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1 Working Paper / Document de travail HJB Equation in Infinite Dimenion and Optimal Control of Stochatic Evolution Equation via Generalized Fukuhima Decompoition Giorgio Fabbri Franceco Ruo WP Nr 04

2 HJB equation in infinite dimenion and optimal control of tochatic evolution equation via generalized Fukuhima decompoition Giorgio Fabbri and Franceco Ruo January 2017 Abtract A tochatic optimal control problem driven by an abtract evolution equation in a eparable Hilbert pace i conidered. Thank to the identification of the mild olution of the tate equation a ν-weak Dirichlet proce, the value procee i proved to be a real weak Dirichlet proce. The uniquene of the correponding decompoition i ued to prove a verification theorem. Through that technique everal of the required aumption are milder than thoe employed in previou contribution about non-regular olution of Hamilton-Jacobi-Bellman equation. KEY WORDS AND PHRASES: Weak Dirichlet procee in infinite dimenion; Stochatic evolution equation; Generalized Fukuhima decompoition; Stochatic optimal control in Hilbert pace AMS MATH CLASSIFICATION: 35Q93, 93E20, 49J20 Aix-Mareille Univ. (Aix-Mareille School of Economic), CNRS, EHESS and Centrale Mareille. 2, Rue de la Charité, Mareille, France. giorgio.fabbri@univ-amu.fr. ENSTA PariTech, Univerité Pari-Saclay, Unité de Mathématique appliquée. 828, Boulevard de Maréchaux, F Palaieau, France. franceco.ruo@entaparitech.fr. 1

3 1 Introduction The goal of thi paper i to how that, if we carefully exploit ome recent development in tochatic calculu in infinite dimenion, we can weaken ome of the hypothee typically demanded in the literature of non-regular olution of Hamilton-Jacobi-Bellman (HJB) equation to prove verification theorem and optimal ynthee of tochatic optimal control problem in Hilbert pace. A well-known, the tudy of a dynamic optimization problem can be linked, via the dynamic programming to the analyi of the related HJB equation, that i, in the context we are intereted in, a econd order infinite dimenion PDE. When thi approach can be uccefully applied, one can prove a verification theorem and expre the optimal control in feedback form (that i, at any time, a a function of the tate) uing the olution of the HJB equation. In thi cae the latter can be identified with the value function of the problem. In the regular cae (i.e. when the value function i C 1,2, ee for intance Chapter 2 of [12]) the tandard proof of the verification theorem i baed on the Itô formula. In thi paper we how that ome recent reult in tochatic calculu, in particular Fukuhima-type decompoition explicitly uited for the infinite dimenional context, can be ued to prove the ame kind of reult for le regular olution of the HJB equation. The idea i the following. In a previou paper ([13]) the author introduced the cla of ν-weak Dirichlet procee (the definition i recalled in Section 2, ν i a Banach pace trictly aociated with a uitable ubpace ν 0 of H) and howed that convolution type procee, and in particular mild olution of infinite dimenional tochatic evolution equation (ee e.g. [6], Chapter 4), belong to thi cla. By applying thi reult to the olution of the tate equation of a cla of tochatic optimal control problem in infinite dimenion we are able to how that the value proce, that i the value of any given olution of the HJB equation computed on the trajectory taken into account 1, i a (real-valued) weak Dirichlet procee (with repect to a given filtration), a notion introduced in [11] and ubequently analyzed in [26]. Such a proce can be written a the um of a local martingale and a martingale orthogonal proce, i.e. having zero covariation with every continuou local martingale. Such decompoition i unique and in Theorem 3.7, we exploit that uniquene property to characterize the martingale part of the value proce a a uitable tochatic integral with repect to a Giranov-tranformed Wiener proce which allow to obtain a ubtitute of the Itô-Dynkin formula for olution of the Hamilton-Jacobi-Bellman equation. Thi i poible when the value proce aociated to the optimal control problem can be expreed by a C 0,1 ([0, T [ H) function of the tate proce, with however a tronger regularity on the firt derivative. We finally ue thi expreion to prove the verification reult tated in Theorem The expreion value proce i ometime ued for the value of the value function computed on the trajectory, often the two definition equal but it not alway the cae. 2 A imilar approach i ued, when H i finite-dimenional, in [25]. In that cae thing are 2

4 We think the interet of our contribution i twofold. On the one hand we how that recent development in tochatic calculu in Banach pace, ee for intance [9, 10], from which we adopt the framework related to generalized covariation and Itô-Fukuhima formulae, but alo other approache a [5, 28, 35] may have important control theory counterpart application. On the other hand the method we preent allow to improve ome previou verification reult weakening a erie of hypothee. We dicu here thi econd point in detail. There are everal way to introduce non-regular olution of econd order HJB equation in Hilbert pace. They are more preciely urveyed in [12] but they eentially are vicoity olution, trong olution and the tudy of the HJB equation through backward SDE. Vicoity olution are defined, a in the finite-dimenional cae, uing tet function that locally touch the candidate olution. The vicoity olution approach wa firt adapted to the econd order Hamilton Jacobi equation in Hilbert pace in [29, 30, 31] and then, for the unbounded cae (i.e. including a poibly unbounded generator of a trongly continuou emigroup in the tate equation, ee e.g. equation (5)) in [34]. Several improvement of thoe pioneering tudie have been publihed, including extenion to everal pecific equation but, differently from what happen in the finite-dimenional cae, there are no verification theorem available at the moment for tochatic problem in infinite-dimenion that ue the notion of vicoity olution. The backward SDE approach can be applied when the mild olution of the HJB equation can be repreented uing the olution of a forward-backward ytem. It wa introduced in [32] in the finite dimenional etting and developed in everal work, among them [7, 15, 16, 17, 18]. Thi method only allow to find optimal feedback in clae of problem atifying a pecific tructural condition, impoing, roughly peaking, that the control act within the image of the noie. The ame limitation concern the L 2 µ approach introduced and developed in [1] and [20]. In the trong olution approach, firt introduced in [2], the olution i defined a a proper limit of olution of regularized problem. Verification reult in thi framework are given in [21, 22, 23, 24]. They are collected and refined in Chapter 4 of [12]. The reult obtained uing trong olution are the main term of comparion for our both becaue in thi context the verification reult are more developed and becaue we partially work in the ame framework by approximating the olution of the HJB equation uing olution of regularized problem. With reference to them our method ha ome advantage 3 : (i) the aumption on the cot tructure are milder, notably they do not include any continuity aumption on the running cot that i only aked to be a meaurimpler and there i not need to ue the notion of ν-weak Dirichlet procee and and reult that are pecifically uited for the infinite dimenional cae. In that cae ν 0 will be iomorphic to the full pace H. 3 Reult for pecific cae, a boundary control problem and reaction-diffuion equation (ee [3, 4]) cannot be treated at the moment with the method we preent here. 3

5 able function; moreover the admiible control are only aked to verify, together with the related trajectorie, a quai-integrability condition of the functional, ee Hypothei 3.3 and the ubequent paragraph; (ii) we work with a bigger et of approximating function becaue we do not require the approximating function and their derivative to be uniformly bounded; (iii) the convergence of the derivative of the approximating olution i not neceary and it i replaced by the weaker condition (16). Thi convergence, in different poible form, i unavoidable in the tandard tructure of the trong olution approach and it i avoided here only thank to the ue of Fukuhima decompoition in the proof. In term of the lat jut mentioned two point, our notion of olution i weaker than thoe ued in the mentioned work, we need neverthele to aume that the gradient of the olution of the HJB equation i continuou a an D(A )-valued function. The paper proceed a follow. Section 2 i devoted to ome preliminary notion, notably the definition of ν-weak-dirichlet proce and ome related reult. Section 3 focue on the optimal control problem and the related HJB equation. It include the key decompoition Theorem 3.7. Section 4 concern the verification theorem. 2 Some preliminary definition and reult Conider a complete probability pace (Ω, F, P). Fix T > 0 and [0, T [. Let {F t } t be a filtration atifying the uual condition. Each time we ue expreion a adapted, martingale, etc... we alway mean with repect to the filtration {F t } t. Given a metric pace S we denote by B(S) the Borel σ-field on S. Conider two real Hilbert pace H and G. By default we aume that all the procee X: [, T ] Ω H are meaurable function with repect to the product σ- algebra B([, T ]) F with value in (H, B(H)). Similar convention are done for G-valued procee. We denote by H ˆ π G the projective tenor product of H and G, ee [33] for detail. Definition 2.1. A real proce X : [, T ] Ω R i called weak Dirichlet proce if it can be written a X = M + A, where M i a local martingale and A i a proce uch that A() = 0 and [A, N] = 0 for every continuou local martingale N. The following reult i proved in Remark 3.5 and 3.2 of [26]. Theorem The decompoition decribed in Definition 2.1 i unique. 2. A emimartingale i a weak Dirichlet proce. We recall that, following [8, 10], a Chi-ubpace (of (H ˆ π G) ) i defined a any Banach ubpace (χ, χ ) which i continuouly embedded into (H ˆ π G) 4

6 and, following [13], given a Chi-ubpace χ we introduce the notion of χ- covariation a follow. Definition 2.3. Given two proce X: [, T ] H and X: [, T ] G, we ay that (X, Y) admit a χ-covariation if the two following condition are atified. H1 For any equence of poitive real number ɛ n 0 there exit a ubequence ɛ nk uch that up k (J (X(r + ɛ nk ) X(r)) (Y(r + ɛ nk ) Y(r))) χ dr < a.., ɛ nk where J : H ˆ π G (H ˆ π G) i the canonical injection between a pace and it bidual. (1) H2 If we denote by [X, Y] ɛ χ the application [X, Y] ɛ χ : χ C([, T ]) J ((X(r + ɛ) X(r)) (Y(r + ɛ) Y(r))) φ φ, dr, χ ɛ χ the following two propertie hold. (2) (i) There exit an application, denoted by [X, Y] χ, defined on χ with value in C([, T ]), atifying [X, Y] ɛ χ(φ) for every φ χ (H ˆ π G). ucp ɛ 0 + [X, Y] χ (φ), (3) (ii) There exit a Bochner meaurable proce [X, Y] χ : Ω [, T ] χ, uch that for almot all ω Ω, [X, Y] χ (ω, ) i a (càdlàg) bounded variation proce, [X, Y] χ (, t)(φ) = [X, Y] χ (φ)(, t) a.. for all φ χ, t [, T ]. If (X, Y) admit a χ-covariation we call [X, Y] χ-covariation of (X, Y). If [X, Y] vanihe we alo write that [X, Y ] χ = 0. We ay that a proce X admit a χ-quadratic variation if (X, X) admit a χ-covariation. In that cae [X, X] i called χ-quadratic variation of X. Definition 2.4. Let H and G be two eparable Hilbert pace. Let ν (H ˆ π G) be a Chi-ubpace. A continuou adapted H-valued proce A: [, T ] Ω H i aid to be ν-martingale-orthogonal if [A, N] ν = 0, for any G-valued continuou local martingale N. 5

7 Lemma 2.5. Let H and G be two eparable Hilbert pace, V: [, T ] Ω H a bounded variation proce. Then the two item below hold. 1. Given any continuou proce Z: [, T ] Ω G and any Chi-ubpace ν (H ˆ π G), we have [V, Z] ν = In particular, for any any Chi-ubpace ν (H ˆ π G), V i ν-martingaleorthogonal. Proof. By Lemma 3.2 of [13] it i enough to how that A(ε) := up Φ ν, Φ ν 1 J ( ) ε 0 (V(t + ε) V(t)) (Z(t + ε) Z(t)), Φ dt 0 in probability (the procee are extended on ]T, T + ε] by defining, for intance, Z(t) = Z(T ) for any t ]T, T + ε]). Now, ince ν i continuouly embedded in (H ˆ π G), there exit a contant C uch that (H ˆ π G) C ν o that A(ε) C C = C up Φ ν, Φ (H ˆ π G) 1 J J ( ) (V(t + ε) V(t)) (Z(t + ε) Z(t)), Φ dt ( (V(t + ε) V(t)) (Z(t + ε) Z(t)) ( (V(t + ε) V(t)) (Z(t + ε) Z(t)) ) (H ˆ πg) ) (H ˆ πg) = C (V(t + ε) V(t)) H (Z(t + ε) Z(t)) G dt, (4) where the lat tep follow by Propoition 2.1 page 16 of [33]. Now, denoting t Y (t) the real total variation function of an H-valued bounded variation function Y defined on the interval [, T ] we get +ε +ε) Y(t + ε) Y(t) = dy (r) d Y (r). So, by uing Fubini theorem in (4), A(ε) Cδ(Z; ε) t +ε t d V (r), where δ(z; ε) i the modulu of continuity of Z. Finally thi converge to zero almot urely and then in probability. Definition 2.6. Let H and G be two eparable Hilbert pace. Let ν (H ˆ π G) be a Chi-ubpace. A continuou H-valued proce X: [, T ] Ω H i called ν-weak-dirichlet proce if it i adapted and there exit a decompoition X = M + A where dt dt 6

8 (i) M i an H-valued continuou local martingale, (ii) A i an ν-martingale-orthogonal proce with A() = 0. The theorem below wa the object of Theorem 3.19 of [13]. Theorem 2.7. Let ν 0 be a Banach ubpace continuouly embedded in H. Define ν := ν 0 ˆ π R and χ := ν 0 ˆ π ν 0. Let F : [, T ] H R be a C 0,1 -function. Denote with x F the Fréchet derivative of F with repect to x and aume that the mapping (t, x) x F (t, x) i continuou from [, T ] H to ν 0. Let X(t) = M(t) + A(t) for t [, T ] be an ν-weak-dirichlet proce with finite χ- quadratic variation. Then Y (t) := F (t, X(t)) i a real weak Dirichlet proce with local martingale part R(t) = F (, X()) + x F (r, X(r)), dm(r), t [, T ]. 3 The etting of the problem and HJB equation In thi ection we introduce a cla of infinite dimenional optimal control problem and we prove a decompoition reult for the trong olution of the related Hamilton-Jacobi-Bellman equation. We refer the reader to [36] and [6] repectively for the claical notion of functional analyi and tochatic calculu in infinite dimenion we ue. 3.1 The optimal control problem Aume from now that H and U are real eparable Hilbert pace, Q L(U), U 0 := Q 1/2 (U). Aume that W Q = {W Q (t) : t T } i an U-valued F t -Q-Wiener proce (with W Q () = 0, P a..) and denote by L 2 (U 0, H) the Hilbert pace of the Hilbert-Schmidt operator from U 0 to H. We denote by A: D(A) H H the generator of the C 0 -emigroup e ta (for t 0) on H. A denote the adjoint of A. Recall that D(A) and D(A ) are Banach pace when endowed with the graph norm. Let Λ be a Polih pace. We formulate the following tandard aumption that will be needed to enure the exitence and the uniquene of the olution of the tate equation. Hypothei 3.1. b: [0, T ] H Λ H i a continuou function and atifie, for ome C > 0, b(, x, a) b(, y, a) C x y, b(, x, a) C(1 + x ), for all x, y H, [0, T ], a Λ. σ : [0, T ] H L 2 (U 0, H) i continuou and, for ome C > 0, atifie, σ(, x) σ(, y) L2(U 0,H) C x y, σ(, x) L2(U 0,H) C(1 + x ), 7

9 for all x, y H, [0, T ]. Given an adapted proce a = a( ) : [, T ] Ω Λ, we conider the tate equation { dx(t) = (AX(t) + b(t, X(t), a(t))) dt + σ(t, X(t)) dwq (t) (5) X() = x. The olution of (5) i undertood in the mild ene: an H-valued adapted proce X( ) i a olution if { } T ( ) P X(r) + b(r, X(r), a(r)) + σ(r, X(r)) 2 L 2(U 0,H) dr < + = 1 and X(t) = e (t )A x + e (t r)a b(r, X(r), a(r)) dr + e (t r)a σ(r, X(r)) dw Q (r) (6) P-a.. for every t [, T ]. Thank to Theorem 3.3 of [19], given Hypothei 3.1, there exit a unique (up to modification) continuou (mild) olution X( ;, x, a( )) of (5). Propoition 3.2. Set ν 0 = D(A ), ν = ν 0 ˆ π R, χ = ν 0 ˆ π ν 0. The proce X( ;, x, a( )) i ν-weak-dirichlet proce admitting a χ-quadratic variation with decompoition M + A where M i the local martingale defined by M(t) = x + t σ(r, X(r)) dw Q(r) and A i a ν-martingale-orthogonal proce. Proof. See Corollary 4.6 of [13]. Hypothei 3.3. Let l : [0, T ] H Λ R (the running cot) be a meaurable function and g : H R (the terminal cot) a continuou function. We conider the cla U of admiible control contituted by the adapted procee a : [, T ] Ω Λ uch that (r, ω) l(r, X(r,, x, a( )), a(r)) + g(x(t,, x, a( ))) i dr dp- i quai-integrable. Thi mean that, either it poitive or negative part are integrable. We conider the problem of minimizing, over all a( ) U, the cot functional [ ] J(, x; a( )) = E l(r, X(r;, x, a( )), a(r)) dr + g(x(t ;, x, a( ))). (7) The value function of thi problem i defined, a uual, a V (, x) = inf J(, x; a( )). (8) a( ) U A uual we ay that the control a ( ) U i optimal at (, x) if a ( ) minimize (7) among the control in U, i.e. if J(, x; a ( )) = V (, x). In thi cae the pair (a ( ), X ( )), where X ( ) := X( ;, x, a ( )), i called an optimal couple at (, x). 8

10 3.2 The HJB equation The HJB equation aociated with the minimization problem above i v + A x v, x T r [ σ(, x)σ (, x) xxv ] 2 { } + inf a Λ x v, b(, x, a) + l(, x, a) = 0, v(t, x) = g(x). (9) In the above equation x v (repectively 2 xxv) i the firt (repectively econd) Fréchet derivative of v with repect to the x variable; it i identified (via Riez Repreentation Theorem, ee [36], Theorem III.3) with element of H, repectively (ee [14], tatement 3.5.7, page 192) with a ymmetric bounded operator on H; v i the derivative with repect to the time variable. The function F CV (, x, p, a) := p, b(, x, a) +l(, x, a), (, x, p, a) [0, T ] H H Λ, (10) i called the current value Hamiltonian of the ytem and it infimum over a Λ F (, x, p) := inf { p, b(, x, a) + l(, x, a)} (11) a Λ i called the Hamiltonian. Uing thi notation the HJB equation (9) can be rewritten a { v + A x v, x T r [ σ(, x)σ (, x) xv ] 2 + F (, x, x v) = 0, (12) v(t, x) = g(x). The hypothei below will be ued in the equel. Hypothei 3.4. The Hamiltonian F (, x, p) i well-defined and finite for all (, x, p) [0, T ] H H and it i continuou in the three variable. We introduce the operator L 0 on C([0, T ] H) defined a { D(L0 ) := { ϕ C 1,2 ([0, T ] H) : x ϕ C([0, T ] H; D(A )) } L 0 (ϕ)(, x) := ϕ(, x) + A x ϕ(, x), x T r [ σ(, x)σ (, x) xxϕ(, 2 x) ], (13) o that the HJB equation (12) can be formally rewritten a { L0 (v)(, x) = F (, x, x v(, x)) (14) v(t, x) = g(x). Recalling that we uppoe the validity of Hypothee 3.3 and 3.4, we conider the two following definition of olution of the HJB equation. Definition 3.5. We ay that v C([0, T ] H) i a trict olution of (14) if v D(L 0 ) and (14) i atified. 9

11 Definition 3.6. Given h C([0, T ] H) and g C(H) we ay that v C 0,1 ([0, T [ H) C 0 ([0, T ] H) with x v UC([0, T [ H; D(A )) i a trong olution of (14) if there exit three equence: {v n } D(L 0 ), {h n } C([0, T ] H) and {g n } C(H) fulfilling the following. (i) For any n N, v n i a trict olution of the problem { L0 (v n )(, x) = h n (, x) v n (T, x) = g n (x). (15) (ii) The following convergence hold: v n v in C([0, T ] H) h n F (,, x v(, )) in C([0, T ] H) g n g in C(H), where the convergence in C([0, T ] H) and C(H) are meant in the ene of uniform convergence on compact et. 3.3 Decompoition for olution of the HJB equation Theorem 3.7. Suppoe Hypothee 3.1 and 3.4 are atified. Suppoe that v C 0,1 ([0, T [ H) C 0 ([0, T ] H) with x v UC([0, T [ H; D(A )) i a trong olution of (14). Let X( ) := X( ; t, x, a( )) be the olution of (5) tarting at time at ome x H and driven by ome control a( ) U. Aume that b i of the form b(t, x, a) = b g (t, x, a) + b i (t, x, a), (16) where b g and b i atify the following condition. the peudo- (i) σ(t, X(t)) 1 b g (t, X(t), a(t)) i bounded (being σ(t, X(t)) 1 invere of σ); (ii) b i atifie Then + lim n x v n (r, X(r)) x v(r, X(r)), b i (r, X(r), a(r)) dr = 0 ucp on [, T 0 ], for each < T 0 < T. v(t, X(t)) v(, X()) = v(t, X(t)) v(, x) = x v(r, X(r)), b(r, X(r), a(r)) dr+ (17) F (r, X(r), x v(r, X(r))) dr x v(r, X(r)), σ(r, X(r)) dw Q (r), t [, T [. (18) 10

12 Proof. We fix T 0 in ], T [. We denote by v n the equence of mooth olution of the approximating problem precribed by Definition 3.6, which converge to v. Thank to Itô formula for convolution type procee (ee e.g. Corollary 4.10 in [13]), every v n verifie v n (t, X(t)) = v n (, x) r v n (r, X(r)) dr A x v n (r, X(r)), X(r) dr + + x v n (r, X(r)), b(r, X(r), a(r)) dr [( Tr σ(r, X(r))Q 1/2) ( σ(r, X(r))Q 1/2) ] 2 xx v n (r, X(r)) x v n (r, X(r)), σ(r, X(r)) dw Q (r), t [, T ]. P a.. (19) Uing Giranov Theorem (ee [6] Theorem 10.14) we can oberve that β Q (t) := W Q (t) + σ(r, X(r)) 1 b g (r, X(r), a(r)) dr, i a Q-Wiener proce with repect to a probability Q equivalent to P on the whole interval [, T ]. We can rewrite (19) a v n (t, X(t)) = v n (, x) r v n (r, X(r)) dr A x v n (r, X(r)), X(r) dr + dr x v n (r, X(r)), b i (r, X(r), a(r)) dr, [( Tr σ(r, X(r))Q 1/2) ( σ(r, X(r))Q 1/2) ] 2 xx v n (r, X(r)) + x v n (r, X(r)), σ(r, X(r)) dβ Q (r). P a.. (20) Since v n i a trict olution of (15), the expreion above give + v n (t, X(t)) = v n (, x) + h n (r, X(r)) dr x v n (r, X(r)), b i (r, X(r), a(r)) dr+ Since we wih to take the limit for n, we define M n (t) := v n (t, X(t)) v n (, x) dr x v n (r, X(r)), σ(r, X(r)) dβ Q (r). h n (r, X(r)) dr (21) x v n (r, X(r)), b i (r, X(r), a(r)) dr. (22) 11

13 {M n } n N i a equence of real Q-local martingale converging ucp, thank to the definition of trong olution and Hypothei (17), to M(t) := v(t, X(t)) v(, x) + F (r, X(r), x v(r, X(r))) dr x v(r, X(r)), b i (r, X(r), a(r)) dr, t [, T 0 ]. (23) Since the pace of real continuou local martingale equipped with the ucp topology i cloed (ee e.g. Propoition 4.4 of [26]) then M i a continuou Q-local martingale indexed by t [, T 0 ]. We have now gathered all the ingredient to conclude the proof. We et ν 0 = D(A ), ν = ν 0 ˆ π R, χ = ν 0 ˆ π ν 0. Propoition 3.2 enure that X( ) i a ν-weak Dirichlet proce admitting a χ-quadratic variation with decompoition M + A where M i the local martingale (with repect to P) defined by M(t) = x + σ(r, X(r)) dw Q(r) and A i a ν-martingale-orthogonal proce. Now X(t) = M(t) + V(t) + A(t), t [, T 0 ], where M(t) = x + σ(r, X(r)) dβ Q(r) and V(t) = b g(r, X(r), a(r))dr, t [, T 0 ] i a bounded variation proce. Thank to [27] Theorem 2.14 page 14-15, M i a Q-local martingale. Moreover V i a bounded variation proce and then, thank to Lemma 2.5, it i a Q ν-martingale orthogonal proce. So V+A i a again (one can eaily verify that the um of two ν-martingale-orthogonal procee i again a ν-martingale-orthogonal proce) a Q ν-martingale orthogonal proce and X i a ν-weak Dirichlet proce with local martingale part M, with repect to Q. Still under Q, ince v C 0,1 ([0, T 0 ] H), Theorem 2.7 enure that the proce v(, X( )) i a real weak Dirichlet proce on [, T 0 ], whoe local martingale part being equal to N(t) = x v(r, X(r)), σ(r, X(r)) dβ Q (r), t [, T 0 ]. On the other hand, with repect to Q, (23) implie that v(t, X(t)) = + [ v(, x) F (r, X(r), x v(r, X(r))) dr ] x v(r, X(r)), b i (r, X(r), a(r)) dr + N(t), t [, T 0 ], (24) i a decompoition of v(, X( )) a Q- emimartingale, which i alo in particular, a Q-weak Dirichlet proce. By Theorem 2.2 uch a decompoition i unique on [, T 0 ] and o M(t) = N(t), t [, T 0 ], o M(t) = N(t), t [, T [. 12

14 Conequently M(t) = x v(r, X(r)), σ(r, X(r)) dβ Q (r) = + x v(r, X(r)), b g (r, X(r), a(r)) dr x v(r, X(r)), σ(r, X(r)) dw Q (r), t [, T ]. (25) Example 3.8. The decompoition (16) with validity of Hypothee (i) and (ii) in Theorem 3.7 are atified if v i a trong olution of the HJB equation in the ene of Definition 3.6 and, moreover the equence of correponding function x v n converge to x v in C([0, T ] H). In that cae we imply et b g = 0 and b = b i. Thi i the typical aumption required in the tandard trong olution literature. Example 3.9. Again the decompoition (16) with validity of Hypothee (i) and (ii) in Theorem 3.7 i fulfilled if the following aumption i atified. σ(t, X(t)) 1 b(t, X(t), a(t)) i bounded, for all choice of admiible control a( ). In thi cae we apply Theorem 3.7 with b i = 0 and b = b g. 4 Verification Theorem In thi ection, a anticipated in the introduction, we ue the decompoition reult of Theorem 3.7 to prove a verification theorem. Theorem 4.1. Aume that Hypothee 3.1, 3.3 and 3.4 are atified and that the value function i finite for any (, x) [0, T ] H. Let v C 0,1 ([0, T [ H) C 0 ([0, T ] H) with x v UC([0, T [ H; D(A )) be a trong olution of (9) uch that x v ha mot polynomial growth in the x variable. Aume that for all initial data (, x) [0, T ] H and every control a( ) U b can be written a b(t, x, a) = b g (t, x, a) + b i (t, x, a) with b i and b g atifying hypothee (i) and (ii) of Theorem 3.7. Then we have the following. (i) v V on [0, T ] H. (ii) Suppoe that, for ome [0, T ), there exit a predictable proce a( ) = a ( ) U uch that, denoting X ( ;, x, a ( )) imply by X ( ), we have F (t, X (t), x v (t, X (t))) = F CV (t, X (t), x v (t, X (t)) ; a (t)), (26) dt dp a.e. Then a ( ) i optimal at (, x); moreover v (, x) = V (, x). 13

15 Proof. We chooe a control a( ) U and call X the related trajectory. We make ue of (18) in Theorem 3.7. Then we need to extend (18) to the cae when t [, T ]. Thi i poible ince v i continuou, x v i locally bounded and F i uniformly continuou on compact et, alo uing Hypothei 3.1 for b and σ. At thi point, etting t = T we can write g(x(t )) = v(t, X(T )) = v(, x) + x v(r, X(r)), b(r, X(r), a(r)) dr+ F (r, X(r), x v(r, X(r))) dr x v(r, X(r)), σ(r, X(r)) dw Q (r). Since both ide of (27) are a.. finite, we can add l(r, X(r), a(r)) dr to them, obtaining g(x(t ))+ + (27) l(r, X(r), a(r)) dr = v(, x)+ x v(r, X(r)), σ(r, X(r)) dw Q (r) ( F (r, X(r), x v(r, X(r))) + F CV (r, X(r), x v(r, X(r)))) dr. (28) Oberve now that, by definition of F and F CV we know that F (r, X(r), x v(r, X(r))) + F CV (r, X(r), x v(r, X(r))) i alway poitive. So it expectation alway exit even if it could be +, but not on an event of poitive probability. Thi how a poteriori that T l(r, X(r), a(r)) dr cannot be on a et of poitive probability. By Propoition 7.4 in [6], all the momenta of up r [,T ] X(r) are finite. On the other hand, σ i Lipchitz-continuou, v(, x) i determinitic and, ince x v ha polynomial growth, then ( E x v(r, X(r)), σ(r, X(r))Q 1/2) ( σ(r, X(r))Q 1/2) x v(r, X(r)) dr i finite. 4.7), Conequently (ee [6] Section 4.3, in particular Theorem 4.27 and x v(r, X(r)), σ(r, X(r)) dw Q (r), i a true martingale vanihing at. Conequently, it expectation i zero. So the expectation of the right-hand ide of (28) exit even if it could be + ; conequently the ame hold for the left-hand ide. 14

16 By definition of J, we have [ J(, x, a( )) = E g(x(t )) + ] l(r, X(r), a(r)) dr = v(, x) ( ) + E F (r, X(r), x v(r, X(r))) + F CV (r, X(r), x v(r, X(r)), a(r)) dr. (29) So minimizing J(, x, a( )) over a( ) i equivalent to minimize ( ) E F (r, X(r), x v(r, X(r))) + F CV (r, X(r), x v(r, X(r)), a(r)) dr, (30) which i a non-negative quantity. A mentioned above, the integrand of uch an expreion i alway nonnegative and then a lower bound for (30) i 0. If the condition of point (ii) are atified uch a bound i attained by the control a ( ), that in thi way i proved to be optimal. Concerning the proof of (i), ince the integrand in (30) i nonnegative, (29) give J(, x, a( )) v(, x). Taking the inf over a( ) we get V (, x) v(, x), which conclude the proof. Remark The firt part of the proof doe not make ue that a belong to U, but only that r l(r, X(,, x, a( )), a( )) i a.. trictly bigger then. Under that only aumption, a( ) i forced to be admiible, i.e. to belong to U. 2. Let v be a trong olution of HJB equation. Oberve that the condition (26) can be rewritten a [ ] a (t) arg min F CV (t, X (t), x v (t, X (t)) ; a). a Λ Suppoe that for any (t, y) [0, T ] H, φ(t, y) = arg min a Λ ( FCV (t, y, x v(t, y); a) ) i meaurable and ingle-valued. Suppoe moreover that l(r, X (r), a (r))dr > a.. (31) Suppoe that the equation { dx(t) = (AX(t) + b(t, X(t), φ(t, X(t)) dt + σ(t, X(t)) dwq (t) X() = x, (32) admit a unique mild olution X. Now (31) and Remark imply that a ( ) i admiible. Then X i the optimal trajectory of the tate variable 15

17 and a (t) = φ(t, X (t)), t [0, T ] i the optimal control. The function φ i the optimal feedback of the ytem ince it give the optimal control a a function of the tate. Remark 4.3. Oberve that, uing exactly the ame argument we ued in thi ection one could treat the (lightly) more general cae in which b ha the form: b(t, x, a) = b 0 (t, x) + b g (t, x, a) + b i (t, x, a). where b g and b i atify condition of Theorem 3.7 and b 0 : [0, T ] H H i continuou. In thi cae the addendum b 0 can be included in the expreion of L 0 that become the following { D(L b 0 0 ) := { ϕ C 1,2 ([0, T ] H) : x ϕ C([0, T ] H; D(A )) } L b0 0 (ϕ) := ϕ + A x ϕ, x + x ϕ, b 0 (t, x) T r [ σ(, x)σ (, x) xxϕ ] 2. (33) Conequently in the definition of regular olution the operator L b0 0 appear intead L 0. ACKNOWLEDGEMENTS: The reearch wa partially upported by the ANR Project MASTERIE 2010 BLAN It wa partially written during the tay of the econd named author at Bielefeld Univerity, SFB 701 (Mathematik). The work of the firt named author wa partially upported ha been developed in the framework of the center of excellence LABEX MME-DII (ANR- 11-LABX ). Reference [1] N. U. Ahmed. Generalized olution of HJB equation applied to tochatic control on Hilbert pace. Nonlinear Anal., 54(3): , [2] V. Barbu and G. Da Prato. Hamilton-Jacobi equation in Hilbert pace. Pitman (Advanced Publihing Program), Boton, MA, [3] S. Cerrai. Optimal control problem for tochatic reaction-diffuion ytem with non-lipchitz coefficient. SIAM J. Control Optim., 39(6): , [4] S. Cerrai. Stationary Hamilton-Jacobi equation in Hilbert pace and application to a tochatic optimal control problem. SIAM J. Control Optim., 40(3): , [5] A. Coo and F. Ruo. Functional Itô veru Banach pace tochatic calculu and trict olution of emilinear path-dependent equation. Infin. Dimen. Anal. Quantum Probab. Relat. Top., 19(4): , 44,

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