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ESAIM: Control, Optimisation and Calculus of Variations

DOI:10.1051/cocv/2014062 www.esaim-cocv.org

OPTIMAL STOCHASTIC CONTROL WITH RECURSIVE COST FUNCTIONALS OF STOCHASTIC DIFFERENTIAL SYSTEMS REFLECTED IN A DOMAIN∗,∗∗

Juan Li

1

and Shanjian Tang

2

Abstract.The paper is concerned with optimal control of a stochastic differential system reflected in a domain. The cost functional is implicitly definedvia a generalized backward stochastic differential equation developed by Pardoux and Zhang [Probab. Theory Relat. Fields110 (1998) 535–558]. The value function is shown to be the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. The proof requires new estimates for the reflected stochastic differential system.

Mathematics Subject Classification. 60H99, 60H30, 35J60, 93E05, 90C39.

Received October 21, 2013. Revised August 25, 2014.

Published online July 6, 2015.

1. Introduction

Let D be an open connected bounded convex subset ofRd such thatD ={φ >0}, ∂D== 0}for some functionφ∈Cb2(Rd) satisfying|∇φ(x)|= 1 at anyx∈∂D.Note that at anyx∈∂D, ∇φ(x) is a unit normal vector on the boundary pointx, pointing towards the interior ofD.

Let U be a metric space. An admissible control process is a U-valued F-progressively measurable process.

The set of all admissible control processes is denoted byU. In this paper, for the initial data (t, x)[0, T]×Rd

Keywords and phrases.Hamilton–Jacobi–Bellman equation, nonlinear Neumann boundary, value function, backward stochastic differential equations, dynamic programming principle, viscosity solution.

Juan Li has been supported by the NSF of P.R. China (Nos. 11071144, 11171187, 11222110), Shandong Province (Nos. BS2011SF010, JQ201202), SRF for ROCS (SEM), Program for New Century Excellent Talents in University (No. NCET-12-0331), 111 Project (No. B12023).

∗∗ Shanjian Tang is supported in part by the National Natural Science Foundation of China (Grants #10325101 and #11171076), by Science and Technology Commission, Shanghai Municipality (Grant No. 14XD1400400), by Basic Re- search Program of China (973 Program) Grant #2007CB814904, by the Science Foundation of the Ministry of Education of China Grant #200900071110001, and by WCU (World Class University) Program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (R31-20007).

1 School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264200, P.R. China.juanli@sdu.edu.cn

2 Institute of Mathematics and Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China.sjtang@fudan.edu.cn

Article published by EDP Sciences c EDP Sciences, SMAI 2015

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we consider the optimal control problem for the following stochastic differential equations (SDEs) reflected in domainD:

Xs=x+s

t b(r, Xr, ur) dr+s

t σ(r, Xr, ur) dBr+s

t ∇φ(Xr) dKr, s∈[t, T];

Ks=s

t I{Xr∈∂D}dKr, K is increasing. (1.1)

Here,u(·)∈ Uis an admissible control, and the driftb: [0, T]×Rd×U Rdand the diffusionσ: [0, T]×Rd×U Rd×d are uniformly Lipschitz continuous and grows linearly in the state variable x. For each u(·) ∈ U, in view of Proposition A.1 in the appendix, the above reflected SDE (1.1) has a unique strong solution, de- noted by (Xt,x;u, Kt,x;u). Consider the following controlled generalized backward stochastic differential equa- tion (GBSDE):

dYs=f

s, Xst,x;u, Ys, Zs, us

ds+g

s, Xst,x;u, Ys

dKst,x;u−ZsdBs, s∈[0, T);

YT =Φ XTt,x;u

. (1.2)

Under suitable conditions on the functionsf, gand Φ(see (H3.2) in Sect. 3 for more details), it has a unique adapted solution (see Pardoux and Zhang [20]), denoted by (Yt,x;u, Zt,x;u) hereafter. The optimal control prob- lem is to maximize the cost functionalJ(t, x;u) := Ytt,x;u over all admissible controlsu ∈ U. The associated Hamilton–Jacobi–Bellman (HJB) equation turns out to have a nonlinear Neumann boundary condition, and reads as follows: ⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

∂tW(t, x) +H

t, x, W, DW, D2W

= 0, (t, x)[0, T)×D,

∂nW(t, x) +g(t, x, W(t, x)) = 0, 0≤t < T, x∈∂D;

W(T, x) =Φ(x), x∈D,¯

(1.3)

where at a pointx∈∂D, ∂n =d

i=1

∂xiφ(x)∂xi, and the HamiltonianH is given by H(t, x, y, p, A) := sup

u∈U

1 2tr

σσT(t, x, u)A

+p, b(t, x, u)+f(t, x, y, p.σ, u)

for (t, x, y, p, A)[0, T]×Rn×R×Rd×Sd. We aim to show that the value function of our optimal control problem is the unique viscosity solution to above HJB equation (1.3).

BSDEs were initially studied by Bismut in 1973 (see Bismut [2–4]), and a general nonlinear version was studied by Pardoux and Peng [17] in 1990. Since then BSDE has received an extensive attention both in the theory and in the application. The reader is referred to, among others, El Karoui et al. [12], Darling and Pardoux [9], Pardoux and Peng [18], Peng [21,22], Hu [13], and Delbaen and Tang [11]. Stochastic differential equations reflected in a domain are referred to Lions [14], Lions and Sznitman [15], Menaldi [16], Pardoux and Williams [19], Saisho [23], among others. Pardoux and Zhang [20] studied BSDEs (1.2), and gave a probabilistic formula for the solution of a semi-linear system of parabolic or elliptic partial differential equation (PDE) with a nonlinear Neumann boundary condition. Other related studies on a PDE with a nonlinear Neumann boundary condition include Boufoussia and Van Casterenb [5], who gave an approximation result to semi-linear parabolic PDEs with Neumann boundary conditions with the help of BSDEs, and Day [10], who studied the Neumann boundary conditions for viscosity solutions of Hamilton–Jacobi equations. In contrast to those works, we study optimal control of stochastic differential systems reflected in a domain, and give the stochastic representation for the solution of the associated HJB equation (1.3) with a nonlinear Neumann boundary condition.

In this paper, the generalized BSDE formulation of dynamic programming given by Peng [21,22] for optimally controlled SDEs, is extended to our controlled stochastic differential systems reflected in a domain. The relevant arguments of Buckdahn and Li [7] is generalized to show that our value functionW (see (3.7)) is deterministic

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J. LI AND S. TANG

(see Prop. 3.1). Since our associated BSDE involves an increasing process which incorporates the reflection of the system state on the boundary of the given domain, we have to resolve some new issues, for example, a new estimate (see Prop. A.3) for the increasing processK, and the linear growth and locally Lipschitz continuity of the value of the system pathY at the initial time with respect to the initial (random) state (see Prop. A.2, which improves the estimates on GBSDE of Pardoux and Zhang [20]). Using these new results, we can prove that the value function is continuous (see Thm. 3.2) and moreover, it is the unique viscosity solution of the associated HJB equation (see Thm. 4.1). On the other hand, Proposition 3.1 allows us to prove the dynamic programming principle (DPP in short, see Thm. 3.1) in a straight forward way by adapting to GBSDEs the method of stochastic backward semigroups introduced by Peng [21]. Furthermore, our proof of Theorem 4.1 contains techniques so as to deal with the Neumann boundary condition, which differs heavily from the counterpart of either Buckdahn and Li [7] or Peng [21]. For more details, the reader is referred to among others Lemmas 4.2 and 4.3 and the constructions of BSDEs (4.10), (4.12), (4.23) and (4.24),etc.

The rest of the paper is organized as follows. In Section 2, we give some preliminary results on BSDEs and GBSDEs. In Section 3, we formulate the optimal stochastic control problem and define the value functionW. We prove thatWis deterministic and satisfies the DPP. Furthermore, we prove thatWis continuous. In Section 4, we prove thatW is the unique viscosity solution to the associated HJB equation. In the end, we give some properties on GBSDEs associated with forward reflected SDEs in the Appendix (Sect. A.1), where Propositions A.2 and A.3 contain new results on GBSDEs. For the reader’s convenience, the proofs of Proposition 3.1 and Theorem 3.1 are given in Section A.2.

2. Preliminaries

We consider the Wiener space (Ω,F, P), whereΩis the set of continuous functions from [0, T] toRdstarting from 0 (Ω =C0([0, T];Rd)), F the completed Borel σ-algebra overΩ, and P the Wiener measure. Let B be the canonical process:Bs(ω) =ωs,s∈[0, T],ω∈Ω. ByF={Fs,0≤s≤T}we denote the natural filtration generated by{Bs}0≤s≤T and augmented by all P-null sets,i.e.,

Fs=σ{Br, r≤s} ∨ N, s∈[0, T],

where N is the set of all P-null subsets, and T > 0 a fixed real time horizon. For any n 1, |z| de- notes the Euclidean norm of z Rn. We introduce the following two spaces of processes: S2(0, T;R) is the collection of (ψt)0≤t≤T which is a real-valued adapted c`adl`ag process such that E[ sup

0≤t≤Tt|2] < +∞; and H2(0, T;Rn) is the collection of (ψt)0≤t≤T which is an Rn-valued progressively measurable process such that

||ψ||22=E[T

0 t|2dt]<+∞.

Let{At, t≥0}be a continuous increasing F-progressively measurable scalar process, satisfyingA0 = 0 and E[eμAT]<∞for allμ >0. We are given a final conditionξ∈L2(Ω,FT, P) such thatE(eμAT|ξ|2)<∞for all μ >0, and two random fieldsf :Ω×[0, T]×R×Rd Randg:Ω×[0, T]×RRsatisfying,

(H2.1)

(i) The processesf(·, y, z) andg(·, y) areF-progressively measurable and ET

0 eμAt|f(t,0,0)|2dt

+ET

0 eμAt|g(t,0)|2dAt

<∞, for allμ >0;

(ii) There is a constantCsuch that, for all (t, y, z)[0, T]×R×Rd,

|f(t, y, z)−f(t, y, z)| ≤C(|y−y|+|z−z|) ;

(iii) There is a constantC such that, for all (t, y)[0, T]×R,

|g(t, y)−g(t, y)| ≤C|y−y|. A solution to the following GBSDE

Yt=ξ+ T

t

f(s, Ys, Zs) ds+ T

t

g(s, Ys) dAs T

t

ZsdBs, 0≤t≤T, (2.1)

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is a pair of F-progressively measurable processes (Yt, Zt)0≤t≤T taking values in R×Rd which satisfies equa- tion (2.1) and

E

0≤t≤Tsup |Yt|2

+E T

0 |Zt|2dt

<∞, 0≤t≤T. (2.2)

From Theorem 1.6 and Proposition 1.1 of [20], we have the following two lemmas.

Lemma 2.1. Let (H2.1)be satisfied. Then GBSDE(2.1)has a unique solution (Y, Z).

Lemma 2.2. Under the assumption(H2.1), we have for any μ >0 E

0≤t≤Tsup eμAt|Yt|2+ T

0 eμAt|Yt|2 dAt+ T

0 eμAt|Zt|2dt

≤CE

eμAT|ξ|2+ T

0 eμAt|f(t,0,0)|2dt+ T

0 eμAt|g(t,0)|2dAt

(2.3) for a positive constantC, which depends on the Lipschitz constant off andg,μ, andT.

Let two sets of data (ξ, f, g, A) and (ξ, f, g, A) satisfy assumption (H2.1). Let (Y, Z) be a solution to GBSDE (2.1) for data (ξ, f, g, A) and (Y, Z) for data (ξ, f, g, A). We define

Y ,¯ Z,¯ ξ,¯ f ,¯g,¯ A¯

= (Y −Y, Z−Z, ξ−ξ, f−f, g−g, A−A).

The following two lemmas are borrowed from Proposition 1.2 and Theorem 1.4 of Pardoux and Zhang [20], respectively.

Lemma 2.3. For any μ >0, there exists a constant C such that E

0≤t≤Tsup eμktYt2+ T

0 eμktZ¯t2dt

≤CE

eμkT ξ¯2+ T

0 eμktf¯(t, Yt, Zt)2dt+ T

0 eμkt|¯g(t, Yt)|2dAt+ T

0 eμkt|g(t, Yt)|2dA¯

t

,(2.4) wherekt:=||A¯||t+At, and||A¯||t is the total variation of the processA¯on the interval [0, t].

For the particular caseA≡A, we have

Lemma 2.4 (Comparison Theorem). Assume that ξ≤ξ, f(t, y, z)≤f(t, y, z), and g(t, y)≤g(t, y), for all (y, z)R×Rd, dP×dt,a.s. Then Yt≤Yt,0≤t≤T, a.s.

Moreover, if Y0 = Y0, then Yt = Yt, 0 t T, a.s. In particular, if in addition either P(ξ < ξ) > 0 or f(t, y, z)< f(t, y, z)for any(y, z)R×Rd holds on a set of positivedt×dP measure, org(t, y)< g(t, y)for any y∈Rholds on a set of positivedAt×dP measure, thenY0< Y0.

3. Formulation of the problem and related DPP

We assume that the two functions b : [0, T]×Rd×U Rd and σ : [0, T]×Rd×U Rd×d satisfy the following three conditions:

(H3.1)

(i) The two functionsbandσ are uniformly continuous in (t, u);

(ii) There is a constatntC >0 such that, for all (t, u)[0, T]×U andx, xRn,

|b(t, x, u)−b(t, x, u)|+|σ(t, x, u)−σ(t, x, u)| ≤C|x−x|;

(iii) There is a constantC >0 such that, for all (t, x, u)[0, T]×Rn×U,

|b(t, x, u)|+|σ(t, x, u)| ≤C(1 +|x|).

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J. LI AND S. TANG

Foru∈ U, the corresponding state process starting fromζ∈L2(Ω,Ft, P; ¯D) at the initial timet, is governed by the following reflected SDE:

⎧⎪

⎪⎪

⎪⎪

⎪⎩

Xst,ζ;u=ζ+s

t b

r, Xrt,ζ;u, ur

dr+s

t σ

r, Xrt,ζ;u, ur dBr +s

t ∇φ Xrt,ζ;u

dKrt,ζ;u, s∈[t, T], Kst,ζ;u=s

t I{Xrt,ζ;u∈∂D}dKrt,ζ;u, Kt,ζ;uis increasing.

(3.1)

In view of Proposition A.1 in the Appendix, SDE (3.1) has a unique strong solution (Xt,ζ;u, Kt,ζ;u). Moreover, for any (t, u)[0, T]× U andζ, ζ∈L2(Ω,Ft, P; ¯D), we have

E

sup

s∈[t,T]|Xst,ζ;u−Xst,ζ;u|4|Ft

≤C|ζ−ζ|4,

E

sup

s∈[t,T]|Xst,ζ;u|4|Ft

≤C

1 +|ζ|4

. (3.2)

Here, the constant C depends only on the Lipschitz and the linear growth constants of b and σ with respect tox.

Assume that three functions Φ:Rd R, f : [0, T]×Rd×R×Rd×U R, andg : [0, T]×Rd×RR satisfy the following conditions:

(H3.2)

(i)f is uniformly continuous in (t, u); g(·)∈C1,2,2([0, T]×Rd×R); and there exists a constantC >0 such that, for allt∈[0, T], u∈U, (x, y, z),(x, y, z)Rd+1+d,

|f(t, x, y, z, u)−f(t, x, y, z, u)|+|g(t, x, y)−g(t, x, y)|

≤C(|x−x|+|y−y|+|z−z|) ;

(ii) There is a constantC >0 such that, for allx, xRd,

|Φ(x)−Φ(x)| ≤C|x−x|;

(iii) There is someC >0 such that, for all (t, u)[0, T]×U andx∈Rn,

|f(t, x,0,0, u)| ≤C(1 +|x|).

Note that conditions (i) and (ii) of assumption (H3.2) imply the globally linear growth in the state variable of the two functions gandΦ: for someC >0,|g(t, x,0)|+|Φ(x)| ≤C(1 +|x|) for all (t, x)∈[0, T]×Rn.

For any u(·) ∈ U, and ζ L2(Ω,Ft, P; ¯D), the mappings ξ := Φ(XTt,ζ;u), g(s, y) := g(s, Xst,ζ;u, y) and f(s, y, z) :=f(s, Xst,ζ;u, y, z, us) satisfy the conditions (H2.1) on the interval [t, T]. Therefore, there is a unique solution to the following GBSDE:

⎧⎪

⎪⎪

⎪⎪

⎪⎩

dYst,ζ;u=f

s, Xst,ζ;u, Yst,ζ;u, Zst,ζ;u, us ds +g

s, Xst,ζ;u, Yst,ζ;u

dKst,ζ;u−Zst,ζ;udBs, YTt,ζ;u=Φ

XTt,ζ;u

,

(3.3)

where (Xt,ζ;u, Kt,ζ;u) solves the reflected SDE (3.1).

Moreover, similar to Proposition A.2, there exists some constant C >0 such that, for allt [0, T], ζ, ζ L2(Ω,Ft, P; ¯D), u∈ U, P-a.s.,

(i) Ytt,ζ;u−Ytt,ζ;u≤C

|ζ−ζ|+|ζ−ζ|12

;

(ii)Ytt,ζ;u≤C(1 +|ζ|). (3.4)

We now define our admissible controls.

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Definition 3.1. An admissible control processu={ur, r∈[t, s]}on [t, s] (withs∈(t, T]) is anFr-progressively measurable process taking values in U. The set of all admissible controls on [t, s] is denoted byUt,s.We identify two processesuand ¯uin Ut,sand write u≡u¯ on [t, s],ifP{u= ¯ua.e. in [t, s]}= 1.

Atu∈ Ut,T, the value of the cost functional is given by

J(t, x;u) :=Ytt,x;u, (t, x)[0, T]×D,¯ (3.5) where the processYt,x;uis defined by GBSDE (3.3).

From TheoremA.7, we have

J(t, ζ;u) =Ytt,ζ;u, (t, ζ)[0, T]×L2

Ω,Ft, P; ¯D

. (3.6)

We define the value function of our stochastic control problem as follows:

W(t, x) := esssup

u∈Ut,T

J(t, x;u), (t, x)[0, T]×D.¯ (3.7) Under assumptions (H3.1) and (H3.2), the value functionW is well-defined on [0, T]×D, and its values at timet are bounded andFt-measurable random variables. In fact, they are all deterministic. We have

Proposition 3.2. For any (t, x) [0, T]×D, we have¯ W(t, x) = E[W(t, x)], P-a.s. Let W(t, x) equal to its deterministic version E[W(t, x)]. ThenW : [0, T]×D¯ Ris a deterministic function.

The proof is an adaptation of relevant arguments of Buckdahn and Li [7]. For the readers’ convenience we give it in the Section A.2 of Appendix.

As an immediate result of (3.4) and (3.7), the value functionW has the following property.

Lemma 3.3. There exists a constant C >0 such that, for all (t, x, x)[0, T]×D¯ ×D,¯ (i) |W(t, x)−W(t, x)| ≤C

|x−x|+|x−x|12

;

(ii)|W(t, x)| ≤C(1 +|x|). (3.8)

We now study the (generalized) DPP for our stochastic control problem (3.1), (3.3), and (3.7). For this we have to define the family of (backward) semigroups related with GBSDE (3.3). Peng [21] first introduced the notion of backward stochastic semigroups to study the DPP for the optimal stochastic control of SDEs. In what follows, it is adapted to the optimal control problem of stochastic differential systemsreflectedin a domain.

Given the initial data (t, x), a positive numberδ≤T−t, an admissible controlu(·)∈ Ut,t+δ, and a random variableη∈L2(Ω,Ft+δ, P;R), we define

Gt,x;us,t+δ[η] := ˜Yst,x;u, s∈[t, t+δ], (3.9) where ( ˜Yst,x;u,Z˜st,x;u)t≤s≤t+δ is the solution of the following GBSDE on the time interval [t, t+δ]:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

−d ˜Yst,x;u=f

s, Xst,x;u,Y˜st,x;u,Z˜st,x;u, us

ds+g

s, Xst,x;u,Y˜st,x;u

dKst,x;u

−Z˜st,x;udBs, s∈[t, t+δ];

Y˜t+δt,x;u=η,

and (Xt,x;u, Kt,x;u) is the solution of reflected SDE (3.1). Then, obviously, for the solution (Yt,x;u, Zt,x;u) of GBSDE (3.3), we have

Gt,x;ut,T Φ

XTt,x;u

=Gt,x;ut,t+δ Yt+δt,x;u

. (3.10)

Furthermore,

J(t, x;u) =Ytt,x;u=Gt,x;ut,T Φ

XTt,x;u

=Gt,x;ut,t+δ Yt+δt,x;u

=Gt,x;ut,t+δ J

t+δ, Xt+δt,x;u;u .

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J. LI AND S. TANG

Remark 3.4. If bothf andgdo not depend on (y, z), we have Gt,x;us,t+δ[η] =E

η+

t+δ

s

f

r, Xrt,x;u, ur dr+

t+δ

s

g

r, Xrt,x;u

dKrt,x;u|Fs

, s∈[t, t+δ].

Theorem 3.5. Under assumptions (H3.1)and (H3.2), the value functionW satisfies the following DPP: For any 0≤t < t+δ≤T,x∈D,¯

W(t, x) = esssup

u∈Ut,t+δ

Gt,x;ut,t+δ W

t+δ, Xt+δt,x;u

. (3.11)

The proof is similar to [4]. For the readers’s convenience we give it in Section A.2.

Lemma 3.3shows that the value function W(t, x) is continuous in x, uniformly int. From Theorem3.5 we can get the continuity ofW(t, x) in t.

Theorem 3.6. Let assumptions (H3.1)and (H3.2)be satisfied. Then the value functionW(t, x)is continuous int.

Proof. Let (t, x)[0, T]×D¯ andδ∈(0, T−t]. We want to prove thatW is continuous int. For this we notice that from (A.33), for an arbitrarily smallε >0,

Iδ1+Iδ2≤W(t, x)−W(t+δ, x)≤Iδ1+Iδ2+Cε, (3.12) where

Iδ1 :=Gt,x;ut,t+δε

W

t+δ, Xt+δt,x;uε

−Gt,x;ut,t+δε[W(t+δ, x)],

Iδ2 :=Gt,x;ut,t+δε[W(t+δ, x)]−W(t+δ, x),

for uε ∈ Ut,t+δ such that (A.33) holds. From Lemma 2.3 (taking μ = 1) and the estimates (A.4), in Ap- pendix, (3.8) we get that, for some constantC which does not depend on the controlsuε,

Iδ1≤C

E W

t+δ, Xt+δt,x;uε

−W(t+δ, x)4Ft

14

≤C

E

Xt+δt,x;uε−x4+Xt+δt,x;uε−x2Ft

14 ,

and sinceE[|Xt+δt,x;uε−x|8|Ft]≤Cδ4(refer to (A.17) in Appendix) we get that|Iδ1| ≤Cδ14. From the definition ofGt,x;ut,t+δε[·] (see (3.9)),

Iδ2=E

W(t+δ, x) + t+δ

t

f

s, Xst,x;uε,Y˜st,x;uε,Z˜st,x;uε, uεs

ds

+ t+δ

t

g

s, Xst,x;uε,Y˜st,x;uε

dKst,x;uε t+δ

t

Z˜st,x;uεdBs|Ft

−W(t+δ, x)

=E t+δ

t

f

s, Xst,x;uε,Y˜st,x;uε,Z˜st,x;uε, uεs

ds+ t+δ

t

g

s, Xst,x;uε,Y˜st,x;uε

dKst,x;uε|Ft

.

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From the Schwartz inequality, Propositions A.2 and A.3 in Appendix and (3.2), we then get Iδ2≤δ12E

t+δ

t

f

s, Xst,x;uε,Y˜st,x;uε,Z˜st,x;uε, uεs2ds|Ft

12

+E

Kt+δt,x;uεFt

12 E

t+δ

t

g

s, Xst,x;uε,Y˜st,x;uε2dKst,x;uεFt

12

≤δ12E t+δ

t

f

s, Xst,x;uε,0,0, uεs+CY˜st,x;uε+CZ˜st,x;uε2 ds|Ft

12

+E

Kt+δt,x;uεFt

12 E

t+δ

t

g

s, Xst,x;uε,0+CY˜st,x;uε2

dKst,x;uεFt

12

≤Cδ12E t+δ

t

1 +Xst,x;uε+Y˜st,x;uε+Z˜st,x;uε2 ds|Ft

12

+CE

Kt+δt,x;uεFt

12 E

t+δ

t

1 +Xst,x;uε+Y˜st,x;uε2

dKst,x;uεFt

12

≤Cδ12 +C

E

Kt+δt,x;uε2Ft

12

≤Cδ12.

Then, from (3.12),|W(t, x)−W(t+δ, x)| ≤Cδ14 +12 +Cε,and lettingε 0 we get W(t, x) is continuous

in t. The proof is complete.

4. Viscosity solutions of related HJB equations

We consider the following PDE:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂tW(t, x) +H

t, x, W, DW, D2W

= 0, (t, x)[0, T)×D,

∂nW(t, x) +g(t, x, W(t, x)) = 0, 0≤t < T, x∈∂D;

W(T, x) =Φ(x), x∈D,¯

(4.1)

where at a pointx∈∂D, ∂n =d

i=1

∂xiφ(x)∂x

i, and the HamiltonianH is defined by H(t, x, y, p, A) := sup

u∈U

1 2tr

σσT(t, x, u)A

+p, b(t, x, u)+f(t, x, y, pσ, u)

,

where (t, x, y, p, A)[0, T]×Rn×R×Rd×Sd withSd being the set of alld×dsymmetric matrices.

In this section we shall prove that the value function W defined by (3.7) is the unique viscosity solution of (4.1). The interested reader is referred to Crandall, Ishii, and Lions [8] for a detailed introduction to viscosity solutions. LetCl,b3 ([0, T]×D) be the set of the real-valued functions that are continuously differentiable up to¯ the third order and whose derivatives of order from 1 to 3 are bounded.

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J. LI AND S. TANG

Definition 4.1. A real-valued continuous functionW ∈C([0, T]×D) is called¯

(i) a viscosity subsolution of (4.1) ifW(T, x)≤Φ(x), for allx∈D, and if for all functions¯ ϕ∈Cl,b3 ([0, T]×D)¯ and (t, x)[0, T)×D¯ such thatW −ϕattains its local maximum at (t, x):

∂ϕ

∂t(t, x) +H

t, x, W, Dϕ, D2ϕ

0, ifx∈D;

max ∂ϕ

∂t(t, x) +H

t, x, W, Dϕ, D2ϕ , ∂ϕ

∂n(t, x) +g(t, x, W)

0, ifx∈∂D;

(ii) a viscosity supersolution of (4.1) ifW(T, x)≥Φ(x), for allx∈D, and if for all functions¯ ϕ∈Cl,b3 ([0, T]×D)¯ and (t, x)[0, T)×D¯ such thatW −ϕattains its local minimum at (t, x):

∂ϕ

∂t(t, x) +H

t, x, W, Dϕ, D2ϕ

0, if x∈D;

min ∂ϕ

∂t(t, x) +H

t, x, W, Dϕ, D2ϕ , ∂ϕ

∂n(t, x) +g(t, x, W)

0, ifx∈∂D;

(iii) a viscosity solution of (4.1) if it is both a viscosity sub- and a supersolution of (4.1).

For simplicity of notations, we define forϕ∈Cl,b3 ([0, T]×D),¯ F(s, x, y, z, u) =

∂sϕ(s, x) +1 2tr

σσT(s, x, u)D2ϕ

+Dϕ.b(s, x, u) +f(s, x, y+ϕ(s, x), z+Dϕ(s, x).σ(s, x, u), u), G(s, x, y) =

∂nϕ(s, x) +g(s, x, y+ϕ(s, x)), (4.2)

for (s, x, y, z, u)[0, T]×D¯ ×R×Rd×U.

Proposition 4.2. Under the assumptions (H3.1) and (H3.2) the value function W is a viscosity subsolution to(4.1).

Proof. Obviously,W(T, x) =Φ(x),x∈D. Suppose that¯ ϕ∈Cl,b3 ([0, T]×D) and (t, x)¯ [0, T)×D¯ is such that W −ϕattains its maximum at (t, x). Without loss of generality, we assume that ϕ(t, x) =W(t, x).

We first consider the casex∈D. We shall prove that sup

u∈UF(t, x,0,0, u)0.

If this is not true, then there exists someθ >0 such that F0(t, x) := sup

u∈UF(t, x,0,0, u)≤ −θ <0. (4.3) Therefore,F(t, x,0,0, u)≤ −θ, for allu∈U.

SinceF0is continuous at (t, x), we can choose ¯α∈(0, T −t] such that

Oα¯(x) :={y˜:|˜y−x| ≤α} ⊂¯ D, (4.4) F(s,y,˜ 0,0, u)≤ −12θ, for all (s,y, u)˜ [t, t+ ¯α]×Oα¯(x)×U. (4.5)

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For anyα∈(0,α], we consider the following BSDE:¯

⎧⎪

⎪⎩

−dYs1,u=F

s, Xst,x;u, Ys1,u, Zs1,u, us

ds+G

s, Xst,x;u, Ys1,u

dKst,x;u

−Zs1,udBs, s∈[t, t+α];

Yt+α1,u = 0,

(4.6)

where the pair of processes (Xt,x,u, Kt,x,u) are given by (3.1) and u(·) ∈ Ut,t+α. It is not hard to check that F(s, Xst,x;u, y, z, us) and G(s, Xst,x;u, y) satisfy (H2.1). Thus, due to Lemma 2.1, GBSDE (4.6) has a unique solution. We have the following observation.

Lemma 4.3. For every s∈[t, t+α], we have the following relationship:

Ys1,u=Gt,x;us,t+α ϕ

t+α, Xt+αt,x;u

−ϕ

s, Xst,x;u

, P-a.s. (4.7)

Proof. We recall thatGt,x;us,t+α[ϕ(t+α, Xt+αt,x;u)] is defined by the solution of the GBSDE

⎧⎪

⎪⎨

⎪⎪

−dYsu=f(s, Xst,x;u, Ysu, Zsu, us) ds+g(s, Xst,x;u, Ysu) dKst,x;u

−ZsudBs, s∈[t, t+α];

Yt+αu =ϕ

t+α, Xt+αt,x;u , with the following formula:

Gt,x;us,t+α ϕ

t+α, Xt+αt,x;u

=Ysu, s∈[t, t+α], (4.8)

(see (3.9)). Hence, we only need to show thatYsu−ϕ(s, Xst,x;u)≡Ys1,u for s∈[t, t+α]. This can be verified directly by applying Itˆo’s formula to ϕ(s, Xst,x;u). Indeed, the stochastic differentials of Ysu−ϕ(s, Xst,x;u) and Ys1,uequal, and with the same terminal condition Yt+αu −ϕ(t+α, Xt+αt,x;u) = 0 =Yt+α1,u. Remark 4.4. Forx∈∂D Lemma 4.1 still holds.

On the other hand, from the DPP (see Thm.3.5), for everyα, ϕ(t, x) =W(t, x) = esssup

u∈Ut,t+α

Gt,x;ut,t+α W

t+α, Xt+αt,x;u ,

and fromW ≤ϕand the monotonicity property of Gt,x;ut,t+δ[·] (see Lem. 2.4) we get esssup

u∈Ut,t+α

Gt,x;ut,t+δ

ϕ

t+α, Xt+αt,x;u

−ϕ(t, x)

0, P-a.s.

Thus, from Lemma4.3, we have esssupu∈Ut,t+αYt1,u0,P-a.s.

Hence, for arbitraryε >0, similar to that of inequality (A.33), there isuε∈ Ut,t+α such that

Yt1,uε ≥ −εα, P-a.s. (4.9)

Remark 4.5. Similarly, (4.9) is still true forx∈∂D.

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J. LI AND S. TANG

For uε∈ Ut,t+α we define τ = inf{s ≥t : |Xst,x;uε−x| ≥α} ∧¯ (t+α). Consequently, on [t, τ] the process (Kt,x;u) is zero and, hence

Ys1;uε =Yτ1;uε+ τ

s

F

r, Xrt,x;uε, Yr1;uε, Zr1;uε, uεr

dr τ

s

Zr1;uεdBr. We consider the following two BSDEs:

dYs2=

CYs2+Zs212θ

ds−Zs2dBs, Yt+α2 = 0,

(4.10) whose unique solution is given by

Ys2= θ 2C

1eC(s−(t+α))

, Zs2= 0, s∈[t, t+α], (4.11)

and −dYs3 =

CYs3+Zs312θ

ds−Zs3dBs, s∈[t, τ];

Yτ3 =Yτ1;uε.

(4.12) Here,C is the Lipschitz constant ofF with respect toy, z; also the Lipschitz constant ofGwith respect toy, in order to be different from the constantCwhich may vary from lines to lines. We have the following lemma.

Lemma 4.6. We haveYt1,uε ≤Yt3and|Yt2−Yt3| ≤Cα32,P-a.s. HereC >0is independent of both the controlu andα.

Proof.

(1) We observe from (4.5) and the definition ofτ that, for all (s, y, z, u)[t, τ]×R×Rd×U, F

s, Xst,x;uε, y, z, uε

≤C(|y|+|z|) +F

s, Xst,x;uε,0,0, uε

≤C(|y|+|z|)−1 2θ.

Consequently, from Lemma 2.2 in [7] (the comparison result for BSDEs) we have that Ys1,uε ≤Ys3, s∈[t, τ], P-a.s.,

where (Y3, Z3) is the solution of BSDE (4.12).

(2) From equation (4.6), Proposition A.1 and Proposition A.2 in the Appendix, we have Yτ1;uε≤C(t+α−τ)12 +C

E

Kt+αt,x;uε−Kτt,x;uε 2

Fτ

12 ,

where C is independent of controls, and Kt+αt,x;uε−Kτt,x;uε = Kt+ατ,Xτt,x;uε;uε by means of the uniqueness of solution of reflected SDE (3.1). Therefore, we have

E

Yτ1;uε2Ft

≤CE[(t+α−τ)|Ft] +CE

Kt+ατ,Xτt,x;uε;uε 2

Ft

. From Proposition A.3 in Appendix, we have

E

Kt+ατ,Xt,x;uετ ;uε 2

Ft

≤C E

(t+α−τ)2|Ft

12

. (4.13)

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