ H × D ([0 , T ]; H ) −→ H, ( t, ˆ µ, ˆ x ) ∂ µ ˆ ϕ ( t, ˆ µ, ˆ x ) such that, for every ( t, ˆ ξ ) ∈ ˆ H , ∂ µ ˆ ϕ ( t, P ˆ ξ , · ) is measurable and D ˆΦ( t, ˆ ξ ) = ∂ µ ˆ ϕ ( t, P ˆ ξ , ˆ ξ ) P -a.s., iscalled the pathwise measure derivative of ˆ ϕ . In the following lemma we state, under general conditions, the existence of the pathwise measurederivative. In order to do it, we proceed similarly to what is done in [51, Section 2.3] (see also [14,Section 5.3]) in the finite-dimensional case. The proof of the lemma is postponed in Appendix C.
Lemma 4.9.
Fix ( t, ˆ ξ ) ∈ ˆ H . Let ˆ ϕ : ˆ H → R be such that its lifting ˆΦ admits a continuous pathwisespace derivative D ˆΦ on the set { t } × O ˆ ξ , where O ˆ ξ is a neighborhood of ˆ ξ in L (Ω; D ([0 , T ]; H )) .Then, there exists a measurable function ˆ g : D ([0 , T ]; H ) → H and D ˆΦ( t, ˆ ξ ) = ˆ g ( ˆ ξ ) , P -a.s. (4.2) Let ˆ ξ ′ be such that P ˆ ξ ′ = P ˆ ξ . If in addition ˆΦ admits a continuous pathwise space derivative on theset { t } × O ˆ ξ ′ , with O ˆ ξ ′ being a neighborhood of ˆ ξ ′ , then (4.2) holds true with ˆ ξ replaced by ˆ ξ ′ .Hence, if the pathwise space derivative D ˆΦ( t, ˆ ξ ) at ( t, ˆ ξ ) exists for every ( t, ˆ ξ ) ∈ ˆ H and if D ˆΦ iscontinuous, then there exists the pathwise measure derivative ∂ µ ˆ ϕ of ˆ ϕ .If in addition the map ∂ µ ˆ ϕ ( t, · , · ) : P ( D ([0 , T ]; H )) × D ([0 , T ]; H ) → R is continuous for every t ∈ [0 , T ] , then ∂ µ ˆ ϕ is uniquely defined.Finally,assume that the pathwise space derivative D ˆΦ exists everywhere and is uniformly con-tinuous. Then ∂ µ ˆ ϕ is measurable. Remark 4.10 ( Non-anticipativity property of ∂ µ ˆ ϕ . ) . Let ˆ ϕ : ˆ H → R be a non-anticipativefunction. Suppose that there exists the pathwise measure derivative of ˆ ϕ . Then, thanks to Remark4.7 and equality (4.1) , ∂ µ ˆ ϕ is a non-anticipative function in the sense that, for every ( t, ˆ ξ ) ∈ ˆ H , ∂ µ ˆ ϕ ( t, P ˆ ξ )( ˆ ξ ) = ∂ µ ˆ ϕ ( t, ( P ˆ ξ ) [0 ,t ] )( ˆ ξ ·∧ t ) P -a.s.or, equivalently, ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) = ∂ µ ˆ ϕ ( t, ˆ µ [0 ,t ] )(ˆ x ·∧ t ) , ˆ µ -a.e.for every ( t, ˆ µ, ˆ x ) ∈ ˆ H × D ([0 , T ]; H ) . Notice that, if a function is pathwise differentiable in measure at some point and the related pathwise spacederivative is continuous (at least in a neighborhood), then it admits the pathwise measure derivative at that point,see e.g. [13, Theorem 6.5] or [15, Proposition 5.25] in finite dimension and our Lemma 4.9. Without the continuityassumption of the pathwise space derivative such a result is not obvious. We stress the fact that ∂ µ ˆ ϕ ( t, P ˆ ξ , · ) is a-priori uniquely determined only P ˆ ξ -a.s. ∂ x ∂ µ ˆ ϕ . Definition 4.11.
Let ˆ ϕ : ˆ H → R be a non-anticipative function and ( t, ˆ µ ) ∈ ˆ H . Suppose that:1) there exists the pathwise measure derivative ∂ µ ˆ ϕ ;2) for every t ∈ [0 , T ] , the map ∂ µ ˆ ϕ ( t, · )( · ) : P ( D ([0 , T ]; H )) × D ([0 , T ]; H ) → H is continuous(hence, by Lemma 4.9, ∂ µ ˆ ϕ is uniquely determined).Given ˆ x ∈ D ([0 , T ]; H ) , we say that ˆ ϕ is pathwise differentiable in measure and space at ( t, ˆ µ, ˆ x ) if there exists an operator ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) ∈ L ( H ) such that lim h → (cid:12)(cid:12) ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x + h [ t,T ] ) − ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) − ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) h (cid:12)(cid:12) H | h | H = 0 . We refer to ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) as the second-order pathwise derivative in measure and space of ˆ ϕ at ( t, ˆ µ, ˆ x ) .If ∂ x ∂ µ ˆ ϕ exists everywhere as function ˆ H × D ([0 , T ]; H ) → L ( H ) , we refer to it as the pathwisederivative in measure and space of ˆ ϕ . Remark 4.12.
Recalling Remark 4.10, we see that ∂ x ∂ µ ˆ ϕ is a non-anticipative function, namelyit holds that ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )( ˆ ξ ) = ∂ x ∂ µ ˆ ϕ ( t, ˆ µ [0 ,t ] )( ˆ ξ ·∧ t ) , P -a.s., for every ( t, ˆ ξ ) ∈ ˆ H , with P ˆ ξ = ˆ µ , or,equivalently, ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) = ∂ x ∂ µ ˆ ϕ ( t, ˆ µ [0 ,t ] )(ˆ x ·∧ t ) , ˆ µ -a.e.for every ( t, ˆ µ, ˆ x ) ∈ ˆ H × D ([0 , T ]; H ) . Definition 4.13.
We denote by C , ( ˆ H ) the set of non-anticipative functions ˆ ϕ : ˆ H → R suchthat: the lifting ˆΦ of ˆ ϕ admits a continuous pathwise space derivative D ˆΦ on ˆ H (hence, by Lemma 4.9,there exists the pathwise measure derivative ∂ µ ˆ ϕ ); ˆ ϕ, ∂ µ ˆ ϕ are continuous; there exist the pathwise time derivative ∂ t ˆ ϕ , the second-order pathwise derivative in measure andspace ∂ x ∂ µ ˆ ϕ , and ∂ t ˆ ϕ, ∂ x ∂ µ ˆ ϕ are continuous. Definition 4.14.
We denote by C , b ( ˆ H ) the set of ˆ ϕ ∈ C , ( ˆ H ) such that ˆ ϕ , ∂ t ˆ ϕ , ∂ µ ˆ ϕ , ∂ x ∂ µ ˆ ϕ are bounded. We end this section with the Itô formula. The proof is postponed in Appendix D.
Theorem 4.15.
Fix t ∈ [0 , T ] and let ξ ∈ S ( F ) . Let also F : [0 , T ] × Ω → H , G : [0 , T ] × Ω →L ( K ; H ) be square-integrable and F -progressively measurable processes, so, in particular, Z T E [ | F s | H ] ds < ∞ , Z T E (cid:2) Tr ( G s G ∗ s ) (cid:3) ds < ∞ . onsider the process X = ( X s ) s ∈ [0 ,T ] given by X s = ξ s ∧ t + Z s ∨ tt F r dr + Z s ∨ tt G r dB r , ∀ s ∈ [0 , T ] . (4.3) If ˆ ϕ : ˆ H → R is in C , b ( ˆ H ) , then the following Itô formula holds: ˆ ϕ ( s, ˆ P X ·∧ s ) = ˆ ϕ ( t, ˆ P ξ ·∧ t ) + Z st ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) dr + Z st E h h F r , ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) i H i dr + 12 Z st E h Tr (cid:16) G r G ∗ r ∂ x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:17)i dr, (4.4) for every s ∈ [ t, T ] (for the definition of ˆ P X ·∧ r see (Ntn ˆ P ) ). Remark 4.16.
Proceeding along the same lines as in the proof of Theorem 4.15, it is possibleto prove Itô’s formula for a larger class of functions than C , b ( ˆ H ) , for example weakening theboundedness assumption of ∂ µ ˆ ϕ by assuming linear growth with respect to ˆ ξ . Remark 4.17.
Following [51], we notice that in the last term of Itô’s formula (4.4) we can re-place ∂ x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) by its symmetrization ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) , where ∂ sym x ∂ µ ˆ ϕ : ˆ H × D ([0 , T ]; H ) → L ( H ) is defined as ∂ sym x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) := 12 (cid:16) ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) + (cid:0) ∂ x ∂ µ ˆ ϕ ( t, ˆ µ )(ˆ x ) (cid:1) ∗ (cid:17) , (4.5) for every ( t, ˆ µ, ˆ x ) ∈ ˆ H × D ([0 , T ]; H ) . ϕ : H → R and Itô’s formula In the present section we use several times the notation (Ntn ˆ P ) , namely we use the superscript ˆ · to denote the natural extension to D ([0 , T ]; H ) of a probability measure on C ([0 , T ]; H ) . Definition 4.18.
Given ϕ : H → R and a non-anticipative map ˆ ϕ : ˆ H → R , we say that ˆ ϕ is consistent with ϕ if (for the definition of ˆ P ξ see (Ntn ˆ P ) ) ϕ ( t, P ξ ) = ˆ ϕ ( t, ˆ P ξ ) , (4.6) for every ( t, ξ ) ∈ H , with ξ ∈ S ( F ) (namely, ξ ∈ L (Ω; C ([0 , T ]; H )) and it is F -progressivelymeasurable). Remark 4.19.
Notice that we can replace S ( F ) with S ( G ) in Definition 4.18, as a matter of factthe sets { P ξ : ξ ∈ S ( F ) } and { P ξ : ξ ∈ S ( G ) } are equal and coincide with P ( C ([0 , T ]; H )) , as itfollows from property (A G ) -ii). This also shows that equality (4.6) characterizes ϕ in terms of ˆ ϕ for every pair ( t, µ ) ∈ H . Next result is crucial in order to define pathwise derivatives for a map ϕ : H → R , as it statesa consistency property for the pathwise derivatives themselves. In what follows, we will always implicitly refer to Itô processes only by continuous versions. emma 4.20. Let ˆ ϕ , ˆ ϕ ∈ C , b ( ˆ H ) . If (for the definition of ˆ P ξ see (Ntn ˆ P ) ) ˆ ϕ ( t, ˆ P ξ ) = ˆ ϕ ( t, ˆ P ξ ) , ∀ ( t, ξ ) ∈ H , with ξ ∈ S ( F ) , then ∂ t ˆ ϕ ( t, ˆ P ξ ) = ∂ t ˆ ϕ ( t, ˆ P ξ ) , (4.7) ∂ µ ˆ ϕ ( t, ˆ P ξ )( ξ ) = ∂ µ ˆ ϕ ( t, ˆ P ξ )( ξ ) , P -a.s. (4.8) ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ )( ξ ) = ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ )( ξ ) , P -a.s. (4.9) for every ( t, ξ ) ∈ H , with ξ ∈ S ( F ) , where ∂ sym x ∂ µ ˆ ϕ is defined by (4.5) . Proof.
See Appendix E.
Remark 4.21.
We do not address here the consistency property of ∂ x ∂ µ ˆ ϕ (for hints on its proof werefer to [51], see the paragraph just after Theorem 2.9), as Itô’s formula (and hence the Hamilton-Jacobi-Bellman equation) only depends on ∂ sym x ∂ µ ˆ ϕ , see Remark 4.17. Remark 4.22.
By the non-anticipativity property of the pathwise derivatives (see Remarks 4.4,4.10, 4.12), it follows that equalities (4.7) - (4.8) - (4.9) hold if and only if ∂ t ˆ ϕ ( t, ˆ P ξ ·∧ t ) = ∂ t ˆ ϕ ( t, ˆ P ξ ·∧ t ) ,∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) = ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) , P -a.s. ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) = ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) , P -a.s. for every ( t, ξ ) ∈ H , with ξ ∈ S ( F ) . Using Lemma 4.20, we can now define the class C , b ( H ) . Definition 4.23.
We denote by C , ( H ) (respectively C , b ( H ) ) the set of maps ϕ : H → R forwhich there exists ˆ ϕ : ˆ H → R such that ˆ ϕ is consistent with ϕ and ˆ ϕ ∈ C , ( ˆ H ) (respectively C , b ( ˆ H ) ). Then, we define (for the definition of ˆ P ξ see (Ntn ˆ P ) ) ∂ t ϕ ( t, P ξ ) := ∂ t ˆ ϕ ( t, ˆ P ξ ) ,∂ µ ϕ ( t, P ξ )( · ) := ∂ µ ˆ ϕ ( t, ˆ P ξ )( · ) ,∂ sym x ∂ µ ϕ ( t, P ξ )( · ) := ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ )( · ) , for every ( t, ξ ) ∈ H , with ξ ∈ S ( F ) . We can finally state the Itô formula.
Theorem 4.24.
Fix t ∈ [0 , T ] and let ξ ∈ S ( F ) . Let also F : [0 , T ] × Ω → H , G : [0 , T ] × Ω →L ( K ; H ) be square integrable and F -progressively measurable process, so, in particular, Z T E [ | F s | H ] ds < ∞ , Z T E (cid:2) Tr ( G s G ∗ s ) (cid:3) ds < ∞ . onsider the process X = ( X s ) s ∈ [0 ,T ] given by X s = ξ s ∧ t + Z s ∨ tt F r dr + Z s ∨ tt G r dB r , ∀ s ∈ [0 , T ] . If ϕ : H → R is in C , b ( H ) , then the following Itô formula holds: ϕ ( s, P X ·∧ s ) = ϕ ( t, P ξ ·∧ t ) + Z st ∂ t ϕ ( r, P X ·∧ r ) dr + Z st E [ h F r , ∂ µ ϕ ( r, P X ·∧ r )( X ·∧ r ) i H ] dr + 12 Z st E [Tr ( G r G ∗ r ∂ x ∂ µ ϕ ( r, P X ·∧ r )( X ·∧ r ))] dr, (4.10) for every s ∈ [ t, T ] . Proof.
By Definition 4.23 there exists ˆ ϕ : ˆ H → R such that ˆ ϕ is consistent with ϕ and ˆ ϕ ∈ C , b ( ˆ H ) . As a consequence, by Theorem 4.15 we have the Itô formula ˆ ϕ ( s, ˆ P X ·∧ s ) = ˆ ϕ ( t, ˆ P ξ ·∧ t ) + Z st ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) dr + Z st E h h F r , ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) i H i dr + 12 Z st E h Tr (cid:16) G r G ∗ r ∂ x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:17)i dr, for every s ∈ [ t, T ] . Using the fact that ˆ ϕ is consistent with ϕ and recalling the definition of pathwisederivatives of ϕ (see Definition 4.23), we obtain the claimed Itô formula for ϕ .In order to apply Itô’s formula to our case we need the following variant of Theorem 4.24 (for asimilar result, see Proposition 1.165 in [25]). Theorem 4.25.
Fix t ∈ [0 , T ] and let ξ ∈ S ( F ) (namely, ξ ∈ L (Ω; C ([0 , T ]; H )) and it is F -progressively measurable). Let A, b, σ be as in Assumption (A A,b,σ ) . Let X = X t,ξ,α be the uniquemild solution of (2.2) . Let ϕ : H → R belong to C , b ( H ) . Assume also that, for all ( t, µ, x ) ∈ H × C ([0 , T ]; H ) , ∂ µ ϕ ( t, µ )( x ) ∈ D ( A ∗ ) and that the map H × C ([0 , T ]; H ) −→ H ( t, µ, x ) A ∗ ∂ϕ µ ( t, µ )( x ) is continuous and bounded .. Then the following variant of Itô formula holds: ϕ ( s, P X ·∧ s ) = ϕ ( t, P ξ ·∧ t ) + Z st ∂ t ϕ ( r, P X ·∧ r ) dr + Z st E [ h X r , A ∗ ∂ µ ϕ ( r, P X ·∧ r )( X ·∧ r ) i H ] dr + Z st E [ h b r ( X, P X ·∧ r , α r , P α r ) , ∂ µ ϕ ( r, P X ·∧ r )( X ·∧ r ) i H ] dr (4.11) + 12 Z st E [Tr ( σ r ( X, P X ·∧ r , α r , P α r ) σ ∗ r ( X, P X ·∧ r , α r , P α r ) ∂ x ∂ µ ϕ ( r, P X ·∧ r )( X ·∧ r ))] dr, for every s ∈ [ t, T ] . Indeed, in view of (2.3) here we could ask only linear growth of ∂ µ ϕ in x roof. The proof can be done proceeding along the same lines as in the proof of Proposition 1.165in [25]. We provide a sketch of proof.First of all if A is a bounded operator from Theorem 4.24 we immediately get (4.11). Now take A possibly unbounded and consider its Yosida approximations A n , for n ∈ N . Call, as in (2.5), X n the solution of the state equation when A is replaced by A n . Then, from (4.11) we get ϕ ( s, P X n ·∧ s ) = ϕ ( t, P ξ ·∧ t ) + Z st ∂ t ϕ ( r, P X n ·∧ r ) dr + Z st E (cid:2) h X nr , A ∗ n ∂ µ ϕ ( r, P X n ·∧ r )( X n ·∧ r ) i H (cid:3) dr + Z st E (cid:2) h b r (cid:0) X n , P X n ·∧ r , α r , P α r (cid:1) , ∂ µ ϕ ( r, P X n ·∧ r )( X n ·∧ r ) i H (cid:3) dr + 12 Z st E (cid:2) Tr (cid:0) σ r (cid:0) X n , P X n ·∧ r , α r , P α r (cid:1) σ ∗ r (cid:0) X n , P X n ·∧ r , α r , P α r (cid:1) ∂ x ∂ µ ϕ ( r, P X n ·∧ r )( X n ·∧ r ) (cid:1)(cid:3) dr, for every s ∈ [ t, T ] . Now the convergence of all terms above follows applying, in a straightforwardway, the result of Proposition 2.10. For every t ∈ [0 , T ] , we introduce the set M t := (cid:8) a : Ω → U : a is F t -measurable (cid:9) . Note that, since the filtration F t is right-continuous, we have M t = ∩ ε> M t + ε . We now considerthe following Hamilton-Jacobi-Bellman (HJB) equation: ∂ t w ( t, µ ) + E h ξ t , A ∗ ∂ µ w ( t, µ )( ξ ) i H + sup a ∈M t (cid:26) E (cid:2) f t (cid:0) ξ, µ, a , P a (cid:1) + (cid:10) b t (cid:0) ξ, µ, a , P a (cid:1) , ∂ µ w ( t, µ )( ξ ) (cid:11) H (cid:3) (5.1) + 12 E h Tr (cid:16) σ t (cid:0) ξ, µ, a , P a (cid:1) σ ∗ t (cid:0) ξ, µ, a , P a (cid:1) ∂ x ∂ µ w ( t, µ )( ξ ) (cid:17)i(cid:27) , for ( t, µ ) ∈ H , t < T , ξ ∈ S ( G ) such that P ξ = µ , with terminal condition w ( T, µ ) = E [ g ( ξ, µ )] , for µ ∈ P ( C ([0 , T ]; H )) , ξ ∈ S ( G ) such that P ξ = µ. (5.2) Definition 5.1.
We say that a function w : H → R belongs to the space C , b,A ∗ ( H ) if it satisfiesthe following regularity assumptions:(i) w : H → R belongs to C , b ( H ) ;(ii) for all ( t, µ, ξ ) ∈ H × S ( F ) , ∂ µ ϕ ( t, µ )( ξ ) ∈ L (Ω; D ( A ∗ )) and the map H × S ( F ) −→ L (Ω; H ) , ( t, µ, ξ ) A ∗ ϕ ( t, µ )( ξ ) is continuous and bounded. efinition 5.2. We say that a function w : H → R is a classical solution to the HJB equation (5.1) with terminal condition (5.2) , if it belongs to the space C , b,A ∗ ( H ) and satisfies (5.1) - (5.2) . Using Theorem 4.25 and Corollary 3.8 we are able to prove the following result.
Theorem 5.3.
Let Assumptions (A A,b,σ ) and (A f,g ) cont hold. Assume also that b, σ, f are uni-formly continuous in t , uniformly with respect to the other variables. Assume that the value function v (see (3.10) ) belongs to the space C , b,A ∗ ( H ) . Then v is a classical solution of (5.1) - (5.2) . Proof.
From Corollary 3.8 we know that, for every ( t, µ ) ∈ H , for every ξ ∈ S ( F ) such P ξ = µ ,for every α ∈ U , and for every h > sufficiently small, α ∈U (cid:26) E (cid:20) h Z t + ht f r (cid:0) X, P X ·∧ r , α r , P α r (cid:1) dr (cid:21) + 1 h (cid:2) v (cid:0) t + h, P X (cid:1) − v ( t, µ ) (cid:3) (cid:27) , (5.3)where, for simplicity, we wrote simply X in place of X t,ξ,α . Now we use (4.11) getting h (cid:2) v (cid:0) t + h, P X t,ξ,α (cid:1) − v ( t, µ ) (cid:3) = 1 h Z t + ht ∂ t v ( r, P X ·∧ r ) dr + 1 h Z t + ht E [ h X r , A ∗ ∂ µ v ( r, P X ·∧ r )( X ·∧ r ) i H ] dr + 1 h Z t + ht E [ h b r ( X, P X ·∧ r , α r , P α r ) , ∂ µ v ( r, P X ·∧ r )( X ·∧ r ) i H ] dr (5.4) + 12 h Z t + ht E [Tr ( σ r ( X, P X ·∧ r , α r , P α r ) σ ∗ r ( X, P X ·∧ r , α r , P α r ) ∂ x ∂ µ v ( r, P X ·∧ r )( X ·∧ r ))] dr, We now show, as in the typical proof of this result, the two inequalities. First take any a ∈ M t andconsider the control α ∈ U defined as α s = 0 , s ∈ [0 , t ) , α s = a , s ∈ [ t, T ] Then, from (5.3) and (5.4) we get ≥ h Z t + ht E f r (cid:0) X, P X ·∧ r , a , P a (cid:1) dr + 1 h Z t + ht ∂ t v ( r, P X ·∧ r ) dr + 1 h Z t + ht E [ h X r , A ∗ ∂ µ v ( r, P X ·∧ r )( X ·∧ r ) i H ] dr + 1 h Z t + ht E [ h b r ( X, P X ·∧ r , a , P a ) , ∂ µ v ( r, P X ·∧ r )( X ·∧ r ) i H ] dr (5.5) + 12 h Z t + ht E [Tr ( σ r ( X, P X ·∧ r , a , P a ) σ ∗ r ( X, P X ·∧ r , a , P a ) ∂ x ∂ µ v ( r, P X ·∧ r )( X ·∧ r ))] dr, Using the regularity of v and the continuity properties of b, σ, f we get, passing to the limit for h → + ≥ ∂ t v ( t, µ ) + E h ξ t , A ∗ ∂ µ v ( t, µ )( ξ ) i H E (cid:2) f t (cid:0) ξ, µ, a , P a (cid:1) + (cid:10) b t (cid:0) ξ, µ, a , P a (cid:1) , ∂ µ v ( t, µ )( ξ ) (cid:11) H (cid:3) + 12 E h Tr (cid:16) σ t (cid:0) ξ, µ, a , P a (cid:1) σ ∗ t (cid:0) ξ, µ, a , P a (cid:1) ∂ x ∂ µ v ( t, µ )( ξ ) (cid:17)i (5.6)27hen the inequality ≥ follows by the arbitrariness of a . We now prove the opposite inequality. Forevery ε > we take α ε ∈ U such that, denoting by X ε the corresponding state trajectory, − ε ≤ E (cid:20) ε Z t + εt f r (cid:0) X ε , P X ε ·∧ r , α εr , P α εr (cid:1) dr (cid:21) + 1 ε (cid:2) v (cid:0) t + ε, P X ε (cid:1) − v ( t, µ ) (cid:3) (cid:27) , (5.7)Now we apply Ito’s formula (5.4) above, for h = ε , getting − ε ≤ E (cid:20) ε Z t + εt f r (cid:0) X ε , P X ε ·∧ r , α εr , P α εr (cid:1) dr (cid:21) + 1 ε Z t + εt ∂ t v ( r, P X ε ·∧ r ) dr + 1 ε Z t + εt E (cid:2) h X εr , A ∗ ∂ µ v ( r, P X ε ·∧ r )( X ε ·∧ r ) i H (cid:3) dr + 1 ε Z t + εt E (cid:2) h b r (cid:0) X ε , P X ε ·∧ r , α r , P α r (cid:1) , ∂ µ v ( r, P X ε ·∧ r )( X ε ·∧ r ) i H (cid:3) dr (5.8) + 12 ε Z t + εt E (cid:2) Tr (cid:0) σ r (cid:0) X ε , P X ε ·∧ r , α r , P α r (cid:1) σ ∗ r (cid:0) X ε , P X ε ·∧ r , α r , P α r (cid:1) ∂ x ∂ µ v ( r, P X ε ·∧ r )( X ε ·∧ r ) (cid:1)(cid:3) dr, By Remark 2.9 we obtain that, as ε → , X ε ·∧ r → ξ ·∧ t and P X ε ·∧ r → P ξ ·∧ t , hence the second andthird integrals of the above right-hand side converge to ∂ t v ( t, µ ) + E [ h ξ t , A ∗ ∂ µ v ( t, µ )( ξ ) i H ] The remaining integrals of the right-hand side of (5.8) can be rewritten as ε Z t + εt (cid:18) E (cid:2) f r (cid:0) X ε , P X ε ·∧ r , α εr , P α εr (cid:1) + h b r (cid:0) X ε , P X ε ·∧ r , α r , P α r (cid:1) , ∂ µ v ( r, P X ε ·∧ r )( X ε ·∧ r ) i H (cid:3) (5.9) E (cid:2) Tr (cid:0) σ r (cid:0) X ε , P X ε ·∧ r , α r , P α r (cid:1) σ ∗ r (cid:0) X ε , P X ε ·∧ r , α r , P α r (cid:1) ∂ x ∂ µ v ( r, P X ε ·∧ r )( X ε ·∧ r ) (cid:1)(cid:3) (cid:19) dr Recall now that, by our assumptions, b and σ are uniformly continuous in ( t, x, µ ) uniformly withrespect to the other variables and that f is locally uniformly continuous in ( x, µ ) uniformly withrespect to the other variables. Hence, using again Remark 2.9 we obtain that (5.9) can be rewrittenas ε Z t + εt (cid:18) E (cid:2) f t (cid:0) ξ, µ, α εr , P α εr (cid:1) + h b t ( ξ, µ, α r , P α r ) , ∂ µ v ( t, µ )( ξ ) i H (cid:3) (5.10) E [Tr ( σ t ( ξ, µ, α r , P α r ) σ ∗ r ( ξ, µ, α r , P α r ) ∂ x ∂ µ v ( t, µ )( ξ ))] (cid:19) dr + ρ ( ε ) where ρ ( ε ) → as ε → . It follows ≤ ε + ρ ( ε ) + ∂ t v ( t, µ ) + E h ξ t , A ∗ ∂ µ v ( t, µ )( ξ ) i H + sup a ∈M t + ε (cid:26) E (cid:2) f t (cid:0) ξ, µ, a , P a (cid:1) + (cid:10) b t (cid:0) ξ, µ, a , P a (cid:1) , ∂ µ v ( t, µ )( ξ ) (cid:11) H (cid:3) (5.11) + 12 E h Tr (cid:16) σ t (cid:0) ξ, µ, a , P a (cid:1) σ ∗ t (cid:0) ξ, µ, a , P a (cid:1) ∂ x ∂ µ v ( t, µ )( ξ ) (cid:17)i(cid:27) Since ∩ ε> M t + ε = M t the claim follows sending ε → .28ow, we provide the definition of viscosity solution that we shall use. Definition 5.4.
We say that a function w : H → R is a viscosity subsolution (respectively super-solution) to the HJB equation (5.1) with terminal condition (5.2) , if: • w ( T, µ ) ≤ ( respectively ≥ ) E [ g ( ξ, µ )] , for µ ∈ P ( C ([0 , T ]; H )) , ξ ∈ S ( G ) such that P ξ = µ ; • for ( t, µ ) ∈ H and for every test function ϕ ∈ C , b,A ∗ ( H ) such that w − ϕ has a maximum at ( t, µ ) (with value ), one has that (5.1) - (5.2) is satisfied with the inequality ≤ ( respectively ≥ ) in place of the equality and with ϕ in place of w .Moreover, w is called a viscosity solution of (5.1) - (5.2) if it is both a viscosity subsolution and aviscosity supersolution. Theorem 5.5.
Let Assumptions (A A,b,σ ) and (A f,g ) cont hold. Assume also that b, σ, f are uni-formly continuous in t , uniformly with respect to the other variables. Then, the value function v isa viscosity solution of (5.1) - (5.2) . Proof.
The proof follows exactly the same lines as in the proof of Theorem 5.3, simply replacing v with ϕ . We derive alternative forms of the Hamilton-Jacobi-Bellman equation (5.1) relying on technicalresults reported in Appendix F. We first need to introduce the following sets:• M G is the set M G := (cid:8) a : Ω → U : a is G -measurable (cid:9) ; • ˇ M is the set of Borel-measurable maps ˇa : C ([0 , T ]; H ) × [0 , → U ;• M is the set of Borel-measurable maps a : C ([0 , T ]; H ) → U .The technical results reported in Appendix F provides the following key proposition. Proposition 5.6.
Suppose that (A A,b,σ ) and (A f,g ) hold. Let ( t, µ ) ∈ H , w ∈ C , b ( H ) , anddefine F : C ([0 , T ]; H ) × U × P (U) → R by F ( x, u, ν ) := f t ( x, µ, u, ν ) + (cid:10) b t ( x, µ, u, ν ) , ∂ µ w ( t, µ )( x ) (cid:11) H + 12 Tr (cid:16) σ t ( x, µ, u, ν ) σ ∗ t ( x, µ, u, ν ) ∂ x ∂ µ w ( t, µ )( x ) (cid:17) , for every ( x, u, ν ) ∈ C ([0 , T ]; H ) × U × P (U) . Let also ξ ∈ S ( G ) with P ξ = µ . Suppose that ξ is such that there exists a G -measurable random variable U ξ having uniformdistribution on [0 , and being independent of ξ (by Lemma F.1 we know that for each µ thereexist at least one ξ , with P ξ = µ , satisfying this property). Then, it holds that sup a ∈M t E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) = sup a ∈M G E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) = sup ˇa ∈ ˇ M E (cid:2) F (cid:0) ξ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1)(cid:3) . (5.12)29) Suppose that F does not depend on its last argument, namely F = F ( x, u ) . Then, it holds that sup a ∈M t E (cid:2) F (cid:0) ξ, a (cid:1)(cid:3) = sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) = E h ess sup u ∈ U F ( ξ, u ) i . (5.13) Proof.
Equalities (5.12) are a direct consequence of Lemma F.2, while (5.13) follows directly fromequalities (F.3) and (F.4) of Lemma F.3.
Remark 5.7.
Notice that the requirement that F does not depend on ν in item 2) of Proposition5.6 is necessary for the validity of the first equality in (5.13) (we do not consider the second equalityin (5.13) in this case, as it is not clear how to write the last quantity in (5.13) when F depends alsoon ν ). As a matter of fact, consider the following example. Example.
Take
U = [0 , , endowed with its Borel σ -algebra, and let F be given by F ( x, u, ν ) = − W ( ν, λ ) , ∀ ( x, u, ν ) ∈ C ([0 , T ]; H ) × [0 , × P ([0 , , with λ being the Lebesgue measure on the unit interval. Let also ξ be constant and identically equalto some fixed path ¯ x ∈ C ([0 , T ]; H ) . Moreover, denote by U G a G -measurable random variable havingdistribution λ , whose existence follows from Lemma 2.1. Then, for every t ∈ [0 , T ] , sup a ∈M t (cid:0) − W (cid:0) P a , λ (cid:1)(cid:1) = 0 and the supremum is attained at a ∗ , where a ∗ := U G . On the other hand, if a ∈ M then a( ξ ) isequal to the constant a(¯ x ) , so, in particular, P a( ξ ) = δ a(¯ x ) . This implies that sup a ∈M (cid:0) − W ( P a( ξ ) , λ ) (cid:1) = sup c ∈ [0 , (cid:0) − W ( δ c , λ ) (cid:1) = − inf c ∈ [0 , (cid:18) Z | c − r | dr (cid:19) = − inf c ∈ [0 , r c − c + 13 = − . Remark 5.8.
Suppose that F = F ( x, u ) and define F ∗ : C ([0 , T ]; H ) → R ∪ { + ∞} as F ∗ ( x ) := sup u ∈ U F ( x, u ) , ∀ x ∈ C ([0 , T ]; H ) . If F ∗ is measurable, then ess sup u ∈ U F ( ξ, u ) = F ∗ ( ξ ) , P -a.s., and the essential supremum appearingin (5.13) can be replaced with the supremum , so that we obtain sup a ∈M t E (cid:2) F ( ξ, a ) (cid:3) = E h sup u ∈ U F ( ξ, u ) i . Notice that, under assumptions (A A,b,σ ) and (A f,g ) , it follows from Proposition 7.47 in [8] that F ∗ is lower semi-analytic (see Definition 7.21 in [8] for the definition of lower semi-analytic ).However, we cannot in general say that F ∗ is measurable (see for instance the discussion at the endof Section B.5 in [8]). Sufficient conditions ensuring the measurability of F ∗ are given for instancein Proposition 7.32 of [8] and read as follows: (a) If F : C ([0 , T ]; H ) × U → R is lower semi-continuous and U is compact, then F ∗ is lowersemi-continuous. If F : C ([0 , T ]; H ) × U → R is upper semi-continuous, then F ∗ is upper semi-continuous. It follows from Proposition 5.6 that, under (A A,b,σ ) and (A f,g ) , the Hamilton-Jacobi-Bellman equa-tion (5.1) can also be written in the following two alternative forms: ∂ t w ( t, µ ) + E h ξ t , A ∗ ∂ µ w ( t, µ )( ξ ) i H + sup a ∈M G (cid:26) E (cid:2) f t (cid:0) ξ, µ, a , P a (cid:1) + (cid:10) b t (cid:0) ξ, µ, a , P a (cid:1) , ∂ µ w ( t, µ )( ξ ) (cid:11) H (cid:3) + 12 E h Tr (cid:16) σ t (cid:0) ξ, µ, a , P a (cid:1) σ ∗ t (cid:0) ξ, µ, a , P a (cid:1) ∂ x ∂ µ w ( t, µ )( ξ ) (cid:17)i(cid:27) = 0 , ( t, µ ) ∈ H , t < T,w ( T, µ ) = E [ g ( ξ, µ )] , µ ∈ P ( C ([0 , T ]; H )) , (5.14)or, alternatively, ∂ t w ( t, µ ) + E h ξ t , A ∗ ∂ µ w ( t, µ )( ξ ) i H + sup ˇa ∈ ˇ M (cid:26) E (cid:2) f t (cid:0) ξ, µ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1) + (cid:10) b t (cid:0) ξ, µ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1) , ∂ µ w ( t, µ )( ξ ) (cid:11) H (cid:3) + 12 E h Tr (cid:16) σ t σ ∗ t (cid:0) ξ, µ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1) ∂ x ∂ µ w ( t, µ )( ξ ) (cid:17)i(cid:27) = 0 , ( t, µ ) ∈ H , t < T,w ( T, µ ) = E [ g ( ξ, µ )] , µ ∈ P ( C ([0 , T ]; H )) , (5.15)where, in both (5.14) and (5.15), ξ ∈ S ( G ) , with P ξ = µ , is such that there exists a G -measurablerandom variable U ξ having uniform distribution on [0 , and being independent of ξ (we recall that,by Lemma F.1, for each µ there exists at least one ξ , with P ξ = µ , satisfying this latter property).Now, suppose that (A A,b,σ ) , (A f,g ) hold and also that the coefficients b , σ , f do not depend ontheir last argument (namely, b = b t ( x, µ, u ) , σ = σ t ( x, µ, u ) , f = f t ( x, µ, u ) ). Then, by Proposition5.6 we deduce that the Hamilton-Jacobi-Bellman equation (5.1) can also be written in the followingtwo alternative forms: ∂ t w ( t, µ ) + E h ξ t , A ∗ ∂ µ w ( t, µ )( ξ ) i H + sup a ∈M (cid:26) E (cid:2) f t (cid:0) ξ, µ, a( ξ ) (cid:1) + (cid:10) b t (cid:0) ξ, µ, a( ξ ) (cid:1) , ∂ µ w ( t, µ )( ξ ) (cid:11) H (cid:3) + 12 E h Tr (cid:16) σ t (cid:0) ξ, µ, a( ξ ) (cid:1) σ ∗ t (cid:0) ξ, µ, a( ξ ) (cid:1) ∂ x ∂ µ w ( t, µ )( ξ ) (cid:17)i(cid:27) = 0 , ( t, µ ) ∈ H , t < T,w ( T, µ ) = E [ g ( ξ, µ )] , µ ∈ P ( C ([0 , T ]; H )) , (5.16)for every ξ ∈ S ( G ) with P ξ = µ , or, alternatively: ∂ t w ( t, µ ) + E h ξ t , A ∗ ∂ µ w ( t, µ )( ξ ) i H + E (cid:20) ess sup u ∈ U (cid:20) f t ( ξ, µ, u ) + (cid:10) b t ( ξ, µ, u ) , ∂ µ w ( t, µ )( ξ ) (cid:11) H + 12 Tr (cid:16) σ t ( ξ, µ, u ) σ ∗ t ( ξ, µ, u ) ∂ x ∂ µ w ( t, µ )( ξ ) (cid:17)(cid:21)(cid:21) = 0 , ( t, µ ) ∈ H , t < T,w ( T, µ ) = E [ g ( ξ, µ )] , µ ∈ P ( C ([0 , T ]; H )) , (5.17)for every ξ ∈ S ( G ) with P ξ = µ . 31 emark 5.9. In the case of the optimal investment problem outlined in Example 2.12-(ii) and inRemark 3.2, equation (3.2) , the HJB equation can be written as follows. ∂ t w ( t, µ ) + E h ξ t , A ∗ ∂ µ w ( t, µ )( ξ ) i H + E (cid:20) e − rt h a , ξ t i H + δ ( ξ, µ ) h ξ t , ∂ µ w ( t, µ )( ξ ) i H + ess sup u ∈ U h h Cu, ∂ µ w ( t, µ )( ξ ) i H − e − rt ( h a , u i H + h M u, u i H ) i + 12 σ Tr (cid:16) ∂ x ∂ µ w ( t, µ )( ξ ) (cid:17)(cid:21) = 0 , ( t, µ ) ∈ H , t < T,w ( T, µ ) = E [ g ( ξ, µ )] , µ ∈ P ( C ([0 , T ]; H )) , (5.18) for every ξ ∈ S ( G ) with P ξ = µ . Note that in the above case the ess sup appearing in the Hamiltoniancan be explicitly computed. A State equation: proofs
We collect in the following lemma some continuity results for contractions in Banach spaces thatwe will use to obtain the needed continuity properties of the mild solution to the state equation.
Lemma A.1.
Let R be a non-empty set, T be a topological space, ( M, d ) be a metric space, Y bea Banach space, γ ∈ [0 , . Let w be a modulus of continuity. Let h : R × T × M × Y → Y be suchthat: | h ( r, x, m, y ) − h ( r, x, m ′ , y ′ ) | ≤ w ( d ( m, m ′ )) + γ | y − y ′ | , ∀ r ∈ R , x ∈ T , m, m ′ ∈ M, y, y ′ ∈ Y. Let
E ⊂ R be a set of subsets of R . Assume that, if { x ι } ι ∈I ⊂ T is a net converging to x ∈ T ,then lim ι sup r ∈ E | h ( r, x ι , m, y ) − ϕ ( r, x, m, y ) | = 0 , ∀ m ∈ M, y ∈ Y, E ∈ E . Denote by ϕ : R × T × M → Y the fixed-point map associated with h , i.e. ϕ is the unique mapsatisfying h ( r, x, m, ϕ ( r, x, m )) = ϕ ( r, x, m ) , ∀ r ∈ R , x ∈ T , m ∈ M. Then, given a net { x ι } ι ∈I ⊂ T converging to x , it holds that lim ι sup r ∈ E | ϕ ( r, x ι , m ) − ϕ ( r, x, m ′ ) | ≤ − γ w ( d ( m, m ′ )) , ∀ m, m ′ ∈ M, E ∈ E . (A.1) Proof.
Write | ϕ ( r, x ι , m ) − ϕ ( r, x, m ′ ) | = | h ( r, x ι , m, ϕ ( r, x ι , m )) − h ( r, x, m ′ , ϕ ( r, x, m ′ )) |≤ | h ( r, x ι , m, ϕ ( r, x ι , m )) − h ( r, x ι , m, ϕ ( r, x, m ′ )) | + | h ( r, x ι , m, ϕ ( r, x, m ′ )) − h ( r, x, m, ϕ ( r, x, m ′ )) | + | h ( r, x, m, ϕ ( r, x, m ′ )) − h ( r, x, m ′ , ϕ ( r, x, m ′ )) |≤ γ | ϕ ( r, x ι , m ) − ϕ ( r, x, m ′ ) | + | h ( r, x ι , m, ϕ ( r, x, m ′ )) − h ( r, x, m, ϕ ( r, x, m ′ )) | + w ( d ( m, m ′ )) . sup r ∈ E | ϕ ( r, x ι , m ) − ϕ ( r, x, m ′ ) | ≤ − γ sup r ∈ E | h ( r, x ι , m, ϕ ( r, x, m ′ )) − h ( r, x, m, ϕ ( r, x, m ′ )) | + 11 − γ w ( d ( m, m ′ )) and the claim follows taking the limit with respect to ι . Proof of Proposition 2.8.
The proof is based, as usual, on a contraction argument. We in-troduce the space L F ( H ) (resp. L F ( L ( K ; H )) ) of all square-integrable F -progressively measur-able processes on [0 , T ] taking values in H (resp. L ( K ; H ) ), normed respectively by k X k L F ( H ) := (cid:0) E (cid:2) R T | X s | H ds (cid:3)(cid:1) / and k Φ k L F ( H ) := (cid:0) E (cid:2) R T | Φ s | L ( K ; H ) ds (cid:3)(cid:1) / . For the sake of brevity, in whatfollows we will denote S t := e At , b αt ( X, P X ) := b t ( X, P X , α t , P α t ) , σ αt ( X, P X ) := σ t ( X, P X , α t , P α t ) .For t ∈ [0 , T ] and α ∈ U , let us define id St : S ( F ) → S ( F ) , ξ ξ · [0 ,t ] ( · ) + S ·− t ξ t ( t,T ] ( · ) F b α : S ( F ) → L F ( H ) , X b α · ( X, P X ) F σ α : S ( F ) → L F ( L ( K ; H )) , X σ α · ( X, P X ) S ⋆ t L F ( H ) → S ( F ) , X [ t,T ] ( · ) Z · t S ·− s X s dsS dB ⋆ t L F ( L ( K ; H )) → S ( F ) , Φ [ t,T ] ( · ) Z · t S ·− s Φ s dB s . (A.2)We briefly explain why the functions above are well-defined. Regarding id St , it holds that id St ( ξ ) = ξ ·∧ t + ( t,T ] ( · )( S ·− t − I ) ξ t , (A.3)which clearly shows that id St ( ξ ) ∈ S ( F ) . Regarding F b α , due to the measurability assumptions on b , we have that F b α ( X ) is progressively measurable. Moreover, recalling Assumption (A A,b,σ ) -(iii),we have k F b α ( X ) k L F ( H ) ≤ L E (cid:20)Z T (cid:0) k X k t (cid:1) dt (cid:21) < ∞ . (A.4)In the same way, we obtain the measurability of F σ α ( X ) and k F σ α ( X ) k L F ( H ) ≤ L E (cid:20)Z T (cid:0) k X k t (cid:1) dt (cid:21) < ∞ . (A.5)Regarding S ⋆ t X , for X ∈ L F ( H ) , it is not difficult to see that it is continuous, F -adapted, andthat k S ⋆ t X k S ≤ e η ( T − t ) ( T − t ) k X k L F ( H ) . (A.6)Finally, regarding S dB ⋆ t Φ , for Φ ∈ L F ( L ( K ; H )) , by [25, Theorem 1.111] we know that the F -adapted process n [ t,T ] ( t ′ ) R t ′ t S t ′ − s Φ s ds o t ′ ∈ [0 ,T ] admits a continuous version, that we name S dB ⋆ t Φ ,and that k S dB ⋆ t Φ k S ≤ C η,T k Φ k L F ( L ( K ; H )) , (A.7)33here C η,T is a constant depending only on η, T .We now define the map ψ : U × [0 , T ] × S ( F ) × S ( F ) → S ( F ) by ψ ( α, t, ξ, X ) = id St ( ξ ) + S ⋆ t F b α ( X ) + S dB ⋆ t F σ α ( X ) , ∀ ( α, t, ξ, X ) ∈ U × [0 , T ] × S ( F ) × S ( F ) . Claim I.
For fixed ξ, X ∈ S ( F ) , ψ ( α, t, ξ, X ) is continuous in t , uniformly in α ∈ U . The continuity of t id St ( ξ ) follows from Lebesgue’s dominated convergence theorem and the factthat, for every fixed ω ∈ Ω , id St ′ ( ξ )( ω ) converges uniformly to id St ( ξ )( ω ) as t ′ → t .Moreover, for t ′ , t ∈ [0 , T ] , t ′ < t , we have S ⋆ t ′ F b α ( X ) − S ⋆ t F b α ( X ) = [ t ′ ,t ] ( · ) Z · t ′ S ·− s F b α ( X ) s ds + [ t,T ] ( · ) Z tt ′ S ·− s F b α ( X ) s ds. Then k S ⋆ t ′ F b α ( X ) − S ⋆ t F b α ( X ) k T ≤ e ηT Z tt ′ | F b α ( X ) s | H ds. This implies, recalling Assumption (A A,b,σ ) (iii), sup α ∈U lim | t ′ − t |→ k S ⋆ t ′ F b α ( X ) − S ⋆ t F b α ( X ) k S = 0 . Finally, take again t ′ , t ∈ [0 , T ] , t ′ < t . Then S dB ⋆ t ′ F σ α ( X ) − S dB ⋆ t F σ α ( X ) = [ t ′ ,t ] ( · ) Z · t ′ S ·− s F σ α ( X ) s dB s + [ t,T ] ( · ) Z tt ′ S ·− s F σ α ( X ) s dB s . By [25, Theorem 1.111], we have k S dB ⋆ t ′ F σ α ( X ) − S dB ⋆ t F σ α ( X ) k S ≤ C ′ η,T k [ t ′ ,t ] F σ α ( X ) k L F ( H ) and then, after recalling Assumption (A A,b,σ ) (iii), sup α ∈U lim | t ′ − t |→ k S dB ⋆ t ′ F σ α ( X ) − S dB ⋆ t F σ α ( X ) k S = 0 . Claim II. ψ ( α, t, ξ, X ) is Lipschitz continuous in ξ , uniformly in t, X, α . For α ∈ U , t ∈ [0 , T ] , ξ, ξ ′ ∈ S ( F ) , X ∈ S ( F ) , we have k ψ ( α, t, ξ, X ) − ψ ( α, t, ξ ′ , X ) k S = k id St ( ξ − ξ ′ ) k S ≤ e ηT ) k ξ − ξ ′ k S . Now, for a, b ∈ [0 , T ] , a < b , let us consider the restriction of ψ to the time interval [ a, b ] and stoppedat time b , namely ψ a,b : U × [ a, b ] × S ( F ) × S ( F ) → S ( F ) , ( α, t, ξ, X ) ψ ( α, t, ξ, X ) b ∧· Claim III.
There exists ε > such that, if b − a ≤ ε , then ψ a,b ( α, t, ξ, X ) is a contraction in X ,uniformly in α, t, ξ, a, b , namely: for some γ ∈ [0 , , k ψ a,b ( α, t, ξ, X ) b ∧· − ψ a,b ( α, t, ξ, X ′ ) b ∧· k S ≤ γ k X − X ′ k S , or all α ∈ U , t ∈ [ a, b ] , ( ξ, X, X ′ ) ∈ S ( F ) and for all a, b ∈ [0 , T ] , a < b, b − a ≤ ε . Let a, b ∈ [0 , T ] , a < b , t ∈ [ a, b ] , α ∈ U , ξ ∈ S ( F ) , X, X ′ ∈ S ( F ) . Notice that ψ ( α, t, ξ, X ) − ψ ( α, t, ξ, X ′ ) = S ⋆ t (cid:0) F b α ( X ) − F b α ( X ′ ) (cid:1) + S dB ⋆ t (cid:0) F σ α ( X ) − F σ α ( X ′ ) (cid:1) (A.8)and k ψ a,b ( α, t, ξ, X ) − ψ a,b ( α, t, ξ, X ′ ) k T = k ψ ( α, t, ξ, X ) − ψ ( α, t, ξ, X ′ ) k b . (A.9)Moreover, recalling Assumption (A A,b,σ ) -(iii), we have k S ⋆ t (cid:0) F b α ( X ) − F b α ( X ′ ) (cid:1) k b ≤ e ηT (cid:18)Z ba (cid:12)(cid:12) F b α ( X ) s − F b α ( X ′ ) s (cid:12)(cid:12) ds (cid:19) ≤ L e ηT ( b − a ) (cid:0) k X − X ′ k T + W ( P X , P X ′ ) (cid:1) . Then E (cid:2) k S ⋆ t (cid:0) F b α ( X ) − F b α ( X ′ ) (cid:1) k b (cid:3) ≤ L e ηT ( b − a ) k X − X ′ k S . (A.10)By [25, Theorem 1.111] there exists a constant C η,T , depending only on η, T , such that E h k S dB ⋆ t (cid:0) F σ α ( X ) − F σ α ( X ′ ) (cid:1) k b i ≤ C η,T E (cid:20)Z ba k F σ α ( X ) s − F σ α ( X ′ ) s k L ( K ; H ) ds (cid:21) ≤ L C η,T ( b − a ) k X − X ′ k S , (A.11)where for the last inequality we have used Assumption (A A,b,σ ) -(iii) again. By (A.8), (A.9), (A.10),and (A.11), we have, if b − a < , k ψ a,b ( α, t, ξ, X ) − ψ a,b ( α, t, ξ, X ′ ) k S ≤ C L,η,T ( b − a ) / k X − X ′ k S , where C L,η,T is a constant depending only on
L, η, T . We can then choose ε ∈ (0 , such that C L,η,T ε / < / . Then the map ψ a,b ( α, t, ξ, · ) is a / -contraction, uniformly in α, t, ξ and in a, b ∈ [0 , T ] , whenever a < b , b − a ≤ ε . Claim IV.
For ε > as in Claim III, and whenever a, b ∈ [0 , T ] , a < b , b − a ≤ ε , there existsa unique mild solution X t,ξ,α to equation (2.2) on the interval [ a, b ] , for any α ∈ U , t ∈ [ a, b ] , ξ ∈ S ( F ) . Let ε be as in Claim III . Then, for any α, t, ξ , by the Banach contraction principle, there exists aunique fixed point X t,ξ,α to ψ a,b ( α, t, ξ, · ) . Clearly X t,ξ,α is a mild solution to (2.2) on the interval [ a, b ] . Claim V.
For ε as in Claim III, and uniformly for a, b ∈ [0 , T ] , a < b , b − a ≤ ε , the map ϕ a,b : U × [ a, b ] × S ( F ) → S ( F ) , ( α, t, ξ ) X t,ξ,α (A.12) is continuous in ( t, ξ ) , uniformly in α , and Lipschitz-continuous in ξ , uniformly in α, t . Let ε be as in Claim III . We apply Lemma A.1 with R = U , T = [ a, b ] , M = S ( F ) , Y = S ( F ) , E = 2 R . Then, by Claims I,II,III , we get lim t → t ′ t ∈ [ a,b ] sup α ∈U k ϕ a,b ( α, t, ξ ) − ϕ a,b ( α, t ′ , ξ ′ ) k S ≤ e ηT ) / k ξ − ξ ′ k S , ∀ t ′ ∈ [ a, b ] , ξ, ξ ′ ∈ S . We say that X t,ξ,α is a mild solution on the interval [ a, b ] if equation (2.2) is solved in the mild sense for s ∈ [ a, b ] . laim VI. For any α ∈ U , t ∈ [0 , T ] , ξ ∈ S ( F ) , there exists a unique mild solution X t,ξ,α to (2.2) ,and the map ϕ : U × [0 , T ] × S ( F ) → S ( F ) , ( α, t, ξ ) X t,ξ,α (A.13) is continuous in t, ξ , uniformly in α , and Lipschitz continuous in ξ , uniformly in t, α . Pick ε > as in Claim III . Choose a = 0 < a < . . . < a n = T with a i +1 − a i ≤ ε . Define ˆ ϕ a i : U × [ a i , T ] × S ( F ) → U × [ a i , T ] × S ( F ) by ˆ ϕ a i ( α, t, ξ ) = ( α, a i +1 , ϕ a i ,a i +1 ( α, t, ξ )) if t ∈ [ a i , a i +1 ] and ˆ ϕ a i ( α, t, ξ ) = ( α, t, ξ a i +1 ∧· ) if t > a i +1 .If we now define ϕ ( α, t, ξ ) to be the second component of ˆ ϕ a n (cid:0) ˆ ϕ a n − ( . . . ˆ ϕ a ( ϕ a ( α, t, ξ )) . . . ) (cid:1) , we can easily check, thanks to Claim V , that ϕ ( α, t, ξ ) is the unique mild solution X t,ξ,α to (2.2),and the map (A.13) has the desired regularity properties. Claim VII. X t,ξ,α = X t,ξ t ∧· ,α . This is due to the fact that, for any α ∈ U , a, b ∈ [0 , T ] , a < b , t ∈ [ a, b ] , ξ ∈ S ( F ) , X ∈ S ( F ) , ψ a,b ( α, t, ξ, X ) = ψ a,b ( α, t, ξ t ∧· , X ) . Hence, the unique fixed point of ψ a,b ( α, t, ξ, · ) has be the same of ψ a,b ( α, t, ξ t ∧· , · ) . Claim VIII.
There exists a constant C such that k X t,ξ,α k S ≤ C (1 + k ξ t ∧· k S ) , ∀ t ∈ [0 , T ] , ξ ∈ S , α ∈ U . Due to
Claim VI , we only need to show that sup α ∈U ,t ∈ [0 ,T ] k X t, ,α k S < ∞ . (A.14)We have, by using (A.4), (A.5), (A.6), (A.7), with T replaced with t ′ ∈ [0 , T ] , E (cid:2) k X t, ,α k t ′ (cid:3) ≤ (cid:16) k S ⋆ t (cid:0) [0 ,t ′ ] F b α ( X t, ,α ) (cid:1) k S + k S dB ⋆ t (cid:0) [0 ,t ′ ] F σ α ( X t, ,α ) (cid:1) k S (cid:17) ≤ (cid:0) L e ηT T + 3 L C η,T (cid:1) Z t ′ (1 + 2 E (cid:2) k X t, ,α k s (cid:3) ) ds. An application of Gronwall’s inequality yields E (cid:2) k X t, ,α k t ′ (cid:3) ≤ C, ∀ t ′ ∈ [0 , T ] , for some C independent of α, t , which proves (A.14) and then (2.3), after recalling Claim VII . Proof of Proposition 2.10.
For α ∈ U , let F b α , F σ α be as in (A.2). Then, let id S n t , S n ⋆ t , S n dB ⋆ t be defined as in (A.2) by replacing S with S n . Denote N = N ∪ {∞} . Let us now define e ψ : U × N × [0 , T ] × S ( F ) × S ( F ) → S ( F ) by e ψ ( α, n, t, ξ, X ) = id S n t ( ξ ) + S n ⋆ t F b α ( X ) + S n dB ⋆ t F σ α ( X ) , α ∈ U , n ∈ N , t ∈ [0 , T ] , ξ ∈ S ( F ) , X ∈ S ( F ) , where we set S ∞ := S . Let e ψ a,b ( α, n, t, ξ, X ) = e ψ ( α, n, t, ξ, X ) b ∧· whenever a, b ∈ [0 , T ] , a < b , t ∈ [ a, b ] . Due to the uniform boundedness of the Yosida approxima-tion, and by arguing as in the proof of Claims II,III of Proposition 2.8, one can show that e ψ ( α, n, t, ξ, X ) is Lipschitz continuous in ξ , uniformly in α, n, t, X . Moreover, thereexists ε > such that, if b − a < ε , then e ψ a,b ( α, n, t, ξ, X ) is a contraction in X ,uniformly in α, n, t, ξ, a, b . (A.15)Now we show that, for α ∈ U , t ′ ∈ [0 , T ] , ξ, X ∈ S ( F ) , lim t → t ′ n →∞ k e ψ ( α, n, t, ξ, X ) − e ψ ( α, ∞ , t ′ , ξ, X ) k S = 0 . (A.16)First, notice that, for P -a.e. ω ∈ Ω , the range of ξ ( ω ) is compact. Since S nt x → S t x uniformlyfor t ∈ [0 , T ] and x ∈ K , whenever K ⊂ H is compact, an application of Lebesgue’s dominatedconvergence theorem provides lim n →∞ k id S n t ′ ( ξ ) − id St ′ ( ξ ) k S ( F ) = 0 , ∀ t ′ ∈ [0 , T ] , ξ ∈ S ( F ) . Secondly, for α ∈ U , X ∈ S ( F ) , after defining f n ( r ) = sup t ∈ [0 ,T ] | ( S nt − S t ) b r ( X, P X , α r , P α r ) | , we have, by Lebesgue’s dominated convergence theorem, lim n →∞ k S n ⋆ t ′ F b α ( X ) − S ⋆ t ′ F b α ( X ) k S = lim n →∞ E " sup t ∈ [ t ′ ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z tt ′ (cid:0) S nt − r − S t − r (cid:1) b r ( X, P X , α r , P α r ) dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim n →∞ E (cid:20)Z Tt ′ f n ( r ) dr (cid:21) = 0 . Thirdly, by [25, Proposition 1.112], we have lim n →∞ k S n dB ⋆ t ′ F σ α ( X ) − S dB ⋆ t ′ F σ α ( X ) k S ( F ) = 0 , ∀ α ∈ U , t ∈ [0 , T ] , X ∈ S ( F ) . Putting together the above partial results, we get lim n →∞ k e ψ ( α, n, t ′ , ξ, X ) − e ψ ( α, ∞ , t ′ , ξ, X ) k S = 0 . Then, to prove (A.16), it is enough to show that lim t → t ′ sup n ∈ N k e ψ ( α, n, t, ξ, X ) − e ψ ( α, n, t ′ , ξ, X ) k S = 0 . But this can be obtained by arguing as in the proof of
Claim I of Proposition 2.8, due to the uniformboundedness k S nt k L ( H ) ≤ e e ηt , for n ∈ N , t ≥ .Now, for any small ε as in (A.15), and when a, b ∈ [0 , T ] , a < b , b − a < ε , denote by e ϕ a,b : U × N × [ a, b ] × S ( F ) → S ( F ) , ( α, n, t, ξ ) X n,t,ξ,α e ψ a,b (similarly as done for ψ a,b , ϕ a,b in the proof of Proposi-tion 2.8). Notice that X n,t,ξ,α is the unique mild solution of (2.5) (resp. (2.2)), when n ∈ N (resp. n = ∞ ), on the interval [ a, b ] . Thanks to (A.15) and (A.16), we can apply Lemma A.1 with R = U , T = N × [ a, b ] , M = S ( F ) , Y = S ( F ) , E = α ∈U , and obtain sup n ∈ N t ∈ [ a,b ] k e ϕ a,b ( α, n, t, ξ ) − e ϕ a,b ( α, n, t, ξ ′ ) k S ≤ C k ξ − ξ ′ k S , ∀ α ∈ U , ξ, ξ ′ ∈ S ( F ) , (A.17)and lim t → t ′ t ∈ [ a,b ] n →∞ k e ϕ a,b ( α, n, t, ξ ) − e ϕ a,b ( α, ∞ , t ′ , ξ ) k S = 0 , ∀ α ∈ U , (A.18)for some constant C , uniformly for a, b ∈ [0 , T ] , a < b , b − a < ε , t ′ ∈ [ a, b ] . To conclude the proofit is enough to use (A.17) and (A.18) iteratively, recalling the relation between the mild solutionon the subinterval [ a, b ] ⊂ [0 , T ] and the global mild solution on [0 , T ] (arguing as in the proof of Claim VI of Proposition 2.8, simply replacing [ a i , T ] with N × [ a i , T ] ). B Law invariance property of V : technical results In the present appendix we prove three technical results that are needed in the proof of the lawinvariance property (Theorem 3.5). The first technical result corresponds to Theorem 6.10 in [39].For the convenience of the reader, we restate it here using the notation adopted in the paper andin the form needed for the proof of Theorem 3.5.
Lemma B.1.
Consider a probability space ( ˆΩ , ˆ F , ˆ P ) , a measurable space ( E, E ) , a Borel space (U , U ) . Consider also two random variables Γ : ˆΩ → E ed α : ˆΩ → U . Suppose that there ex-ists a random variable ˆ U : ˆΩ → R , having uniform distribution on [0 , , such that Γ and ˆ U areindependent. Then, there exists a measurable function a : E × [0 , → U satisfying (cid:0) Γ , a(Γ , ˆ U ) (cid:1) L ˆΩ = (cid:0) Γ , α (cid:1) , where L ˆΩ = stands for equality in law (between random objects defined on ( ˆΩ , ˆ F , ˆ P ) ). Proof.
See Theorem 6.10 in [39].Before stating next result, we introduce the following notation. For every t ∈ [0 , T ] , let F B,t =( F B,ts ) s ≥ be the P -completion of the filtration generated by ( B s ∨ t − B t ) s ≥ . Let also P rog ( F B,t ) bethe σ -algebra of [0 , T ] × Ω of all F B,t -progressive sets (recall that a set C ⊂ [0 , T ] × Ω is called F B,t -progressive if the corresponding indicator function C is an F B,t -progressively measurable process;notice that the family of all F B,t -progressive sets is a σ -algebra). Lemma B.2.
Let t ∈ [0 , T ] , α ∈ U , ξ ∈ S ( F ) , with ξ being B ([0 , T ]) ⊗ F t -measurable. Suppose thatthere exists an F t -measurable random variable U ξ , having uniform distribution on [0 , and beingindependent of ξ . Then, there exists a measurable function a : (cid:0) [0 , T ] × Ω × C ([0 , T ]; H ) × [0 , , P rog ( F B,t ) ⊗ B ⊗ B ([0 , (cid:1) −→ (U , U ) uch that (a s ( ξ, U ξ )) s ∈ [0 ,T ] ∈ U , a s ( · , · ) is constant for every s < t , moreover (cid:16) ( ξ s ) s ∈ [0 ,T ] , (a s ( ξ, U ξ )) s ∈ [ t,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) L = (cid:16) ( ξ s ) s ∈ [0 ,T ] , ( α s ) s ∈ [ t,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) , (B.1) where L = stands for equality in law (between random objects defined on (Ω , F , P ) ). Proof.
Denote ˆΩ = [0 , T ] × Ω , ˆ F = B ([0 , T ]) ⊗ F , ˆ P = λ T ⊗ P , with λ T being the uniform distribution on ([0 , T ] , B ([0 , T ])) . Then, consider the canonical extensionof U ξ to ˆΩ , which will be denoted by ˆ U ξ (notice that ˆ U ξ has uniform distribution on [0 , andis independent of ξ ). Let also ( ¯ E, ¯ E ) be the measurable space defined as: ¯ E = [0 , T ] × Ω and ¯ E = P rog ( F B,t ) . Then, let I B,t : ˆΩ → ¯ E be the identity map. Finally, define Γ = ( I B,t , ξ ) .Notice that Γ is a random variable taking values in the measurable space ( E, E ) , with E = ¯ E × C ([0 , T ]; H ) = [0 , T ] × Ω × C ([0 , T ]; H ) and E = ¯ E ⊗ B = P rog ( F B,t ) ⊗ B . We also observe that Γ and ˆ U ξ are independent. We can then apply Lemma B.1, from which it follows the existence of amap ¯a : [0 , T ] × Ω × C ([0 , T ]; H ) × [0 , → U , measurable with respect to the σ -algebra P rog ( F B,t ) ⊗ B ⊗ B ([0 , , such that (cid:0) Γ , ¯a(Γ , ˆ U ξ ) (cid:1) L ˆΩ = (cid:0) Γ , α (cid:1) , where L ˆΩ stands for equality in law between random objects defined on ( ˆΩ , ˆ F , ˆ P ) . Then, we deduce (cid:16) ( ξ s ) s ∈ [0 ,T ] , (¯a s ( ξ, U ξ )) s ∈ [0 ,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) L = (cid:16) ( ξ s ) s ∈ [0 ,T ] , ( α s ) s ∈ [0 ,T ] , ( B s − B t ) s ∈ [ t,T ] (cid:17) , where we recall that L stands for equality in law between random objects defined on (Ω , F , P ) .Finally, let a : [0 , T ] × Ω × C ([0 , T ]; H ) × [0 , → U be the map given by a s ( ω, x, u ) = u [0 ,t ) ( s ) + ¯a s ( ω, x, u ) [ t,T ] ( s ) , ∀ ( s, ω, x, u ) ∈ [0 , T ] × Ω × C ([0 , T ]; H ) × U , where u ∈ U is arbitrarily chosen. We have that (a s ( ξ, U ξ )) s ∈ [0 ,T ] is F -progressively measurable(here we use that a s ( · , · ) is constant for every s < t , ξ is B ([0 , T ]) ⊗ F t -measurable and U ξ is F t -measurable). So, in particular, (a s ( ξ, U ξ )) s ∈ [0 ,T ] ∈ U and equality (B.1) holds. Lemma B.3.
Let t ∈ [0 , T ] and ξ ∈ S ( F ) , with ξ being B ([0 , T ]) ⊗ F t -measurable. Suppose thatthere exists { x , . . . , x m } ⊂ C ([0 , T ]; H ) , with x i = x j if i = j , such that P ξ = m X i =1 p i δ x i , where δ x i is the Dirac measure at x i and p i > , with P mi =1 p i = 1 . Then, there exists an F t -measurable random variable U ξ having uniform distribution on [0 , and being independent of the F t -measurable map ˜ ξ : Ω → C ([0 , T ]; H ) , ω ξ ( ω ) . Proof.
We recall from Lemma 2.1 that there exists a G -measurable (so, in particular, F t -measurable)random variable U G with uniform distribution on [0 , . If U G is independent of ˜ ξ , then we take U ξ = U G , otherwise we proceed as follows. Denote E i := (cid:8) ω ∈ Ω : ξ ( ω ) = x i (cid:9) , ∀ i = 1 , . . . , m. ξ is B ([0 , T ]) ⊗F t -measurable, it follows that each E i belongs to F t . Now, for each i = 1 , . . . , m ,define the function F i : [0 , → [0 , as follows F i ( r ) := P (cid:0) { U G ≤ r } ∩ E i (cid:1) , ∀ r ∈ [0 , . Notice that F i satisfies the following properties:• F i is a non-decreasing function;• F i is continuous;• F i (0) = 0 and F i (1) = P ( E i ) = p i > .For every i = 1 , . . . , m ed n ∈ N , define r i, k n := min (cid:26) r ∈ [0 ,
1] : F i ( r ) = k n p i (cid:27) , k = 1 , . . . , n . Then, for each n ∈ N define the random variable X n : Ω → R as X n ( ω ) := m X i =1 2 n − X k =1 (cid:26) r i, k − n
We devote this appendix to extend a useful result (firstly proved in [13, Section 6.1] in the case ofthe set R d ) to the case of the set C ([0 , T ]; H ) . More precisely, our aim is to prove that D ˆΦ( t, ˆ ξ ) only depends on the law of ˆ ξ . Here we substantially follow the idea of [50] and [51, Section 2.3].However, since our setting is more general, we present the full proof. C.1 The discrete case
We first consider the case where the random variable ˆ ξ takes a countable number of values. Weassume the following. Assumption (A ξ ). Consider a sequence { ˆ x i } i ∈ N ⊂ D ([0 , T ]; H ) , ˆ x i = ˆ x j if i = j . We assume thatthe random variable ˆ ξ has law P ˆ ξ = N X i =1 p i δ ˆ x i where N ∈ N \ { } , δ ˆ x i denotes the Dirac measure at ˆ x i and p i > , with P Ni =1 p i = 1 . emma C.1. Let ( t, ˆ ξ ) ∈ ˆ H , with ˆ ξ as in Assumption (A ξ ) . Let ˆ ϕ : ˆ H → R be such that its lifting ˆΦ is pathwise differentiable in space at ( t, ˆ ξ ) . Then there exists a measurable function ˆ g : D ([0 , T ] , H ) → H such that ˆ g ( ˆ ξ ) ∈ L (Ω; H ) and D ˆΦ( t, ˆ ξ ) = ˆ g ( ˆ ξ ) . (C.1) The function ˆ g can be defined by ˆ g (ˆ x ) := 0 if ˆ x
6∈ { ˆ x i } i =1 ,...,N , and h ˆ g (ˆ x i ) , h i H := p − i E hD D ˆΦ( t, ˆ ξ ) , h ˆ ξ − (ˆ x i ) E H i = lim ε → + εp i (cid:20) ˆ ϕ (cid:18) t, X j =1 ,...,Nj = i p j δ ˆ x j + p i δ ˆ x i + εh [ t,T ] (cid:19) − ˆ ϕ (cid:18) t, N X j =1 p j δ ˆ x j (cid:19)(cid:21) , (C.2) for all h ∈ H and i = 1 , . . . , N . Proof.
Fix i = 1 , . . . , N and h ∈ H, h = 0 . Take any measurable set A ′ ⊂ A i := { ˆ ξ = ˆ x i } and set ˆ η := h A ′ ∈ L (cid:0) Ω; H ) . Notice that, for any ε > we have P ˆ ξ + ε ˆ η = X j = i p j δ ˆ x j + P ( A ′ ) δ ˆ x i + εh [ t,T ] + (cid:0) p i − P ( A ′ ) (cid:1) δ ˆ x i which depends only on P ˆ ξ and on P ( A ′ ) . We get E h D D ˆΦ( t, ˆ ξ ) , h A ′ E H i = lim ε → + ˆ ϕ (cid:16) t, X j = i p j δ ˆ x j + P ( A ′ ) δ ˆ x i + εh [ t,T ] + (cid:0) p i − P ( A ′ ) (cid:1) δ ˆ x i (cid:17) − ˆ ϕ (cid:16) t, N X j =1 p j δ ˆ x j (cid:17) ε . (C.3)Notice that the map H × L (Ω; R ) −→ L (Ω; H ) , ( h, ζ ) ζh is bilinear and continuous, hence Λ : H × L (Ω; R ) −→ R , ( h, ζ ) E [ h D ˆΦ( t, ˆ ξ ) , ζh i H ] is a bilinear and continuous form. By the Riesz representation theorem, there exists a (unique)linear and continuous map T : H −→ L (Ω; R ) representing Λ , namely Λ( h, ζ ) = E [ h D ˆΦ( t, ˆ ξ ) , ζh i H ] = E [ T ( h ) ζ ] ∀ ( h, ζ ) ∈ H × L (Ω; R ) . (C.4)By (C.4) and (C.3), it follows that E [ h D ˆΦ( t, ˆ ξ ) , h A ′ i H ] = E [ T ( h ) A ′ ] depends only on the law of ˆ ξ and on P ( A ′ ) for every measurable A ′ ⊂ { ˆ ξ = ˆ x i } . Recall from Lemma 2.1 that the probabilityspace (Ω , F , P ) supports a random variable with uniform distribution on [0 , . We can then applyLemma 2 of [50], getting that T ( h ) is P -a.s. constant on { ˆ ξ = ˆ x i } . For i = 1 , . . . , N , define the map ψ i : H → R by ψ i ( h ) := 1 p i E [ T ( h ) A i ] , ∀ h ∈ H. ψ i ( h ) = T ( h ) , P -a.s. on A i . Notice that, by the very definition of ψ i and the linearity of Λ ,we have N X i =1 ψ i ( h ) E [ ζ A i ] = N X i =1 E [ T ( h ) ζ A i ] = N X i =1 Λ( h, ζ A i ) = Λ( h, ζ ) = E [ h D ˆΦ( t, ˆ ξ ) , hζ i H ] , for all h ∈ H , ζ ∈ L (Ω; R ) . The linearity and continuity of T entails ψ i ∈ H ′ . Therefore, it can beidentified to some φ i ∈ H . Define ˆ g : D ([0 , T ]; H ) → H by ˆ g (ˆ x ) := N X i =1 φ i δ ˆ x i (ˆ x ) ∀ ˆ x ∈ D ([0 , T ]; H ) . Notice that, P -a.s. on Ω and for all h ∈ H , h ˆ g ( ˆ ξ ) , h i H = (cid:28) N X i =1 φ i δ ˆ x i ( ˆ ξ ) , h (cid:29) H = (cid:28) N X i =1 φ i A i , h (cid:29) H = N X i =1 ψ i ( h ) A i = N X i =1 T ( h ) A i = T ( h ) . (C.5)Now let Y := P Mk =1 a k B k be a simple function, where M ∈ N , a k ∈ H, B k ∈ F . By (C.5) it follows E [ h D ˆΦ( t, ˆ ξ ) , Y i H ] = M X k =1 E [ h D ˆΦ( t, ˆ ξ ) , a k B k i H ] = M X k =1 E [ T ( a k ) B k ] (C.6) = M X k =1 E h h ˆ g ( ˆ ξ ) , a k i H B k i = E h h ˆ g ( ˆ ξ ) , Y i H i . Since (C.6) holds for any simple function Y , we conclude D ˆΦ( t, ˆ ξ ) = ˆ g ( ˆ ξ ) P -a.s.which is (C.1). Finally, (C.2) comes from (C.1) and (C.3). C.2 The general case
Proof of Lemma 4.9.
We proceed by approximation. For n ∈ N , let { D ni } i ∈ N be a partition of D ([0 , T ]; H ) made by Borel sets such that diam d Sk ( D ni ) < − n . Such a partition exists because ofthe separability of ( D ([0 , T ]; H ) , d Sk ) . For i ∈ N , choose d ni ∈ D ni . Define, for every n ∈ N , ˆ ζ n := X i ∈ N d ni D ni ( ˆ ξ ) . Then d Sk ( ˆ ξ ( ω ) , ˆ ζ n ( ω )) ≤ − n for all ω ∈ Ω and all n ∈ N . Therefore ˆ ζ n ∈ L (Ω; D ([0 , T ] , H )) and ˆ ζ n → ˆ ξ uniformly. By a diagonal argument, we can choose N n ∈ N such that the sequence { ˆ ξ n } n ∈ N ,defined by ˆ ξ n := N n X i =1 d ni D ni ( ˆ ξ ) (C.7)42onverges to ˆ ξ both P -a.s. and in L (Ω; D ([0 , T ]; H )) . By Lemma C.1, there exists for each n afunction ˆ g n : D ([0 , T ]; H ) → H such that ˆ g n ( ˆ ξ n ) ∈ L (Ω; H ) and D ˆΦ( t, ˆ ξ n ) = ˆ g n ( ˆ ξ n ) . (C.8)Define ˜ g n := N n X i =1 ˆ g n ( d ni ) D ni . Notice that ˆ g n ( ˆ ξ n ) = ˜ g n ( ˆ ξ ) . Then, by (C.8) and by continuity of D ˆΦ( t, · ) in a neighborhood of ˆ ξ , wehave ˜ g n ( ˆ ξ ) → D ˆΦ( t, ˆ ξ ) in L (Ω; H ) . Let { ˜ g n k ( ˆ ξ ) } k be a subsequence converging P -a.s. to D ˆΦ( t, ˆ ξ ) .Define S := (cid:8) ˆ x ∈ D ([0 , T ]; H ) : { ˜ g n k (ˆ x ) } k is convergent (cid:9) , and ˆ g : D ([0 , T ]; H ) → H by ˆ g := S lim k ˜ g n k . Clearly ˆ g is measurable. Notice that P ( ˆ ξ ∈ S ) = 1 ,then lim k →∞ ˜ g n k ( ˆ ξ ) = ˆ g ( ˆ ξ ) P -a.s. in D ([0 , T ]; H ) (C.9)which provides ˆ g ( ˆ ξ ) = D ˆΦ( t, ˆ ξ ) .Finally, if ˆ ξ ′ is distributed as ˆ ξ , then we can perform exactly the same steps by replacing ˆ ξ, ˆ ξ n by ˆ ξ ′ , ˆ ξ ′ n and by choosing the same N n in (C.7) and then the same ˜ g n , ˆ g . Moreover, P ( ˆ ξ ′ ∈ S ) = 1 aswell. Then (C.9) holds with the same ˆ g and with ˆ ξ replaced by ˆ ξ ′ , which entails ˆ g ( ˆ ξ ′ ) = D ˆΦ( t, ˆ ξ ′ ) .The remaining part of the proof goes exactly as in the proof of Corollary 2.3 of [51]. D Proof of Itô’s formula
Proof of Theorem 4.15 . Let ξ, ˆ ϕ be as in the statement. In what follows, we will tacitly makeuse of the non-anticipative property of ˆ ϕ and of its derivatives. Let t ∈ [0 , T ) , s ∈ ( t, T ] . For n ∈ N ,let t n − := t, t nk := t + kn ( s − t ) ∀ k = 1 , . . . , n and X n = P nk =1 X t nk − [ t nk − ,t nk ) . Clearly E [ k X n k T ] < ∞ . Observe that X nt nk ∧· = X nt nk − ∧· + ( X nt nk − X nt nk − ) [ t nk ,T ] ∀ k = 1 , . . . , n. (D.1)By continuity of X , we clearly have lim n →∞ sup r ∈ [0 ,T ] k X r ( ω ) − X nr ( ω ) k H = 0 ∀ ω ∈ Ω . (D.2)Denote X n,θ,k = X nt nk − ∧· + θ ( X nt nk − X nt nk − ) ∀ θ ∈ [0 , . (D.3)By continuity of X , if n → ∞ and if { k n } n ∈ N ⊂ N is a sequence such that k n ∈ { , . . . , n } and t nk n → r in [ t, s ] , then lim n →∞ sup θ ∈ [0 , k X r ∧· ( ω ) − X n,θ,k n t kn ∧· ( ω ) k T = 0 ∀ ω ∈ Ω . (D.4)43onsider the difference ˆ ϕ ( s, X n ) − ˆ ϕ ( t, X n ) , written as ˆ ϕ ( s, ˆ P X n ) − ˆ ϕ ( t, ˆ P X n ) = n X k =1 (cid:18) ˆ ϕ ( t nk , ˆ P X n ) − ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) (cid:19) + n X k =1 (cid:18) ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) − ˆ ϕ ( t nk − , ˆ P X n ) (cid:19) . (D.5)Let us firt take in consideration the quantity ˆ ϕ ( t nk , ˆ P X n ) − ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) . By (D.1) and by the thefact that ˆ ϕ ∈ C , b ( ˆ H ) , we have ˆ ϕ ( t nk , ˆ P X n ) − ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) = ˆΦ( t nk , X n ) − ˆΦ( t nk , X nt nk − ∧· )= Z E h h D ˆΦ (cid:16) t nk , X n,θ,k , (cid:17) , X nt nk − X nt nk − i i dθ = Z E h h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) , X nt nk − X nt nk − i i dθ = (after recalling (4.3)) = Z E " h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) , Z t nk t nk − F r dr i dθ + Z E " h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) , Z t nk t nk − G r dB r i dθ = I n,kF + I n,kB . (D.6)Observe that, by the boundedness assumption on ∂ µ ˆ ϕ and the integrability assumption on F , wecan commute the integral R t nk t nk − · dr first with E and then (since the map [0 , T ] L (Ω; H ) , r F r is measurable and integrable) with R · dθ . By summing over k the quantity I n,kF , we then have n X k =1 I n,kF = n X k =1 Z t nk t nk − Z E h h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) , F r i i dθdr = Z st Z E " h n X k =1 ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) [ t nk − ,t nk ) ( r ) , F r i dθdr. (D.7)By continuity of ∂ µ ˆ ϕ and by (D.4), we have lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k ( ω ) (cid:17) [ t nk − ,t nk ) ( r ) − ∂ µ ˆ ϕ (cid:16) r, ˆ P X (cid:17) ( X ( ω )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H = 0 ∀ ω ∈ Ω . (D.8)By Lebesgue’s dominated convergence theorem and by (D.8),(D.7), we conclude lim n →∞ n X k =1 I n,kF = Z st E h h ∂ µ ˆ ϕ (cid:16) r, ˆ P X (cid:17) ( X ) , F r i i dr. (D.9) But notice that the integral in R · dθ cannot be commuted with E before showing the existence of a jointlymeasurable representant ( θ, ω ) Z ( θ, ω ) of θ ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,θ,k )( ω ) .
44e now address I n,kB . In this case, we cannot immediately commute R t nk t nk − · dB r with E , because X n,θ,k is a-priori only F t nk -measurable. We then add a null term and then expand the sum. Indeed,we have Z E " h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k − (cid:17) , Z t nk t nk − G r dB r i dθ == Z E "Z t nk t nk − h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k − (cid:17) , G r dB r i dθ = 0 . Then, by using the continous second-order pathwise derivative in mesure and space of ˆ ϕ I n,kB = Z E " h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) , Z t nk t nk − G r dB r i dθ = Z E " h ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k (cid:17) − ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,θ,k − (cid:17) , Z t nk t nk − G r dB r i dθ = Z E " h Z ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) (cid:16) X nt nk − X nt nk − (cid:17) θdε, Z t nk t nk − G r dB r i dθ = Z E " h Z ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) Z t nk t nk − F r dr ! θdε, Z t nk t nk − G r dB r i dθ + Z E " h Z ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) Z t nk t nk − G r dB r ! θdε, Z t nk t nk − G r dB r i dθ = I n,kBF + I n,kBB . (D.10)Notice that n X k =1 E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) Z t nk t nk − F r dr ! , Z t nk t nk − G r dB r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ M n X k =1 E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t nk t nk − F r dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t nk t nk − G r dB r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H / ≤ M n X k =1 s − tn E "Z t nk t nk − | F r | H dr E "Z t nk t nk − | G r | L ( K ; H ) dr / ≤ M (cid:18) s − tn (cid:19) / n X k =1 E "Z t nk t nk − | F r | H dr E "Z t nk t nk − | G r | L ( K ; H ) dr / ≤ M (cid:18) s − tn (cid:19) / (cid:18) E (cid:20)Z st | F r | H dr (cid:21) E (cid:20)Z st | G r | L ( K ; H ) dr (cid:21)(cid:19) / , which goes to as n → ∞ . By Lebesgue’s dominated convergence theorem, it then follows lim n →∞ n X k =1 I n,kBF = 0 . (D.11)45ow, in order to compute I n,kBB , consider first that, for m ∈ N and n ≥ m , n X k =1 E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h (cid:16) ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) − ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) (cid:17) Z t nk t nk − G r dB r ! , Z t nk t nk − G r dB r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n X k =1 E (cid:12)(cid:12)(cid:12) ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) − ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) (cid:12)(cid:12)(cid:12) L ( H ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t nk t nk − G r dB r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ≤ E sup n ∈ N , n ≥ m,k =1 ,...,n (cid:12)(cid:12)(cid:12) ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) − ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) (cid:12)(cid:12)(cid:12) L ( H ) (cid:12)(cid:12)(cid:12)(cid:12)Z st G r dB r (cid:12)(cid:12)(cid:12)(cid:12) H . By continuity of ∂ x ∂ µ ˆ ϕ and by (D.4) it follows lim m →∞ sup n ∈ N , n ≥ m,k =1 ,...,n (cid:12)(cid:12)(cid:12) ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ( ω )) − ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ( ω )) (cid:12)(cid:12)(cid:12) = 0 , for every ω ∈ Ω , and then, by Lebesgue’s dominated convergence theorem, lim n →∞ n X k =1 E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h (cid:16) ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k ) − ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) (cid:17) Z t nk t nk − G r dB r ! , Z t nk t nk − G r dB r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (D.12)By (D.12) and by Lebesgue’s dominated convergence theorem, we can then write lim n →∞ n X k =1 I n,kBB == lim n →∞ n X k =1 Z E " h Z ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) Z t nk t nk − G r dB r ! θdε, Z t nk t nk − G r dB r i dθ = lim n →∞ n X k =1 Z Z θ E "Z t nk t nk − Tr (cid:16) G r G ∗ r ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) (cid:17) dr dεdθ = lim n →∞ Z Z θ E "Z st n X k =1 Tr (cid:16) G r G ∗ r ∂ x ∂ µ ˆ ϕ ( t nk , ˆ P X n,θ,k )( X n,εθ,k − ) (cid:17) [ t nk − ,t nk ) ( r ) dr dεdθ. (D.13)Moreover, by continuity of ∂ x ∂ µ ˆ ϕ and by (D.4), we have lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 ∂ x ∂ µ ˆ ϕ (cid:16) t nk , ˆ P X n,θ,k (cid:17) (cid:16) X n,εθ,k ( ω ) (cid:17) [ t nk − ,t nk ) ( r ) − ∂ x ∂ µ ˆ ϕ (cid:16) r, ˆ P X (cid:17) ( X ( ω )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H = 0 ∀ ω ∈ Ω . This, together with (D.13), provides lim n →∞ n X k =1 I n,kBB = 12 Z st E h Tr (cid:16) G r G ∗ r ∂ x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:17)i dr. (D.14)46e now consider the quantity P nk =1 ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) − ˆ ϕ ( t nk − , ˆ P X n ) appearing in (D.5). By usingthe continuity of the pathwise time derivative ∂ t ˆ ϕ , we can write n X k =1 (cid:18) ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) − ˆ ϕ ( t nk − , ˆ P X ntnk − ∧· ) (cid:19) = n X k =1 Z t nk t nk − ∂ t ˆ ϕ ( r, ˆ P X ntnk − ∧· ) dr = Z st n X k =1 ∂ t ˆ ϕ ( r, ˆ P X ntnk − ∧· ) [ t nk − ,t nk ) ( r ) dr (D.15)By continuity of ∂ t ˆ ϕ and by (D.4), we have lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 ∂ t ˆ ϕ ( r, ˆ P X ntnk − ∧· ) [ t nk − ,t nk ) ( r ) − ∂ t ˆ ϕ ( r, ˆ P X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (D.16)By (D.15),(D.16), and by Lebesgue’s dominated convergence theorem, we obtain lim n →∞ n X k =1 (cid:18) ˆ ϕ ( t nk , ˆ P X ntnk − ∧· ) − ˆ ϕ ( t nk − , ˆ P X ntnk − ∧· ) (cid:19) = Z st ∂ t ˆ ϕ ( r, ˆ P X ) dr. (D.17)Putting together (D.5), (D.6), (D.9), (D.10), (D.11), (D.14), (D.17), we finally obtain, by recallingalso (D.4), ˆ ϕ ( s, ˆ P X ) − ˆ ϕ ( t, ˆ P X ) = lim n →∞ (cid:16) ˆ ϕ ( s, ˆ P X n ) − ˆ ϕ ( t, ˆ P X n ) (cid:17) = Z st ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) dr + Z st E h h F r , ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) i H i dr + 12 Z st E h Tr (cid:16) G r G ∗ r ∂ x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:17)i dr, which concludes the proof. E Consistency property of pathwise derivatives
Proof of Lemma 4.20.
The consistency of pathwise time derivatives is a direct consequence oftheir definition (Definition 4.3).We now prove the consistency of the other two derivatives using Itô’s formula (Theorem 4.15).Fix t ∈ [0 , T ] and let ξ ∈ S ( F ) . Let also F : [0 , T ] × Ω → H (resp. G : [0 , T ] × Ω → L ( K ; H ) ) be aintegrable (resp. square-integrable) and F -progressively measurable process, so, in particular, Z T E [ | F s | H ] ds < ∞ , Z T E (cid:2) Tr ( G s G ∗ s ) (cid:3) ds < ∞ . Consider the process X = ( X s ) s ∈ [0 ,T ] given by X s = ξ s ∧ t + Z s ∨ tt F r dr + Z s ∨ tt G r dB r , ∀ s ∈ [0 , T ] . s ∈ [ t, T ] , ˆ ϕ ( s, ˆ P X ·∧ s ) = ˆ ϕ ( t, ˆ P ξ ·∧ t ) + Z st ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) dr + Z st E h h F r , ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) i H i dr + 12 Z st E h Tr (cid:16) G r G ∗ r ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:17)i dr and ˆ ϕ ( s, ˆ P X ·∧ s ) = ˆ ϕ ( t, ˆ P ξ ·∧ t ) + Z st ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) dr + Z st E h h F r , ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) i H i dr + 12 Z st E h Tr (cid:16) G r G ∗ r ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:17)i dr. Since ˆ ϕ ( r, ˆ P X ·∧ r ) = ˆ ϕ ( r, ˆ P X ·∧ r ) and ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) = ∂ t ˆ ϕ ( r, ˆ P X ·∧ r ) , for every r ∈ [0 , T ] , we find Z st E h(cid:10) F r , (cid:0) ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) − ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:1)(cid:11) H i dr (E.1) + 12 Z st E (cid:2) Tr (cid:0) G r G ∗ r (cid:0) ∂ sym x ∂ µ ˆ ϕ ( r, P X ·∧ r )( X ·∧ r ) − ∂ sym x ∂ µ ˆ ϕ ( r, P X ·∧ r )( X ·∧ r ) (cid:1)(cid:1)(cid:3) dr = 0 , for every s ∈ [ t, T ] . Consistency of ∂ µ ˆ ϕ and ∂ µ ˆ ϕ . Let Z : Ω → H be an F t -measurable random variable in L (Ω; H ) ,and define F s := Z [ t,T ] ( s ) , G s := 0 , ∀ s ∈ [0 , T ] . Then, from (E.1) we obtain Z st E h(cid:10) Z, (cid:0) ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) − ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:1)(cid:11) H i dr = 0 , ∀ s ∈ [ t, T ] . (E.2)From the continuity of the map r ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) − ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) , we get in particularthat the integrand of (E.2) is equal to zero at r = t , namely E h(cid:10) Z, (cid:0) ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) − ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) (cid:1)(cid:11) H i = 0 . From the arbitrariness of the F t -measurable random variable Z ∈ L (Ω; H ) , this yields ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) = ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) , P -a.s.Recalling Remark 4.22, we see that this proves the consistency of ∂ µ ˆ ϕ and ∂ µ ˆ ϕ . Consistency of ∂ sym x ∂ µ ˆ ϕ and ∂ sym x ∂ µ ˆ ϕ . Let
Λ : Ω → L ( K ; H ) be an F t -measurable randomvariable in L (Ω; L ( K ; H )) , and define F s := 0 , G s := Λ [ t,T ] ( s ) , ∀ s ∈ [0 , T ] . Then, from (E.1) we obtain Z st E h Tr (cid:16) ΛΛ ∗ (cid:0) ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) − ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) (cid:1)(cid:17)i dr = 0 , ∀ s ∈ [ t, T ] . r ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) − ∂ sym x ∂ µ ˆ ϕ ( r, ˆ P X ·∧ r )( X ·∧ r ) , we get E h Tr (cid:16) ΛΛ ∗ (cid:0) ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) − ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) (cid:1)(cid:17)i = 0 . From the arbitrariness of the F t -measurable random variable Λ ∈ L (Ω; L ( K ; H )) , we concludethat ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) = ∂ sym x ∂ µ ˆ ϕ ( t, ˆ P ξ ·∧ t )( ξ ·∧ t ) , P -a.s.which, together with Remark 4.22, gives the claimed consistency of ∂ sym x ∂ µ ˆ ϕ and ∂ sym x ∂ µ ˆ ϕ . F Hamilton-Jacobi-Bellman equation: technical results
In the present appendix we prove three technical results which are used in Section 5 to derivealternative forms of the Hamilton-Jacobi-Bellman equation (5.1).
Lemma F.1.
Given µ ∈ P ( C ([0 , T ]; H )) there exists ξ ∈ S ( G ) and a G -measurable randomvariable U ξ : Ω → R such that:1) P ξ = µ ;2) U ξ has uniform distribution on [0 , ;3) ξ and U ξ are independent. Proof.
Fix µ ∈ P ( C ([0 , T ]; H )) and consider the probability space ([0 , , B ([0 , , λ ) , where λ denotes the Lebesgue measure on the unit interval. Denote ¯Ω = [0 , × C ([0 , T ]; H ) , ¯ F = B ([0 , ⊗ B , ¯ P = λ ⊗ µ. Fix an orthonormal basis { e n } n ∈ N of H . Then, let J : [0 , × C ([0 , T ]; H ) → C ([0 , T ]; H ) be themap defined as J ( r, x ) := re + ∞ X n =2 h x, e n − i H e n , ∀ ( r, x ) ∈ [0 , × C ([0 , T ]; H ) . Let ˜ µ denote the image measure (or push-forward) of ¯ P = λ ⊗ µ by J . Notice that ˜ µ ∈ P ( C ([0 , T ]; H )) .Then, from property (A G ) -ii) it follows that there exists a continuous and B ([0 , T ]) ⊗ G -measurableprocess ˜ ξ : [0 , T ] × Ω → H with law equal to ˜ µ .Now, define the maps P : C ([0 , T ]; H ) → R and Q : C ([0 , T ]; H ) → C ([0 , T ]; H ) as P ( x ) := h x , e i H , Q ( x ) := ∞ X n =1 h x, e n +1 i H e n , ∀ x ∈ C ([0 , T ]; H ) , where we recall that x is the value of the path x at time t = 0 . Then, denote U ξ := P ( ˜ ξ ) , ξ := Q ( ˜ ξ ) . It is then easy to see that U ξ and ξ satisfy the claimed properties.49 emma F.2. Fix t ∈ [0 , T ] , ξ ∈ S ( G ) and let F : C ([0 , T ]; H ) × U × P (U) → R be a measurablefunction. Suppose that E [ | F ( ξ, a , P a ) | ] < + ∞ , ∀ a ∈ M t . Suppose also that there exists a G -measurable random variable U ξ : Ω → R having uniform distribution on [0 , and being independentof ξ . Then, it holds that sup a ∈M t E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) = sup a ∈M G E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) = sup ˇa ∈ ˇ M E (cid:2) F (cid:0) ξ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1)(cid:3) . (F.1) Proof.
Since ˇa( ξ, U ξ ) ∈ M G and M G ⊂ M t , we immediately get sup a ∈M t E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) ≥ sup a ∈M G E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) ≥ sup ˇa ∈ ˇ M E (cid:2) F (cid:0) ξ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1)(cid:3) . It remains to prove the inequality sup ˇa ∈ ˇ M E (cid:2) F (cid:0) ξ, ˇa( ξ, U ξ ) , P ˇa( ξ,U ξ ) (cid:1)(cid:3) ≥ sup a ∈M t E (cid:2) F (cid:0) ξ, a , P a (cid:1)(cid:3) . (F.2)To this end, denote ˆΩ = [0 , T ] × Ω , ˆ F = B ([0 , T ]) ⊗ F , ˆ P = λ T ⊗ P , with λ T being the uniform distribution on ([0 , T ] , B ([0 , T ])) . Given a ∈ M t , consider the canonicalextensions of a and U ξ to ˆΩ , which will be denoted respectively by ˆ a and ˆ U ξ (notice that ˆ U ξ hasuniform distribution on [0 , and is independent of ξ ). We can now apply Lemma B.1, with ( E, E ) , Γ , ˆ U being respectively ( C ([0 , T ]; H ) , B ) , ξ , ˆ U ξ . Then, it follows the existence of a measurable map ˇa : C ([0 , T ]; H ) × [0 , → U such that (cid:0) ξ, ˇa( ξ, ˆ U ξ ) (cid:1) L ˆΩ = (cid:0) ξ, ˆ a (cid:1) , where L ˆΩ stands for equality in law between random objects defined on ( ˆΩ , ˆ F , ˆ P ) . Then, we deduce (cid:16) ( ξ s ) s ∈ [0 ,T ] , ˇa( ξ, U ξ ) (cid:17) L = (cid:16) ( ξ s ) s ∈ [0 ,T ] , a (cid:17) , where L stands for equality in law between random objects defined on (Ω , F , P ) . This implies thevalidity of inequality (F.2) and concludes the proof of (F.1). Lemma F.3.
Fix t ∈ [0 , T ] , ξ ∈ S ( G ) and let F : C ([0 , T ]; H ) × U → R be a measurable function.Suppose that E [ | F ( ξ, a ) | ] < + ∞ , ∀ a ∈ M t , and also that sup u ∈ U F ( x, u ) < + ∞ , ∀ x ∈ C ([0 , T ]; H ) .Then, it holds that sup a ∈M t E (cid:2) F ( ξ, a ) (cid:3) = sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) (F.3) and sup a ∈M t E (cid:2) F ( ξ, a ) (cid:3) = E h ess sup u ∈ U F ( ξ, u ) i . (F.4) Remark F.4.
Suppose that ξ and U ξ are as in Lemma F.1, namely ξ ∈ S ( G ) , P ξ = µ , and ξ issuch that there exists a G -measurable random variable U ξ having uniform distribution on [0 , andbeing independent of ξ . Then, formula (F.3) holds without assuming that sup u ∈ U F ( x, u ) < + ∞ , x ∈ C ([0 , T ]; H ) . As a matter of fact, in this case, thanks to formula (F.1) , it is enough to provethat sup ˇa ∈ ˇ M E (cid:2) F ( ξ, ˇa( ξ, U ξ )) (cid:3) = sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) . Clearly, it holds that sup ˇa ∈ ˇ M E [ F ( ξ, ˇa( ξ, U ξ ))] ≥ sup a ∈M E [ F ( ξ, a( ξ ))] . On the other hand, for everyfixed ˇa ∈ ˇ M we have E (cid:2) F ( ξ, ˇa( ξ, U ξ )) (cid:3) = Z E (cid:2) F ( ξ, ˇa( ξ, r )) (cid:3) dr ≤ sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) . From the arbitrariness of ˇa we get the other inequality and we conclude that (F.3) holds. Proof of Lemma F.3.
Since a( ξ ) ∈ M t , we immediately get sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) ≤ sup a ∈M t E (cid:2) F ( ξ, a ) (cid:3) . In addition, for every fixed a ∈ M t we have E (cid:2) F (cid:0) ξ, a (cid:1)(cid:3) ≤ E h ess sup u ∈ U F ( ξ, u ) i . From the arbitrariness of a , we find sup a ∈M t E [ F ( ξ, a )] ≤ E [ess sup u ∈ U F ( ξ, u )] . Then, in order toprove the validity of both (F.3) and (F.4), it remains to prove that E h ess sup u ∈ U F ( ξ, u ) i ≤ sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) . (F.5)Let µ = P ξ denote the law of ξ . Suppose for a moment that for every ε > there exists a εµ ∈ M and a µ -null set N ε ∈ B such that F ( x, u ) ≤ F ( x, a εµ ( x )) + ε, ∀ x ∈ C ([0 , T ]; H ) \ N ε , ∀ u ∈ U . (F.6)Then, in particular, there exists a P -null set ¯ N ε ∈ F such that F ( ξ ( ω ) , u ) ≤ F ( ξ ( ω ) , a εµ ( ξ ( ω ))) + ε, ∀ ω ∈ Ω \ ¯ N ε , ∀ u ∈ U . As a consequence, using the definition of essential supremum for the family of real-valued randomvariables { F ( ξ, u ) } u ∈ U , we find ess sup u ∈ U F ( ξ, u ) ≤ F ( ξ, a εµ ( ξ )) + ε, P -a.s.This yields E h ess sup u ∈ U F ( ξ, u ) i ≤ E (cid:2) F ( ξ, a εµ ( ξ )) (cid:3) + ε ≤ sup a ∈M E (cid:2) F ( ξ, a( ξ )) (cid:3) + ε. From the arbitrariness of ε we get inequality (F.5) and we conclude that (F.3) holds.It remains to prove (F.6). To this end, we use that U is a Borel space and we implementProposition 7.50 in [8] (in particular, X , Y , D , f in the statement of this latter proposition aregiven respectively by C ([0 , T ]; H ) , U , C ([0 , T ]; H ) × U , F ). By Proposition 7.50 in [8] it follows51hat, for every ε > , there exists an analytically measurable function (we refer to Definition 7.20in [8] for the definition of analytically measurable function) a ε : C ([0 , T ]; H ) → U such that F ( x, a ε ( x )) ≥ ( sup u ∈ U F ( x, u ) − ε, if sup u ∈ U F ( x, u ) < + ∞ , /ε, if sup u ∈ U F ( x, u ) = + ∞ , for all x ∈ C ([0 , T ]; H ) . Since it holds that sup u ∈ U F ( x, u ) < + ∞ , ∀ x ∈ C ([0 , T ]; H ) , we can rewritethe above inequality simply as sup u ∈ U F ( x, u ) ≤ F ( x, a ε ( x )) + ε, ∀ x ∈ C ([0 , T ]; H ) . (F.7)Using Lemma 7.27 in [8] and the fact that U is Borel-isomorphic to a Borel subset of [0 , , we seethat there exists a Borel-measurable function a εµ : C ([0 , T ]; H ) → U such that a ε ( x ) = a εµ ( x ) for µ -a.e. x ∈ C ([0 , T ]; H ) . Hence, from (F.7) we obtain (F.6). References [1] B. Acciaio, J. Backhoff-Veraguas, and R. Carmona. Extended mean field control problems: stochasticmaximum principle and transport perspective.
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