# Necessary Conditions for Optimality for Stochastic Evolution Equations

Authors: AbdulRahman Al-Hussein
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with providing the maximum principle for a control problem governed by a stochastic evolution system on a separable Hilbert space. In particular, necessary conditions for optimality for this stochastic optimal control problem are derived by using the adjoint backward stochastic evolution equation. Moreover, all coefficients appearing in this system are allowed to depend on the control variable. We achieve our results through the semigroup approach.


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## Body

## 1. Introduction

Consider a stochastic controlled problem governed by the following stochastic evolution equation (SEE):(1)dX(t)=(AX(t)+b(X(t),ν(t)))dt+σ(X(t),ν(t))dW(t),t∈(0,T],X(0)=x0.
We will be interested in trying to minimize the cost functional, which is given by (5), over a set of admissible controls.This system is driven mainly by a possibly unbounded linear operatorA on a separable Hilbert space H and a cylindrical Wiener process W on H. Here, ν(·) denotes a control process.We will derive the maximum principle for this control problem. More precisely, we will concentrate on providing necessary conditions for optimality for this optimal control problem, which gives this minimization. For this purpose we will apply the theory ofbackward stochastic evolution equations (BSEEs) shortly as in (10) in Section 3. These equations together with backward stochastic differential equations (BSDEs) have become of great importance in a number of fields. For example, in [1–7], one can find applications of BSDEs to stochastic optimal control problems. Some of these references have also studied the maximum principle to find either necessary or sufficient conditions for optimality for stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs). Necessary conditions for optimality of the control process ν(·) and its corresponding solution Xν(·) but for the case when the noise term σ does not depend on ν(t) can be found in [3].In our work here, we allowσ to depend on the control variable and study a stochastic control problem associated with the former SEE. This control problem is explained in details in Section 2, and the main theorem is stated in Section 3 and is proved together with all necessary estimates in Section 4. Sufficient conditions for optimality for this optimal control problem can be found in [2]. We refer the reader also to [1].On the other hand, we recall that control problems governed by SPDEs that are driven by martingales are studied in [8]. In fact in [8], we derived the maximum principle (necessary conditions) for optimality of stochastic systems governed by SPDEs. The technique used there relies heavily on the variational approach. The reason beyond that is that the only known way until now to find solutions to the resulting adjoint BSPDEs is achieved through the same variational approach and is established in details in [9]. Thus, the semigroup approach to get mild solutions (as done here in Theorem 2 and in Section 3) cannot be used to study such adjoint BSPDEs considered in [8]. Moreover, it is not obvious how one can allow the control variable ν(t) to enter in the noise term and in particular in the mapping G in equation (1.1) of [8] and obtain a result like Theorem 3. This problem is still open and is also pointed out in [8, Remark 6.4].In the present work, we will show how to handle this open problem in great success, and as we stated earlier, we can and will allow all coefficients in (1) and especially in the diffusion term to depend on the control variable ν(t). We emphasize that our work here does not need to go through neither the technique of Hamilton-Jacobi-Bellman equations nor the technique of viscosity solutions. We refer the reader to [10] for this business and to [11] and some of the related references therein for the semigroup technique. Thus, our results here are new. In this respect, we thank the anonymous referee for pointing out the recent and relevant work of Fuhrman et al. in [12].


## 2. Statement of the Problem

Let(Ω,ℱ,ℙ) be a complete probability space, and denote by 𝒩 the collection of ℙ-null sets of ℱ. Let {W(t),0≤t≤T} be a cylindrical Wiener process on H with its completed natural filtration ℱt=σ{ℓ∘W(s),0≤s≤t,ℓ∈H*}∨𝒩, t≥0; see [13] for more details.For a separable Hilbert spaceE, denote by Lℱ2(0,T;E) to the space of all {ℱt,0≤t≤T} progressively measurable processes f with values in E such that
(2)𝔼[∫0T|f(t)|E2dt]<∞.This space is Hilbert with respect to the norm(3)∥f∥=(𝔼[∫0T|f(t)|E2dt])1/2.Moreover, iff∈Lℱ2(0,T;L2(H)), where L2(H) is the space of all Hilbert-Schmidt operators on H, the stochastic integral ∫f(t)dW(t) can be defined and is a continuous stochastic martingale in H. The norm and inner product on L2(H) will be denoted, respectively, by ∥·∥2 and 〈·,·〉2.Let us assume that𝒪 is a separable Hilbert space equipped with an inner product 〈·,·〉𝒪 and U is a convex subset of 𝒪. We say that ν(·):[0,T]×Ω→𝒪 is admissible if ν(·)∈Lℱ2(0,T;𝒪) and ν(t)∈Ua.e., a.s. The set of admissible controls will be denoted by 𝒰ad.Suppose thatb:H×𝒪→H and σ:H×𝒪→L2(H) are two continuous mappings, and consider the following controlled SEE:
(4)dX(t)=(AX(t)+b(X(t),ν(t)))dt+σ(X(t),ν(t))dW(t),X(0)=x0,
where ν(·)∈𝒰ad. A solution (in the sense of the following theorem) of (4) will be denoted by Xν(·) to indicate the presence of the control process ν(·).Letℓ:H×𝒪→ℝ and ϕ:H→ℝ be two measurable mappings such that the following cost functional is defined:
(5)J(ν(·))∶=𝔼[∫0Tℓ(Xν(·)(t),ν(t))dt+ϕ(Xν(·)(T))],ν(·)∈𝒰ad.For example, one can takeℓ and ϕ to satisfy the assumptions of Theorem 3 in Section 3.The optimal control problem of system (4) is to find the value function(6)J*∶=inf{J(ν(·)):ν(·)∈𝒰ad}
and an optimal controlν*(·)∈𝒰ad such that
(7)J*=J(ν*(·)).If this happens, the corresponding solutionXν*(·) is called an optimal solution of the stochastic control problem (4)–(7) and (Xν*(·),ν*(·)) is called an optimal pair.We close this section by the following theorem.Theorem 1.
Assume thatA is an unbounded linear operator on H that generates a C0-semigroup {S(t),t≥0} on H, b, σ are continuously Fréchet differentiable with respect to x, and their derivatives bx, σx are uniformly bounded. Then, for every ν(·)∈𝒰ad there exists a unique mild solution Xν(·) on [0,T] to (4). That is, Xν(·) is a progressively measurable stochastic process such that X(0)=x0, and for all t∈[0,T],
(8)Xν(·)(t)=S(t)x0+∫0tS(t-s)b(Xν(·)(s),ν(s))ds+∫0tS(t-s)σ(Xν(·)(s),ν(s))dW(s).The proof of this theorem can be derived in a similar way to those in [14, Chapter 7] or [15].From here on, we will assume thatA is the infinitesimal generator of a C0-semigroup {S(t),t≥0} on H. Its adjoint operator A*:𝒟(A*)⊂H→H is then the infinitesimal generator of the adjoint semigroup {S*(t),t≥0} of {S(t),t≥0}.


## 3. Stochastic Maximum Principle

It is known from the literature that BSDEs play a fundamental role in deriving the maximum principle for SDEs. In this section, we will search for such a role for SEEs like (4). To prepare for this business, let us first define the Hamiltonian by the following formula:
(9)ℋ:H×𝒪×H×L2(H)⟶ℝ,ℋ(x,ν,y,z)∶=ℓ(x,ν)+〈b(x,ν),y〉H+〈σ(x,ν),z〉2.Then, we consider the following BSEE onH:
(10)-dYν(·)(t)=(A*Yν(·)(t)+∇xℋ(Xν(·)(t),ν(t),Yν(·)(t),Zν(·)(t)))dt-Zν(·)(t)dW(t),0≤t<T,Yν(·)(T)=∇ϕ(Xν(·)(T)),
where ∇ϕ denotes the gradient of ϕ, which is defined by using the directional derivative Dϕ(x)(h) of ϕ at a point x∈H in the direction of h∈H, as 〈∇ϕ(x),h〉H=Dϕ(x)(h)(=ϕx(h)). This equation is the adjoint equation of (4).As in the previous section, amild solution (or a solution) of (10) is a pair (Y,Z)∈Lℱ2(0,T;H)×Lℱ2(0,T;L2(H)) such that we have ℙ—a.s. for all t∈[0,T](11)Yν(·)(t)=S*(T-t)∇ϕ(Xν(·)(T))+∫tTS*(s-t)∇xℋ(Xν(·)(s),ν(s),Yν(·)(s),Zν(·)(s))ds-∫tTS*(s-t)Zν(·)(s)dW(s).Theorem 2.
Assume thatb, σ, ℓ, and ϕ are continuously Fréchet differentiable with respect to x, the derivatives bx, σx, σν, and ℓx are uniformly bounded, and
(12)|ϕx|L(H,H)≤k(1+|x|H)
for some constant k>0.
Then, there exists a unique (mild) solution(Yν(·),Zν(·)) of BSEE (10).The proof of this theorem can be found in [16] or [17]. An alternative proof by using finite dimensional framework through the Yosida approximation of A can be found in [18].Our main result is the following.Theorem 3.
Suppose that the following two conditions hold.(i)
b, σ, and ℓ are continuously Fréchet differentiable with respect to x, ν, ϕ is continuously Fréchet differentiable with respect to x, the derivatives bx, bν, σx, σν, ℓx, and ℓν are uniformly bounded, and
(13)|ϕx|L(H,H)≤k(1+|x|H)
for some constant k>0.(ii)
ℓx is Lipschitz with respect to u uniformly in x.
If(Xν*(·),ν*(·)) is an optimal pair for the control problem (4)–(7), then there exists a unique solution (Yν*(·),Zν*(·)) to the corresponding BSEE (10) such that the following inequality holds:
(14)〈∇νℋ(Xν*(·)(t),ν*(t),Yν*(·)(t),Zν*(·)(t)),ν*(t)-ν〉𝒪≤0a.e.t∈[0,T],a.s.∀ν∈U.The proof of this theorem will be given in Section4. Now, to illustrate this theorem, let us present an example.Example 4.
LetH and 𝒪 be two separable Hilbert spaces as considered earlier, and let U=𝒪. We will study in this example a special case of the control problem (4)–(7). In particular, given ϕ as in Theorem 3, we would like to minimize the cost functional
(15)J(ν(·))=𝔼[∫0T|ν(t)|𝒪2dt]+𝔼[ϕ(Xν(·)(T))]
subject to
(16)dXν(·)(t)=(AXν(·)(t)+Bν(t))dt+Dν(t)dW(t),t∈(0,T],Xν(·)(0)=x0∈H,
where B is a bounded linear operator from 𝒪 into H and D is another bounded linear operator from 𝒪 into L2(H).
The Hamiltonian is then given by the formula:(17)ℋ(x,ν,y,z)=|ν|𝒪2+〈Bν,y〉H+〈Dν,z〉L2(H),
where (x,ν,y,z)∈H×𝒪×H×L2(H) and the adjoint BSEE is
(18)-dYν(·)(t)=A*Yν(·)(t)dt-Zν(·)(t)dW(t),t∈[0,T),Yν(·)(T)=∇ϕ(Xν(·)(T)).
From the construction of the solution of (18), as for example, in [16, Lemma 3.1], this BSEE attains an explicit solution:
(19)Yν(·)(t)=𝔼[S*(T-t)∇ϕ(Xν(·)(T))∣ℱt],Zν(·)(t)=S*(T-t)Rν(·)(t),
where Rν(·) is the unique element of Lℱ2(0,T;L2(H)) satisfying
(20)∇ϕ(Xν(·)(T))=𝔼[∇ϕ(Xν(·)(T))]+∫0TRν(·)(t)dW(t).
On the other hand, for fixed(x,y,z), we note that the function ν↦ℋ(x,ν,y,z) attains its minimum at ν=(1/2)(B*y+D*z)(∈U), where B*:H→𝒪 and D*:L2(H)→𝒪 are the adjoint operators of B and D, respectively. So, we elect
(21)ν*(t,ω)=12(B*Yν*(·)(t,ω)+D*Zν*(·)(t,ω))
as a candidate optimal control.
It is easy to see that with these choices all the requirements of Theorem3 are verified. Hence, this candidate ν*(·) given in (21) is an optimal control for the problem (15)-(16), and its corresponding optimal solution Xν*(·) is the solution of the following SEE:
(22)dXν*(·)(t)=(AXν*(·)(t)+12B(B*Yν*(·)(t)+D*Zν*(·)(t))12)dt+12D(B*Yν*(·)(t)+D*Zν*(·)(t))dW(t),t∈(0,T],Xν*(·)(0)=x0.
Finally, the value function attains the formula(23)J*=14𝔼[∫0T|B*Yν*(·)(t)+D*Zν*(·)(t)|𝒪2dt]+𝔼[ϕ(Xν*(·)(T))].Remark 5.
A concrete example in the setting of Example4 can be constructed by taking H=𝒪=L2(ℝd), d≥1, A=(1/2)Δ (half-Laplacian), B=idH, Dν:=〈v,h〉H𝒬1/2, and ϕ(x)=〈ρ,x〉H, for some fixed elements h, ρ of H and a positive definite nuclear operator 𝒬 on H.
The computations in this case ofℋ, Y*, Z*, ν*(·), and X* become direct from the corresponding equations in Example 4.


## 4. Proofs

Letν*(·) be an optimal control, and let X*≡Xν*(·) be the corresponding solution of (4). Let ν(·) be an element of Lℱ2(0,T;𝒪) such that ν*(·)+ν(·)∈𝒰ad. For a given 0≤ε≤1, consider the variational control:
(24)νε(t)=ν*(t)+εν(t),t∈[0,T].We note that the convexity ofU implies that νε(·)∈𝒰ad. Considering this control νε(·), we will let Xνε(·) be the solution of the SEE (4) corresponding to νε(·) and denote it briefly by Xε.Letp be the solution of the following linear equation:
(25)dp(t)=(Ap(t)+bx(X*(t),ν*(t))p(t)+bν(X*(t),ν*(t))ν(t))dt+(σx(X*(t),ν*(t))p(t)+σν(X*(t),ν*(t))ν(t))dW(t),p(0)=0.The following three lemmas contain estimates that will play a vital role in deriving the desired variational equation and the maximum principle for our control problem.Lemma 6.
Assume condition (i) of Theorem3. Then,
(26)supt∈[0,T]𝔼[|p(t)|2]<∞.Proof.
The solution of (25) is given by the formula
(27)p(t)=∫0tS(t-s)(bx(X*(s),ν*(s))p(s)+bν(X*(s),ν*(s))ν(s))ds+∫0tS(t-s)(σx(X*(s),ν*(s))p(s)+σν(X*(s),ν*(s))ν(s))dW(s).
By using Minkowski's inequality (triangle inequality), Holder's inequality, Burkholder's inequality for stochastic convolution together with assumption (i), and Gronwall's inequality, we obtain easily(28)supt∈[0,T]𝔼[|p(t)|2]≤C
for some constant C>0.Lemma 7.
Assuming condition (i) of Theorem3, one has
(29)supt∈[0,T]𝔼[|Xε(t)-X*(t)|2]=O(ε2).Proof.
Observe first from (8) that
(30)Xε(t)-X*(t)=∫0tS(t-s)(b(Xε,νε(s))-b(X*(s),ν*(s)))ds+∫0tS(t-s)(σ(Xε,νε(s))-σ(X*(s),ν*(s)))dW(s).
Hence,(31)𝔼[|Xε(t)-X*(t)|2]≤2M2T𝔼[∫0t|b(Xε(s),νε(s))-b(X*(s),ν*(s))|2ds∫0t|b(Xε(s),νε(s))]+2M2𝔼[∫0t∥σ(Xε,νε(s))-σ(X*(s),ν*(s))∥22ds∫0t∥σ(Xε,νε(s))],
where M:=supt∈[0,T]∥S(t)∥L(H,H).
Secondly, from condition (i), we get(32)𝔼[∫0t|b(Xε(s),νε(s))-b(X*(s),ν*(s))|2ds]≤2𝔼[∫0t|b(Xε(s),νε(s))-b(X*(s),νε(s))|2ds]+2𝔼[∫0t|b(X*(s),νε(s))-b(X*(s),ν*(s))|2ds]=2𝔼[∫0t|b~x(s,ε)(Xε(s)-X*(s))|2ds]+2𝔼[∫0t|δεb(s)|2ds]≤2C1𝔼[∫0t|Xε(s)-X*(s)|2ds]+2C2ε2,
where for y∈H(33)b~x(s,ε)(y)=∫01‍bx(X*(s)+θ(Xε(s)-X*(s)),νε(s))(y)dθ,δεb(s)=b(X*(s),νε(s))-b(X*(s),ν*(s)),C1 is a positive constant, and C2 is another positive constant coming thanks to (i) from the following inequality:
(34)𝔼[∫0T|δεb(s)|2ds]=𝔼[∫0T|b(X*(s),νε(s))-b(X*(s),ν*(s))|2ds∫0T|b(X*(s),νε(s))]=𝔼[∫0T|∫01bν(X*(s),ν*(s)+θ(νε(s)-ν*(s)))×(νε(s)-ν*(s))dθ∫|2ds]≤C2ε2.
Similarly,(35)𝔼[∫0t|σ(Xε(s),νε(s))-σ(X*(s),ν*(s))|2ds]≤2C3𝔼[∫0t|Xε(s)-X*(s)|2ds]+2C4ε2,
for some positive constants C3, C4.
Finally, by applying (32), (35) in (30) and then using Gronwall's inequality, we find that
(36)𝔼[|Xε(t)-X*(t)|2]≤C5ε2
for some constant C5>0 that depends in particular on Ci, i=1,…,4, and M. Hence, the proof is complete.Keeping the notationsb~x and δεb used in the preceding proof, let us state the following lemma.Lemma 8.
Letηε(t)=((Xε(t)-X*(t))/ε)-p(t). Then, under condition (i) of Theorem 3,
(37)limε→0+supt∈[0,T]𝔼[|ηε(t)|2]=0.Proof.
From the corresponding equations (4) and (25), we deduce that
(38)ηε(t)=∫0tS(t-s)[1ε(b(Xε(s),νε(s))-b(X*(s),νε(s)))-bx(X*(s),ν*(s))p(s)∫]ds+∫0tS(t-s)[1εδεb(s)-bν(X*(s),ν*(s))ν(s)∫]ds+∫0t‍S(t-s)[1ε(σ(Xε(s),νε(s))-σ(X*(s),νε(s)))-σx(X*(s),ν*(s))p(s)1ε]dW(s)+∫0tS(t-s)[1εδεσ(s)-σν(X*(s),ν*(s))ν(s)1ε]ds=∫0tS(t-s)[-bx(X*(s),ν*(s)))p(s)b~x(s,ε)ηε(s)+(-bx(X*(s),ν*(s))b~x(s,ε)-bx(X*(s),ν*(s)))p(s)b~x(s,ε)ηε(s)]ds+∫0tS(t-s)[1εδεb(s)-bν(X*(s),ν*(s))ν(s)1ε]ds+∫0tS(t-s)[(s,ε)σ~x(s,ε)ηε(s)+(σ~x(s,ε)-σx(X*(s),ν*(s)))p(s)]dW(s)+∫0tS(t-s)[1εδεσ(s)-σν(X*(s),ν*(s))ν(s)1ε]dW(s),
where
(39)δεσ(s)=σ(X*(s),νε(s))-σ(X*(s),ν*(s)),σ~x(s,ε)(y)=∫01σx(X*(s)+θ(Xε(s)-X*(s)),νε(s))×(y)dθ,y∈H.
Consequently, from (i) and as in the proof of Lemma 7, it follows that
(40)𝔼[|ηε(t)|2]≤C6∫0t𝔼[|ηε(s)|2]ds+ρ(ε),
for all t∈[0,T], where
(41)ρ(ε)=8MT𝔼[∫0T|(b~x(s,ε)-bx(X*(s),ν*(s))(b~x(s,ε))p(s)(b~x(s,ε)|2ds∫0T|(b~x(s,ε)∫]+8M𝔼[∫0T∥-σx(X*(s),ν*(s)))p(s)(σ~x(s,ε)-σx(X*(s),ν*(s)))p(s)∥22ds∫0T∥(σ~x(s,ε)∫]+4MT𝔼[∫0T|1εδεb(s)-bν(X*(s),ν*(s))ν(s)1ε|2ds∫∫]+4M𝔼[∫0T∥1εδεσ(s)-σν(X*(s),ν*(s))ν(s)1ε∥22ds∫∫].
But (i), (28), and the dominated convergence theorem give
(42)𝔼[∫0T|(b~x(s,ε)-bx(X*(s),ν*(s)))p(s)|2ds]=𝔼[∫0T|∫01(bx(X*(s)+θ(Xε(s)-X*(s)),νε(s))-bx(X*(s),ν*(s)))p(s)dθ∫|2ds∫∫]≤∫0T∫01𝔼[×p(s)|2|×p(s)(bx(X*(s)+θ(Xε(s)-X*(s)),νε(s))-bx(X*(s),ν*(s)))×p(s)|2]dθds⟶0,
as ε→0+.
Similarly, we have(43)𝔼[∫0T∥(σ~x(s,ε)-σx(X*(s)))p(s)∥22ds]⟶0,
as ε→0+.
On the other hand, as done for (34),
(44)𝔼[∫0T|1εδεb(s)-bν(X*(s),ν*(s))ν(s)1ε|2ds∫∫]≤∫0T∫01𝔼[-bν(X*(s),ν*(s)))ν(s)|2|-bν(X*(s),ν*(s)))ν(s)(bν(X*(s),ν*(s)+θ(νε(s)-ν*(s)))-bν(X*(s),ν*(s)))ν(s)|2]dθds⟶0
if ε→0+, by using (i) and the dominated convergence theorem. Similarly,
(45)𝔼[∫0T∥1εδεσ(s)-σν(X*(s),ν*(s))ν(s)∥22ds]⟶0
if ε→0+.
Finally applying (43)–(45) in (41) shows that
(46)ρ(ε)⟶0,asε⟶0+.
Hence, from (40) and Gronwall's inequality, we obtain
(47)supt∈[0,T]𝔼[|ηε(t)|2]⟶0,
as ε→0+.The following theorem contains our main variational equation, which is one of the main tools needed for deriving the maximum principle stated in Theorem3.Theorem 9.
Suppose that (i) and (ii) in Theorem3 hold. For each ε>0, we have
(48)J(νε(·))-J(ν*(·))=ε𝔼[ϕx(X*(T))p(T)]+ε𝔼[∫0Tℓx(X*(s),ν*(s))×p(s)ds∫∫0T‍]+𝔼[∫0T(ℓ(X*(s),νε(s))-ℓ(X*(s),ν*(s)))ds∫∫]+o(ε).Proof.
We can writeJ(νε(·))-J(ν*(·)) as
(49)J(νε(·))-J(ν*(·))=I1(ε)+I2(ε),
with
(50)I1(ε)=𝔼[ϕ(Xε(T))-ϕ(X*(T))],I2(ε)=𝔼[∫0T(ℓ(Xε(s),νε(s))-ℓ(X*(s),ν*(s)))ds].
Note that with the help of our assumptions and by making use of Lemmas8, 7, and 6 and the dominated convergence theorem, we deduce that
(51)1εI1(ε)=1ε𝔼[∫01ϕx(X*(T)+θ(Xε(T)-X*(T)))(Xε(T)-X*(T))dθ∫01ϕx(X*(T)+θ(Xε(T)]=𝔼[∫01ϕx(X*(T)+θ(Xε(T)-X*(T)))×(p(T)+ηε(T))dθ∫∫]⟶𝔼[ϕx(X*(T))p(T)],asε⟶0+.
Hence,
(52)I1(ε)=ε𝔼[ϕx(X*(T))p(T)]+o(ε).
Similarly,
(53)1εI2(ε)=1ε𝔼[∫0T(ℓ(Xε(s),νε(s))-ℓ(X*(s),νε(s)))ds]+1ε𝔼[∫0T(ℓ(X*(s),νε(s))-ℓ(X*(s),ν*(s)))ds∫0T(ℓ(X*(s),νε(s))]=𝔼[∫0T∫01ℓx(+θ(Xε(s)-X*(s)X*(s)+θ(Xε(s)-X*(s)),νε(s))×(p(s)+ηε(s))dθds∫0T∫01ℓx(+θ(Xε(s)-X*(s)X*(s)]+1ε𝔼[∫0T(ℓ(X*(s),νε(s))-ℓ(X*(s),ν*(s)))ds∫0T(ℓ(X*(s),νε(s))].
On the other hand, applying Lemmas8, 7, and 6 and using the continuity and boundedness of ℓx in (i), (ii) and the dominated convergence theorem imply that
(54)𝔼[∫0T∫01ℓx(X*(s)+θ(Xε(s)-X*(s)),ν*(s)+εν(s))(p(s)+ηε(s))dθds∫0T∫01ℓx(X*(s)+θ(Xε(s)-X*(s)),]⟶𝔼[∫0Tℓx(X*(s),ν*(s))p(s)ds].
In particular, we obtain
(55)I2(ε)=ε𝔼[∫0Tℓx(X*(s),ν*(s))p(s)ds]+𝔼[∫0T(ℓ(X*(s),νε(s))-ℓ(X*(s),ν*(s)))ds∫0T(ℓ(X*(s),νε(s))]+o(ε).
As a result, the theorem follows from (49), (52) and (55).Let us next introduce an important variational inequality.Lemma 10.
Let hypotheses (i), (ii) in Theorem3 hold. Let (Y*,Z*)≡(Yν*(·),Zν*(·)) be the solution of BSEE (10) corresponding to the optimal pair (X*,ν*(·)). Then,
(56)ε𝔼〈Y*(T),p(T)〉+ε𝔼[∫0Tℓx(X*(s),ν*(s))p(s)ds]+𝔼[∫0T(δεℋ(s)-〈δεb(s),Y*(s)〉-〈δεσ(s),Z*(s)〉2)ds∫0T(δεℋ(s)-〈δεb(s),Y*(s)〉]≥o(ε),
where
(57)δεℋ(s)=ℋ(X*(s),νε(s),Y*(s),Z*(s))-ℋ(X*(s),ν*(s),Y*(s),Z*(s)).Proof.
Sinceν*(·) is an optimal control, then J(νε(·))-J(ν*(·))≥0. Hence, the result follows from (48) and (9).The following duality relation between (25) and (10) is also needed in order to establish the proof of Theorem 3.Lemma 11.
Under hypothesis (i) in Theorem3, we have
(58)𝔼〈Y*(T),p(T)〉=-𝔼[∫0Tℓx(X*(s),ν*(s))p(s)ds]+𝔼[∫0T〈bν(X*(s),ν*(s))ν(s),Y*(s)〉ds∫0T〈bν(X*(s),ν*(s))ν(s),]+𝔼[∫0T〈σν(X*(s),ν*(s))ν(s),Z*(s)〉2ds∫0T〈σν(X*(s),ν*(s))ν(s),].Proof.
The proof is done by using Yosida approximation of the operatorA and Itô's formula for the resulting SDEs and can be gleaned directly from the proof of Theorem 2.1 in [18].We are now ready to establish (or complete in particular) the proof of Theorem3.Proof of Theorem3.
Recall the BSEE (10):
(59)-dYν(·)(t)=(A*Yν(·)(t)+∇xℋ(Xν(·)(t),ν(t),Yν(·)(t),Zν(·)(t)))dt-Zν(·)(t)dW(t),0≤t<T,Yν(·)(T)=∇ϕ(Xν(·)(T)).
From Theorem 2, there exists a unique solution (Y*,Z*) to it. Thereby, it remains to prove (15).
Applying (56) and (58) gives
(60)𝔼[∫0T(δεℋ(s)+〈εbν(X*(s),ν*(s))ν(s)-δεb(s),Y*(s)〉+〈εσν(X*(s),ν*(s))ν(s)-δεσ(s),Z*(s)εσν(X*(s),ν*(s))ν(s)〉2δεℋ(s))ds∫0T(δεℋ(s)]≥o(ε).
But, as done for (44), by using the continuity and boundedness of bν in assumption (i) and the dominated convergence theorem, one can find that
(61)1ε𝔼[∫0T〈εbν(X*(s),ν*(s))ν(s)-δεb(s),Y*(s)〉ds∫01(bν(X*(s),ν*(s)]=-𝔼[∫0T〈Y*(s),∫01(bν(X*(s),ν*(s)+θ(νε(s)-ν*(s)))-bν(X*(s),ν*(s)))×ν(s)dθ∫01(bν(X*(s),ν*(s)〉ds∫0T〈Y*(s),∫01(bν(X*(s),ν*(s)]⟶0,
as ε→0+. This means that
(62)𝔼[∫0T〈εbν(X*(s),ν*(s))ν(s)-δεb(s),Y*(s)〉ds]=o(ε).
Similarly,
(63)𝔼[∫0T〈εσν(X*(s),ν*(s))ν(s)-δεσ(s),Z*(s)〉2ds]=o(ε).
Now, by applying (62) and (63) in (60), we deduce that
(64)𝔼[∫0Tδεℋ(s)ds]≥o(ε).
Therefore, by dividing (64) by ε and letting ε→0+, the following inequality holds:
(65)𝔼[∫0T〈∇νℋ(t,X*(t),ν*(t),Y*(t),Z*(t)),ν(t)〉𝒪dt]≥0.
Finally, (14) follows by arguing, if necessary, as in [19, page 280], for instance.



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*Source: 469390-2013-10-03.xml*