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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 709682, 16 pages
http://dx.doi.org/10.1155/2012/709682
Research Article

Maximum Principle for Stochastic Recursive Optimal Control Problems Involving Impulse Controls

1School of Mathematics, Shandong University, Jinan 250100, China
2School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

Received 13 January 2012; Accepted 11 April 2012

Academic Editor: Zhenya Yan

Copyright © 2012 Zhen Wu and Feng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a stochastic recursive optimal control problem in which the control variable has two components: the regular control and the impulse control. The control variable does not enter the diffusion coefficient, and the domain of the regular controls is not necessarily convex. We establish necessary optimality conditions, of the Pontryagin maximum principle type, for this stochastic optimal control problem. Sufficient optimality conditions are also given. The optimal control is obtained for an example of linear quadratic optimization problem to illustrate the applications of the theoretical results.

1. Introduction

The nonlinear backward stochastic differential equations (BSDEs for short) were first introduced by Pardoux and Peng [1]. Independently, Duffie and Epstein [2] introduced BSDEs under economic background. In [2], they presented a stochastic recursive utility which is an extension of the standard additive utility with the instantaneous utility depending not only on the instantaneous consumption rate but also on the future utility. Actually, it corresponds to the solution of a particular BSDE whose generator does not depend on the variable 𝑧. And then, El Karoui et al. [3] gave the formulation of recursive utilities from the BSDE point of view. The problem that the cost function of the control system is described by the solution of BSDE is called the stochastic recursive optimal control problem. In this case, the control systems become forward-backward stochastic differential equations (FBSDEs).

One fundamental research direction for optimal control problem is to establish the necessary optimality conditions—Pontryagin maximum principle. Stochastic maximum principle for forward, backward, and forward-backward systems has been studied by many authors, including Peng [4, 5], Tang and Li [6], Wang and Yu [7], Wu [8], and Xu [9] for full information and Huang et al. [10], Wang and Wu [11], Wang and Yu [12], and Wu [13] for partial information case. However, in these papers, there are only regular controls in the control systems and impulse controls are not included.

Stochastic impulse control problems have received considerable research attention in recent years due to wide applicability in a number of different areas, especially in mathematical finance; see, for example, [1417]. In most cases, the optimal impulse control problem was studied through dynamic programming principle. It was shown in particular that the value function is a solution of some quasi-variational inequalities.

The first result in stochastic maximum principle for singular control problem was obtained by Cadenillas and Haussmann [18], in which linear dynamics, convex cost criterion, and convex state constraint are assumed. Bahlali and Chala [19] generalized [18] to the nonlinear dynamics case with a convex state constraint. Bahlali and Mezerdi [20] considered a stochastic singular control problem in which the control system is governed by a stochastic differential equation where the regular control enters the diffusion coefficient and the control domain is not necessarily convex. The stochastic maximum principle was obtained with the approach developed by Peng [4]. Dufour and Miller [21] studied a stochastic singular control problem in which the admissible control is of bounded variation. It is worth pointing out that the control systems in these works are stochastic differential equations with singular control, and few examples are given to illustrate the theoretical results. Wu and Zhang [22] were the first to study stochastic optimal control problems of forward-backward systems involving impulse controls, and they obtained both the maximum principle and sufficient optimality conditions for the optimal control problem.

In this paper, we continue to study stochastic optimal control problem involving impulse controls, in which the control system is described by a forward-backward stochastic differential equation and the control variable consists of regular control and impulse control. Different from [22], it is assumed in this paper that the domain of the regular controls is not necessarily convex and the control variable does not enter the diffusion coefficient. Thus the result of this paper and that of [22] do not contain each other. We obtain the stochastic maximum principle by using a spike variation on the regular part of the control and a convex perturbation on the impulsive one. Sufficient optimality conditions are also obtained which can help to find the optimal control in applications.

The rest of this paper is organized as follows. In Section 2 we give some preliminary results and the formulation of our stochastic optimal control problem. In Section 3 we obtain the maximum principle for our stochastic optimal control problem. Sufficient optimality conditions for the optimal control problem is established in Section 4, and an example of linear quadratic optimization problem is also given to illustrate the applications of our theoretical results.

2. Formulation of the Stochastic Optimal Control Problem

Firstly we introduce some notations. Let (Ω,,) be a probability space and 𝔼 the expectation with respect to . Let 𝑇 be a finite time horizon and 𝑡 the natural filtration of a 𝑑-dimensional standard Brownian motion {𝐵𝑡,0𝑡𝑇} augmented by the -null sets of . For 𝑛 and 𝑝>1, denote by 𝑆𝑝(𝑛) the set of 𝑛-dimensional adapted processes {𝜑𝑡,0𝑡𝑇} such that 𝔼[sup0𝑡𝑇|𝜑𝑡|𝑝]<, and denote by 𝐻𝑝(𝑛) the set of 𝑛-dimensional adapted processes {𝜓𝑡,0𝑡𝑇} such that 𝔼[(𝑇0|𝜓𝑡|2𝑑𝑡)𝑝/2]<.

Let 𝑈 be a nonempty subset of 𝑘 and 𝐾 a nonempty convex subset of 𝑛. Let {𝜏𝑖} be a given sequence of increasing 𝑡-stopping times such that 𝜏𝑖+ as 𝑖. We denote by the class of right continuous processes 𝜂()=𝑖1𝜂𝑖𝟙[𝜏𝑖,𝑇]() such that each 𝜂𝑖 is an 𝜏𝑖-measurable random variable. It is worth noting that the assumption 𝜏𝑖+ implies that at most finitely many impulses may occur on [0,𝑇]. Denote by 𝒰 the class of adapted processes 𝑣[0,𝑇]×Ω𝑈 such that 𝔼[sup0𝑡𝑇|𝑣𝑡|3]<, and denote by 𝒦 the class of 𝐾-valued impulse processes 𝜂() such that 𝔼[(𝑖1|𝜂𝑖|)3]<. We call 𝒜=𝒰×𝒦 the admissible control set. In what follows, for a continuous function 𝑙(), the integration 𝑇0𝑙(𝑡)𝑑𝜂𝑡 is understood as follows: 𝑇0𝑙(𝑡)𝑑𝜂𝑡=0𝜏𝑖𝑇𝑙𝜏𝑖𝜂𝑖.(2.1)

Given 𝜂() and 𝑥𝑛, we consider the following SDE with impulses: 𝑑𝑋𝑡=𝑏𝑡,𝑋𝑡𝑑𝑡+𝜎𝑡,𝑋𝑡𝑑𝐵𝑡+𝐶𝑡𝑑𝜂𝑡,𝑋0=𝑥,(2.2) where 𝑏[0,𝑇]×Ω×𝑛𝑛, 𝜎[0,𝑇]×Ω×𝑛𝑛×𝑑, and 𝐶[0,𝑇]𝑛×𝑛 are measurable mappings. Similar to [22, Proposition 2.1], we have the following.

Proposition 2.1. Let 𝐶 be continuous and 𝑏, 𝜎 uniformly Lipschitz in 𝑥. Assume that 𝑏(,0)𝐻𝑝(𝑛), 𝜎(,0)𝐻𝑝(𝑛×𝑑), and 𝔼[(𝑖1|𝜂𝑖|)𝑝]< for some 𝑝2. Then SDE (2.2) admits a unique solution 𝑋()𝑆𝑝(𝑛).

For 𝜂(), let us consider the following BSDE with impulses: 𝑑𝑌𝑡=𝑓𝑡,𝑌𝑡,𝑍𝑡𝑑𝑡+𝑍𝑡𝑑𝐵𝑡𝐷𝑡𝑑𝜂𝑡,𝑌𝑇=𝜁,(2.3) where 𝜁𝑇, 𝑓[0,𝑇]×Ω×𝑚×𝑚×𝑑𝑚 and 𝐷[0,𝑇]𝑚×𝑛 are measurable mappings. Similar to [22, Proposition 2.2], we have the following.

Proposition 2.2. Let 𝐷 be continuous and 𝑓 Lipschitz in (𝑦,𝑧). Assume that 𝔼|𝜁|𝑝<, 𝔼[(𝑖1|𝜂𝑖|)𝑝]<, and 𝑓(,0,0)𝐻𝑝(𝑚) for some 𝑝2. Then BSDE (2.3) admits a unique solution (𝑌(),𝑍())𝑆𝑝(𝑚)×𝐻𝑝(𝑚×𝑑).

The control system of our stochastic optimal control problem is subject to the following FBSDE: 𝑑𝑥𝑡𝑣,𝜂=𝑏𝑡,𝑥𝑡𝑣,𝜂,𝑣𝑡𝑑𝑡+𝜎𝑡,𝑥𝑡𝑣,𝜂𝑑𝐵𝑡+𝐶𝑡𝑑𝜂𝑡,𝑑𝑦𝑡𝑣,𝜂=𝑓𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑧𝑡𝑣,𝜂,𝑣𝑡𝑑𝑡+𝑧𝑡𝑣,𝜂𝑑𝐵𝑡𝐷𝑡𝑑𝜂𝑡,𝑥0𝑣,𝜂=𝑎𝑛,𝑦𝑇𝑣,𝜂𝑥=𝑔𝑇𝑣,𝜂,(2.4) where 𝑏[0,𝑇]×𝑛×𝑈𝑛, 𝜎[0,𝑇]×𝑛𝑛×𝑑, 𝑓[0,𝑇]×𝑛×𝑚×𝑚×𝑑×𝑈𝑚, 𝑔𝑛𝑚 are measurable mappings, and 𝐶[0,𝑇]𝑛×𝑛, 𝐷[0,𝑇]𝑚×𝑛 are continuous functions. The objective is to minimize the following cost functional over the class 𝒜: 𝜙𝑥𝐽(𝑣(),𝜂())=𝔼𝑇𝑣,𝜂𝑦+𝛾0𝑣,𝜂+𝑇0𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑣𝑡𝑑𝑡+𝑖1𝑙𝜏𝑖,𝜂𝑖,(2.5) where 𝜙𝑛, 𝛾𝑚, [0,𝑇]×𝑛×𝑚×𝑈, and 𝑙[0,𝑇]×𝑛 are measurable mappings.

In what follows we assume the following.(𝐻1)𝑏, 𝜎, 𝑓, 𝑔 are continuous, and they are continuously differentiable in (𝑥,𝑦,𝑧), with derivatives continuous and uniformly bounded. Moreover, assume that 𝑏 and 𝑓 have linear growth in (𝑥,𝑦,𝑧,𝑣).(𝐻2)𝜙, 𝛾, , 𝑙 are continuous, and they are continuously differentiable in (𝑥,𝑦,𝜂), with derivatives continuous and bounded by 𝑐(1+|𝑥|), 𝑐(1+|𝑦|), 𝑐(1+|𝑥|+|𝑦|+|𝑣|), and 𝑐(1+|𝜂|), respectively. Moreover, we assume |(𝑡,0,0,𝑣)|𝑐(1+|𝑣|3) for any (𝑡,𝑣).

From Propositions 2.1 and 2.2, it follows that FBSDE (2.4) admits a unique solution (𝑥𝑣,𝜂(),𝑦𝑣,𝜂(),𝑧𝑣,𝜂())𝑆3(𝑛)×𝑆3(𝑚)×𝐻3(𝑚×𝑑) for any (𝑣(),𝜂())𝒜, and the functional 𝐽 is well defined.

3. Stochastic Maximum Principle for the Optimal Control Problem

Let (𝑢(),𝜉()=𝑖1𝜉𝑖𝟙[𝜏𝑖,𝑇]())𝒜 be an optimal control and (𝑥𝑢,𝜉(),𝑦𝑢,𝜉(),𝑧𝑢,𝜉()) the corresponding trajectory. We introduce the spike variation with respect to 𝑢() as follows: 𝑢𝜀𝑡=𝑢𝑣,if𝜏𝑡𝜏+𝜀,𝑡,otherwise,(3.1) where 𝜏[0,𝑇) is an arbitrarily fixed time, 𝜀>0 is a sufficiently small constant, and 𝑣 is an arbitrary 𝑈-valued 𝜏-measurable random variable such that 𝔼|𝑣|3<. Let 𝜂() be such that 𝜉()+𝜂()𝒦. Then it is easy to check that 𝜉𝜀()=𝜉()+𝜀𝜂(), 0𝜀1 is also an element of 𝒦. Let us denote by (𝑥𝜀(),𝑦𝜀(),𝑧𝜀()) the trajectory associated with (𝑢𝜀(),𝜉𝜀()). For convenience, denote 𝜑(𝑡)=𝜑(𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑢𝑡), 𝜑(𝑢𝜀𝑡)=𝜑(𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑢𝜀𝑡) for 𝜑=𝑏,𝜎,𝑓,,𝑏𝑥,𝜎𝑥,𝑓𝑥,𝑓𝑦,𝑓𝑧,𝑥,𝑦. In what follows, we use 𝑐 to denote a positive constant which can be different from line to line.

Let us introduce the following FBSDE (called the variational equation): 𝑑𝑥1𝑡=𝑏𝑥(𝑡)𝑥1𝑡𝑢+𝑏𝜀𝑡𝑏(𝑡)𝑑𝑡+𝜎𝑥(𝑡)𝑥1𝑡𝑑𝐵𝑡+𝜀𝐶𝑡𝑑𝜂𝑡,𝑑𝑦1𝑡𝑓=𝑥(𝑡)𝑥1𝑡+𝑓𝑦(𝑡)𝑦1𝑡+𝑓𝑧(𝑡)𝑧1𝑡𝑢+𝑓𝜀𝑡𝑓(𝑡)𝑑𝑡+𝑧1𝑡𝑑𝐵𝑡𝜀𝐷𝑡𝑑𝜂𝑡,𝑥10=0,𝑦1𝑇=𝑔𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇.(3.2) By Propositions 2.1 and 2.2, FBSDE (3.2) admits a unique solution (𝑥1(),𝑦1(),𝑧1())𝑆3(𝑛)×𝑆3(𝑚)×𝐻3(𝑚×𝑑).

Similar to [9, Lemma 1], we can easily obtain the following.

Lemma 3.1. We have sup0𝑡𝑇𝔼||𝑥1𝑡||3+sup0𝑡𝑇𝔼||𝑦1𝑡||3+𝔼𝑇0|𝑧1𝑡|2𝑑𝑡3/2𝑐𝜀3.(3.3)

We proceed to give the following lemma.

Lemma 3.2. The following estimations hold: sup0𝑡𝑇𝔼||𝑥𝜀𝑡𝑥𝑡𝑢,𝜉𝑥1𝑡||2𝐶𝜀𝜀2,(3.4)sup0𝑡𝑇𝔼||𝑦𝜀𝑡𝑦𝑡𝑢,𝜉𝑦1𝑡||2𝐶𝜀𝜀2𝔼,(3.5)𝑇0||𝑧𝜀𝑡𝑧𝑡𝑢,𝜉𝑧1𝑡||2𝑑𝑡𝐶𝜀𝜀2,(3.6) where 𝐶𝜀0 as 𝜀0.

Proof. It is easy to check that 𝑥𝜀𝑡𝑥𝑡𝑢,𝜉𝑥1𝑡=𝑡0𝐶𝜀𝑠𝑥𝜀𝑠𝑥𝑠𝑢,𝜉𝑥1𝑠+𝐴𝜀𝑠+𝑑𝑠𝑡0𝐷𝜀𝑠𝑥𝜀𝑠𝑥𝑠𝑢,𝜉𝑥1𝑠+𝐵𝜀𝑠𝑑𝐵𝑠,(3.7) where 𝐴𝜀𝑠=10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑢𝜀𝑠𝑏𝑥(𝑠)𝑑𝜆𝑥1𝑠,𝐵𝜀𝑠=10𝜎𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠𝜎𝑥(𝑠)𝑑𝜆𝑥1𝑠,𝐶𝜀𝑠=10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝑥1𝑠𝑥+𝜆𝜀𝑠𝑥𝑠𝑢,𝜉𝑥1𝑠,𝑢𝜀𝑠𝐷𝑑𝜆,𝜀𝑠=10𝜎𝑥𝑠,𝑥𝑠𝑢,𝜉+𝑥1𝑠𝑥+𝜆𝜀𝑠𝑥𝑠𝑢,𝜉𝑥1𝑠𝑑𝜆.(3.8) Since 𝑏𝑥, 𝜎𝑥 are uniformly bounded, we have sup0𝑠𝑇(|𝐶𝜀𝑠|+|𝐷𝜀𝑠|)𝑐. Hence, if we can obtain sup0𝑡𝑇𝔼𝑡0𝐴𝜀𝑠𝑑𝑠+𝑡0𝐵𝜀𝑠𝑑𝐵𝑠2𝐶𝜀𝜀2,(3.9) then the estimation (3.4) can be obtained from Gronwall's lemma and (3.7). Let us take the 𝐴𝜀 term for example. By the definition of 𝑢𝜀 and Hölder's inequality, we have sup0𝑡𝑇𝔼𝑡0𝐴𝜀𝑠𝑑𝑠22𝔼𝑇0||||10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑢𝑠𝑏𝑥(𝑠)𝑑𝜆𝑥1𝑠||||𝑑𝑠2+2𝔼𝜏𝜏+𝜀||||10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑣𝑏𝑥(𝑠)𝑑𝜆𝑥1𝑠||||𝑑𝑠2=2𝐼+2𝐼𝐼.(3.10) From Hölder's inequality, Lemma 3.1, and the dominated convergence theorem, it follows that 𝐼𝑇𝔼𝑇0||||10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑢𝑠𝑏𝑥(𝑠)𝑑𝜆𝑥1𝑠||||2𝑑𝑠𝑇𝑇0𝔼||𝑥1𝑠||32/3𝔼||||10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑢𝑠𝑏𝑥||||(𝑠)𝑑𝜆61/3𝑑𝑠𝑇5/3sup0𝑠𝑇𝔼||𝑥1𝑠||32/3𝑇0𝔼||||10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑢𝑠𝑏𝑥||||(𝑠)𝑑𝜆6𝑑𝑠1/3𝐶𝜀𝜀2.(3.11) Since 𝑏𝑥 is uniformly bounded, by Lemma 3.1 we get 𝐼𝐼𝜀𝜏𝜏+𝜀𝔼||||10𝑏𝑥𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑣𝑏𝑥(𝑠)𝑑𝜆𝑥1𝑠||||2𝑑𝑠𝑐𝜀2sup0𝑠𝑇𝔼||𝑥1𝑠||2𝑐𝜀4.(3.12) Thus we obtain sup0𝑡𝑇𝔼[𝑡0𝐴𝜀𝑠𝑑𝑠]2𝐶𝜀𝜀2. In the same way we can get sup0𝑡𝑇𝔼𝑡0𝐵𝜀𝑠𝑑𝐵𝑠2𝐶𝜀𝜀2.(3.13) Hence, the estimation (3.4) is proved.
Now we prove (3.5) and (3.6). Set 𝑋𝜀𝑠=𝑥𝜀𝑠𝑥𝑠𝑢,𝜉𝑥1𝑠,𝑌𝜀𝑠=𝑦𝜀𝑠𝑦𝑠𝑢,𝜉𝑦1𝑠,𝑍𝜀𝑠=𝑧𝜀𝑠𝑧𝑠𝑢,𝜉𝑧1𝑠,Π𝜀𝑠=𝑠,𝑥𝑠𝑢,𝜉+𝑥1𝑠+𝜆𝑋𝜀𝑠,𝑦𝑠𝑢,𝜉+𝑦1𝑠+𝜆𝑌𝜀𝑠,𝑧𝑠𝑢,𝜉+𝑧1𝑠+𝜆𝑍𝜀𝑠,𝑢𝜀𝑠,Λ𝜀𝑠=𝑠,𝑥𝑠𝑢,𝜉+𝜆𝑥1𝑠,𝑦𝑠𝑢,𝜉+𝜆𝑦1𝑠,𝑧𝑠𝑢,𝜉+𝜆𝑧1𝑠,𝑢𝜀𝑠.(3.14) It is easy to obtain 𝑌𝜀𝑡𝑥=𝑔𝜀𝑇𝑥𝑔𝑇𝑢,𝜉𝑔𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇𝑇𝑡𝑍𝜀𝑠𝑑𝐵𝑠+𝑇𝑡𝐸𝑠1,𝜀𝑥1𝑠+𝐸𝑠2,𝜀𝑦1𝑠+𝐸𝑠3,𝜀𝑧1𝑠+𝑑𝑠𝑇𝑡𝐹𝑠1,𝜀𝑋𝜀𝑠+𝐹𝑠2,𝜀𝑌𝜀𝑠+𝐹𝑠3,𝜀𝑍𝜀𝑠𝑑𝑠,(3.15) where 𝐸𝑠1,𝜀=10𝑓𝑥Λ𝜀𝑠𝑓𝑥(𝑠)𝑑𝜆,𝐸𝑠2,𝜀=10𝑓𝑦Λ𝜀𝑠𝑓𝑦𝐸(𝑠)𝑑𝜆,𝑠3,𝜀=10𝑓𝑧Λ𝜀𝑠𝑓𝑧(𝑠)𝑑𝜆,𝐹𝑠1,𝜀=10𝑓𝑥Π𝜀𝑠𝐹𝑑𝜆,𝑠2,𝜀=10𝑓𝑦Π𝜀𝑠𝑑𝜆,𝐹𝑠3,𝜀=10𝑓𝑧Π𝜀𝑠𝑑𝜆.(3.16) We have 𝑔𝑥𝜀𝑇𝑥𝑔𝑇𝑢,𝜉𝑔𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇=𝑔𝑥𝜀𝑇𝑥𝑔𝑇𝑢,𝜉+𝑥1𝑇+𝑔𝑥𝑇𝑢,𝜉+𝑥1𝑇𝑥𝑔𝑇𝑢,𝜉𝑔𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇=10𝑔𝑥𝑥𝑇𝑢,𝜉+𝑥1𝑇+𝜆𝑋𝜀𝑇𝑑𝜆𝑋𝜀𝑇+10𝑔𝑥𝑥𝑇𝑢,𝜉+𝜆𝑥1𝑇𝑔𝑥𝑥𝑇𝑢,𝜉𝑑𝜆𝑥1𝑇=𝐼+𝐼𝐼.(3.17) Since 𝑔𝑥 is uniformly bounded, it follows from (3.4) that 𝔼|𝐼|2𝑐𝔼|𝑋𝜀𝑇|2𝐶𝜀𝜀2. Since 𝑔𝑥 is continuous and uniformly bounded, from Lemma 3.1 and the dominated convergence theorem it follows that 𝔼||||𝐼𝐼2sup0𝑡𝑇𝔼||𝑥1𝑡||32/3𝔼||||10𝑔𝑥𝑥𝑇𝑢,𝜉+𝜆𝑥1𝑇𝑔𝑥𝑥𝑇𝑢,𝜉||||𝑑𝜆61/3𝐶𝜀𝜀2.(3.18) Consequently, 𝔼|||𝑔𝑥𝜀𝑇𝑥𝑔𝑇𝑢,𝜉𝑔𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇|||2||𝐼||2𝔼2||||+2𝔼𝐼𝐼2𝐶𝜀𝜀2.(3.19) From Lemma 3.1 and the dominated convergence theorem, it follows that sup0𝑡𝑇𝔼𝑇𝑡𝐸𝑠1,𝜀𝑥1𝑠+𝐸𝑠2,𝜀𝑦1𝑠+𝐸𝑠3,𝜀𝑧1𝑠𝑑𝑠2𝐶𝜀𝜀2.(3.20) Since 𝑓𝑥, 𝑓𝑦, and 𝑓𝑧 are uniformly bounded, we have sup0𝑡𝑇||𝐹𝑠1,𝜀||+||𝐹𝑠2,𝜀||+||𝐹𝑠3,𝜀||𝑐.(3.21) Similar to the proof of Lemma 1 in [9] for the BSDE part, we can obtain (3.5) and (3.6) with the iterative method.

We are now ready to state the variational inequality.

Lemma 3.3. The following variational inequality holds: 𝔼𝜙𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇+𝛾𝑦𝑦0𝑢,𝜉𝑦10+𝜀𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖𝜂𝑖+𝔼𝑇0𝑥(𝑡)𝑥1𝑡+𝑦(𝑡)𝑦1𝑡𝑢+𝜀𝑡(𝑡)𝑑𝑡𝑜(𝜀).(3.22)

Proof. From the optimality of (𝑢(),𝜉()), we have 𝐽(𝑢𝜀(),𝜉𝜀())𝐽(𝑢(),𝜉())0.(3.23) From Lemmas 3.1 and 3.2, it follows that 𝔼𝜙𝑥𝜀𝑇𝑥𝜙𝑇𝑢,𝜉+𝑥1𝑇𝔼𝜙𝑥=𝑜(𝜀),𝑇𝑢,𝜉+𝑥1𝑇𝑥𝜙𝑇𝑢,𝜉𝜙=𝔼𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇+𝑜(𝜀).(3.24) Hence, 𝔼𝜙𝑥𝜀𝑇𝑥𝜙𝑇𝑢,𝜉𝜙=𝔼𝑥𝑥𝑇𝑢,𝜉𝑥1𝑇+𝑜(𝜀).(3.25) Similarly we get 𝔼𝛾𝑦𝜀0𝑦𝛾0𝑢,𝜉𝛾=𝔼𝑦𝑦0𝑢,𝜉𝑦10𝔼+𝑜(𝜀),𝑖1𝑙𝜏𝑖,𝜉𝑖+𝜀𝜂𝑖𝑖1𝑙𝜏𝑖,𝜉𝑖=𝜀𝔼𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖𝜂𝑖+𝑜(𝜀),(3.26) while 𝔼𝑇0𝑡,𝑥𝜀𝑡,𝑦𝜀𝑡,𝑢𝜀𝑡(𝑡)𝑑𝑡=𝔼𝑇0𝑡,𝑥𝜀𝑡,𝑦𝜀𝑡,𝑢𝜀𝑡𝑡,𝑥𝑡𝑢,𝜉+𝑥1𝑡,𝑦𝑡𝑢,𝜉+𝑦1𝑡,𝑢𝜀𝑡𝑑𝑡+𝔼𝑇0𝑡,𝑥𝑡𝑢,𝜉+𝑥1𝑡,𝑦𝑡𝑢,𝜉+𝑦1𝑡,𝑢𝜀𝑡𝑢𝜀𝑡𝑑𝑡+𝔼𝑇0𝑢𝜀𝑡(𝑡)𝑑𝑡=𝐼+𝐼𝐼+𝔼𝑇0𝑢𝜀𝑡.(𝑡)𝑑𝑡(3.27) Since 𝑥,𝑦,𝑧 have linear growth, it follows from Lemma 3.2 and Hölder's inequality that 𝐼=𝔼𝑇010𝑥Π𝜀𝑡𝑋𝜀𝑡+𝑦Π𝜀𝑡𝑌𝜀𝑡𝑑𝜆𝑑𝑡=𝑜(𝜀).(3.28) By Lemma 3.1 and the dominated convergence theorem, we have 𝐼𝐼=𝔼𝑇010𝑥Λ𝜀𝑡𝑥1𝑡+𝑦Λ𝜀𝑡𝑦1𝑡𝑑𝜆𝑑𝑡=𝔼𝑇0𝑥𝑢𝜀𝑡𝑥1𝑡+𝑦𝑢𝜀𝑡𝑦1𝑡𝑑𝑡+𝑜(𝜀)=𝔼𝑇0𝑥𝑢𝜀𝑡𝑥(𝑥𝑡)1𝑡+𝑦𝑢𝜀𝑡𝑦(𝑦𝑡)1𝑡𝑑𝑡+𝔼𝑇0𝑥(𝑡)𝑥1𝑡+𝑦(𝑡)𝑦1𝑡𝑑𝑡+𝑜(𝜀)=𝔼𝜏𝜏+𝜀𝑥(𝑡,𝑣)𝑥𝑥(𝑡)1𝑡+𝑦(𝑡,𝑣)𝑦𝑦(𝑡)1𝑡𝑑𝑡+𝔼𝑇0𝑥(𝑡)𝑥1+𝑦(𝑡)𝑦1𝑑𝑡+𝑜(𝜀),(3.29) where 𝜑(𝑡,𝑣)=𝜑(𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑣), 𝜑=𝑥,𝑦. It follows from Hölder's inequality that 𝔼𝐼𝐼𝜏𝜏+𝜀||𝑥(𝑡,𝑣)𝑥||(𝑡)2𝑑𝑡1/2𝔼𝑇0||𝑥1𝑡||2𝑑𝑡1/2+𝔼𝜏𝜏+𝜀||𝑦(𝑡,𝑣)𝑦||(𝑡)2𝑑𝑡1/2𝔼𝑇0||𝑦1𝑡||2𝑑𝑡1/2+𝔼𝑇0𝑥(𝑡)𝑥1𝑡+𝑦(𝑡)𝑦1𝑡𝑑𝑡+𝑜(𝜀).(3.30) Using Lemma 3.1 again, we get 𝐼𝐼𝔼𝑇0𝑥(𝑡)𝑥1𝑡+𝑦(𝑡)𝑦1𝑡𝑑𝑡+𝑜(𝜀).(3.31) Consequently, 𝔼𝑇0𝑡,𝑥𝜀𝑡,𝑦𝜀𝑡,𝑢𝜀𝑡(𝑡)𝑑𝑡=𝔼𝑇0𝑥(𝑡)𝑥1𝑡+𝑦(𝑡)𝑦1𝑡𝑢+𝜀𝑡(𝑡)𝑑𝑡+𝑜(𝜀).(3.32) The variational inequality follows from (3.25)–(3.32).

Now we introduce the following FBSDE (called the adjoint equation): 𝑑𝑝𝑡=𝑓𝑦(𝑡)𝑝𝑡𝑦(𝑡)𝑑𝑡+𝑓𝑧(𝑡)𝑝𝑡𝑑𝐵𝑡,𝑑𝑞𝑡=𝑓𝑥(𝑡)𝑝𝑡𝑏𝑥(𝑡)𝑞𝑡𝜎𝑥(𝑡)𝑘𝑡𝑥(𝑡)𝑑𝑡+𝑘𝑡𝑑𝐵𝑡,𝑝0=𝛾𝑦𝑦0𝑢,𝜉,𝑞𝑇=𝑔𝑥𝑥𝑇𝑢,𝜉𝑝𝑇+𝜙𝑥𝑥𝑇𝑢,𝜉.(3.33)

It is easy to check that the adjoint equation admits a unique solution (𝑝(),𝑞(),𝑘())𝑆3(𝑚)×𝑆3(𝑛)×𝐻3(𝑚×𝑑).

We are now in a position to state the stochastic maximum principle.

Theorem 3.4. Let (𝑢(),𝜉()) be an optimal control, (𝑥𝑢,𝜉(),𝑦𝑢,𝜉(),𝑧𝑢,𝜉()) the corresponding trajectory, and (𝑝(),𝑞(),𝑘()) the solution of the adjoint equation. Then for any 𝑣𝑈 and 𝜂()𝒦 it holds that 𝐻𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑣,𝑝𝑡,𝑞𝑡,𝑘𝑡𝐻𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑢𝑡,𝑝𝑡,𝑞𝑡,𝑘𝑡𝔼0,a.e.,a.s.,(3.34)𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖+𝑞𝜏𝑖𝐶𝜏𝑖𝑝𝜏𝑖𝐷𝜏𝑖𝜂𝑖𝜉𝑖0,(3.35) where 𝐻[0,𝑇]×𝑛×𝑚×𝑚×𝑑×𝑈×𝑚×𝑛×𝑛×𝑑 is defined by 𝐻(𝑡,𝑥,𝑦,𝑧,𝑣,𝑝,𝑞,𝑘)=𝑝,𝑓(𝑡,𝑥,𝑦,𝑧,𝑣)+𝑞,𝑏(𝑡,𝑥,𝑣)+𝑘,𝜎(𝑡,𝑥)+(𝑡,𝑥,𝑦,𝑣).(3.36)

Proof. Applying Itô's formula to 𝑝𝑡,𝑦1𝑡+𝑞𝑡,𝑥1𝑡, by Lemma 3.3 we derive 𝔼𝑇0𝐻𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑢𝜀𝑡,𝑝𝑡,𝑞𝑡,𝑘𝑡𝐻𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑢𝑡,𝑝𝑡,𝑞𝑡,𝑘𝑡𝑑𝑡+𝜀𝔼𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖+𝑞𝜏𝑖𝐶𝜏𝑖𝑝𝜏𝑖𝐷𝜏𝑖𝜂𝑖𝑜(𝜀),(3.37) where 𝜂() satisfies 𝜉()+𝜂()𝒦. Dividing (3.37) by 𝜀 and letting 𝜀 go to 0, we obtain 𝔼𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑣,𝑝𝜏,𝑞𝜏,𝑘𝜏𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑢𝜏,𝑝𝜏,𝑞𝜏,𝑘𝜏+𝔼𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖+𝑞𝜏𝑖𝐶𝜏𝑖𝑝𝜏𝑖𝐷𝜏𝑖𝜂𝑖[].0,a.e.𝜏0,𝑇(3.38) By choosing 𝑣=𝑢𝜏 in (3.38) we obtain the conclusion (3.35). If we choose 𝜂()0, then for 𝑣𝜏 satisfying 𝔼|𝑣|3< we have 𝔼𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑣,𝑝𝜏,𝑞𝜏,𝑘𝜏𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑢𝜏,𝑝𝜏,𝑞𝜏,𝑘𝜏0.(3.39) Now let us set 𝑣𝜏=𝑣𝟙𝐴+𝑢𝜏𝟙A for any 𝑣𝑈 and 𝐴𝜏. Then it is obvious that 𝑣𝜏𝜏 and 𝔼|𝑣𝜏|3<. So from (3.39) it follows that, for any 𝐴𝜏, 𝔼𝟙𝐴𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑣,𝑝𝜏,𝑞𝜏,𝑘𝜏𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑢𝜏,𝑝𝜏,𝑞𝜏,𝑘𝜏0.(3.40) Hence, 𝔼𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑣,𝑝𝜏,𝑞𝜏,𝑘𝜏𝐻𝜏,𝑥𝜏𝑢,𝜉,𝑦𝜏𝑢,𝜉,𝑧𝜏𝑢,𝜉,𝑢𝜏,𝑝𝜏,𝑞𝜏,𝑘𝜏𝜏0,𝑣𝑈.(3.41) Since the quantity inside the conditional expectation is 𝜏-measurable, the conclusion (3.34) can be obtained easily.

Similar to [22, Corollary 3.1], by Theorem 3.4 we can easily obtain the following

Corollary 3.5. Assume 𝐾=𝑛. Then for the optimal control (𝑢(),𝜉()) it holds that 𝐻𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑣,𝑝𝑡,𝑞𝑡,𝑘𝑡𝐻𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑧𝑡𝑢,𝜉,𝑢𝑡,𝑝𝑡,𝑞𝑡,𝑘𝑡𝑙0,𝑣𝑈,a.e.,a.s.,𝜉𝜏𝑖,𝜉𝑖+𝑞𝜏𝑖𝐶𝜏𝑖𝑝𝜏𝑖𝐷𝜏𝑖=0,𝑖1,a.𝑠..(3.42)

Remark 3.6. We can still obtain the stochastic maximum principle if the assumptions are relaxed in the following way.(i)The regular control process 𝑣() and the impulse control process 𝜂() are assumed to satisfy 𝔼[sup0𝑡𝑇|𝑣𝑡|𝑝]< and 𝔼[𝑖1|𝜂𝑖|𝑝]< for some 𝑝(2,3).(ii)The assumption |(𝑡,0,0,𝑣)|𝑐(1+|𝑣|3) in Hypothesis (H2) can be weakened as |(𝑡,0,0,𝑣)|𝑐(1+|𝑣|𝑝).(iii)In the spike variation setting, the random variable 𝑣 is assumed to satisfy 𝔼|𝑣|𝑝<.
In fact, under these new assumptions both the solutions of the control system (2.4) and the variational equation (3.2) belong to 𝑆𝑝(𝑛)×𝑆𝑝(𝑚)×𝐻𝑝(𝑚×𝑑). The conclusion of Lemma 3.1 becomes sup0𝑡𝑇𝔼||𝑥1𝑡||𝑝+sup0𝑡𝑇𝔼||𝑦1𝑡||𝑝+𝔼𝑇0||𝑧1𝑡||2𝑑𝑡𝑝/2𝑐𝜀𝑝.(3.43) And Lemmas 3.2 and 3.3 still hold true.

4. Sufficient Optimality Conditions for Optimal Controls

We still denote by (𝑥𝑣,𝜂(),𝑦𝑣,𝜂(),𝑧𝑣,𝜂()) the trajectory corresponding to (𝑣(),𝜂())𝒜. Let us first introduce an additional assumption.

(H3) The control domain 𝑈 is a convex body in 𝑘. The maps 𝑏, 𝑓, and are locally Lipschitz in the regular control variable 𝑣.

Theorem 4.1. Let (H1)–(H3) hold. Assume that the functions 𝜙, 𝛾, 𝜂𝑙(𝑡,𝜂) and (𝑥,𝑦,𝑧,𝑣)𝐻(𝑡,𝑥,𝑦,𝑧,𝑣,𝑝,𝑞,𝑘) are convex. Moreover, 𝑦𝑇𝑣,𝜂 has the following particular form: 𝑦𝑇𝑣,𝜂=𝐾𝑥𝑇𝑣,𝜂+𝜁 for 𝐾𝑚×𝑛 and 𝜁𝐿3(Ω,𝑇,;𝑚). Let (𝑝𝑢,𝜉,𝑞𝑢,𝜉,𝑘𝑢,𝜉) be the solution of the adjoint equation associated with (𝑢,𝜉)𝒜. Then (𝑢,𝜉) is an optimal control of the stochastic optimal control problem if it satisfies (3.34) and (3.35).

Proof. Set 𝐽=𝐽(𝑣(),𝜂())𝐽(𝑢(),𝜉()). Since 𝜙, 𝛾, 𝜂𝑙(𝑡,𝜂) are convex, we have 𝜙𝑥𝑇𝑣,𝜂𝑥𝜙𝑇𝑢,𝜉𝜙𝑥𝑥𝑇𝑢,𝜉𝑥𝑇𝑣,𝜂𝑥𝑇𝑢,𝜉,𝛾𝑦0𝑣,𝜂𝑦𝛾0𝑢,𝜉𝛾𝑦𝑦0𝑢,𝜉𝑦0𝑣,𝜂𝑦0𝑢,𝜉,𝑖1𝑙𝜏𝑖,𝜂𝑖𝑖1𝑙𝜏𝑖,𝜉𝑖𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖𝜂𝑖𝜉𝑖.(4.1) Thus, 𝜙𝐽𝔼𝑥𝑥𝑇𝑢,𝜉𝑥𝑇𝑣,𝜂𝑥𝑇𝑢,𝜉+𝛾𝑦𝑦0𝑢,𝜉𝑦0𝑣,𝜂𝑦0𝑢,𝜉+𝔼𝑇0𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑣𝑡𝑡,𝑥𝑡𝑢,𝜉,𝑦𝑡𝑢,𝜉,𝑢𝑡𝑑𝑡+𝔼𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖𝜂𝑖𝜉𝑖.(4.2) Set 𝑣,𝜂(𝑡)=𝐻(𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑧𝑡𝑣,𝜂,𝑣𝑡,𝑝𝑡𝑢,𝜉,𝑞𝑡𝑢,𝜉,𝑘𝑡𝑢,𝜉). Then by Itô's formula applied to 𝑞𝑡𝑢,𝜉,(𝑥𝑡𝑣,𝜂𝑥𝑡𝑢,𝜉)+𝑝𝑡𝑢,𝜉,(𝑦𝑡𝑣,𝜂𝑦𝑡𝑢,𝜉), we get 𝐽Ξ+Θ, where Ξ=𝔼𝑖1𝑙𝜉𝜏𝑖,𝜉𝑖+𝑞𝜏𝑖𝐶𝜏𝑖𝑝𝜏𝑖𝐷𝜏𝑖𝜂𝑖𝜉𝑖,Θ=𝔼𝑇0𝑣,𝜂(𝑡)𝑢,𝜉(𝑡)𝑥𝑢,𝜉𝑥(𝑡)𝑡𝑣,𝜂𝑥𝑡𝑢,𝜉𝑦𝑢,𝜉(𝑦𝑡)𝑡𝑣,𝜂𝑦𝑡𝑢,𝜉𝑧𝑢,𝜉(𝑧𝑡)𝑡𝑣,𝜂𝑧𝑡𝑢,𝜉.𝑑𝑡(4.3) From (3.35) we have Ξ0. By (3.34) and [23, Lemma 2.3-(iii); Chapter 3], we have 0𝜕𝑢𝑢,𝜉(𝑡). By [23, Lemma 2.4; Chapter 3], we further conclude that 𝑥𝑢,𝜉(𝑡),𝑦𝑢,𝜉(𝑡),𝑧𝑢,𝜉(𝑡),0𝜕𝑥,𝑦,𝑧,𝑢𝑢,𝜉(𝑡).(4.4) Then, by [23, Lemma 2.3-(v); Chapter 3] and the convexity of 𝐻(𝑡,.,.,.,.,𝑝,𝑞,𝑘), we obtain 𝑣,𝜂(𝑡)𝑢,𝜉(𝑡)𝑥𝑢,𝜉𝑥(𝑡)𝑡𝑣,𝜂𝑥𝑡𝑢,𝜉+𝑦𝑢,𝜉𝑦(𝑡)𝑡𝑣,𝜂𝑦𝑡𝑢,𝜉+𝑧𝑢,𝜉𝑧(𝑡)𝑡𝑣,𝜂𝑧𝑡𝑢,𝜉,(4.5) from which it follows immediately that Θ0. Thus we obtain 𝐽0 and the proof is complete.

We now give an example of linear quadratic optimal control problem involving impulse controls to illustrate the application of our theoretical results.

Example 4.2. For simplicity, assume that the variables and coefficients are scalar-valued. Let us take 𝑈={1,1} and 𝐾=. There are only two values −1 and 1 in 𝑈 which is a usual case in practice and represents only two control states: “on” and “off”. For (𝑣(),𝜂())𝒜, the controlled system is subject to the following linear FBSDE:𝑑𝑥𝑡=𝐴𝑥𝑡+𝐵𝑣𝑡𝑑𝑡+𝐶𝑥𝑡𝑑𝐵𝑡+𝐻𝑑𝜂𝑡,𝑑𝑦𝑡=𝐷𝑥𝑡+𝐸𝑦𝑡+𝐹𝑧𝑡+𝐺𝑣𝑡𝑑𝑡+𝑧𝑡𝑑𝐵𝑡𝑅𝑑𝜂𝑡,𝑥0=𝑎,𝑦𝑇=𝑔𝑥𝑇,(4.6) and the cost functional is given by1𝐽(𝑣(),𝜂())=2𝔼𝑊𝑥2𝑇+𝛾𝑦20+𝑇0𝑀𝑥2𝑡+𝑁𝑦2𝑡+𝑄𝑣2𝑡𝑑𝑡+𝐿𝑖1𝜂2𝑖.(4.7) The coefficients are deterministic constants such that 𝑊,𝛾,𝑀,𝑁0 and 𝑄,𝐿>0. By Propositions 2.1 and 2.2 we know that the control system admits a unique solution (𝑥(),𝑦(),𝑧())𝑆3()×𝑆3()×𝐻3() for any (𝑣,𝜂)𝒜. And the functional 𝐽 is well defined from 𝒜 into .
Let (𝑢(),𝜉()=𝑖1𝜉𝑖𝟙[𝜏𝑖,𝑇]())𝒜 be an optimal control and (𝑥(),𝑦(),𝑧()) the corresponding trajectory. Then the following adjoint equation 𝑑𝑝𝑡=𝐸𝑝𝑡𝑁𝑦𝑡𝑑𝑡+𝐹𝑝𝑡𝑑𝐵𝑡,𝑑𝑞𝑡=𝐷𝑝𝑡𝐴𝑞𝑡𝐶𝑘𝑡𝑀𝑥𝑡𝑑𝑡+𝑘𝑡𝑑𝐵𝑡,𝑝0=𝛾𝑦0,𝑞𝑇=𝑔𝑝𝑇+𝑊𝑥𝑇(4.8) admits a unique solution (𝑝(),𝑞(),𝑘())𝑆3()×𝑆3()×𝐻3(). The Hamiltonian 𝐻 is given by 1𝐻(𝑡,𝑥,𝑦,𝑧,𝑣,𝑝,𝑞,𝑘)=𝑝(𝐷𝑥+𝐸𝑦+𝐹𝑧+𝐺𝑣)+𝑞(𝐴𝑥+𝐵𝑣)+𝑘𝐶𝑥+2𝑀𝑥2+𝑁𝑦2+𝑄𝑣2.(4.9) Then by Corollary 3.5 we obtain 𝐺𝑝𝑡+𝐵𝑞𝑡1𝑣+2𝑄𝑣2𝐺𝑝𝑡+𝐵𝑞𝑡𝑢𝑡+12𝑄𝑢2𝑡,𝑣𝑈,a.e.,a.s.,(4.10)𝐿𝜉𝑖+𝐻𝑞𝜏𝑖𝑅𝑝𝜏𝑖=0,𝑖1,a.s..(4.11) From (4.10) we get 𝑢𝑡=1,if𝐺𝑝𝑡𝐵𝑞𝑡0,1,otherwise.(4.12) From (4.11) we obtain that 𝜉𝑖=𝐿1𝑅𝑝𝜏𝑖𝐻𝑞𝜏𝑖,𝑖1,a.s..(4.13) Hence, if (𝑢,𝜉)𝒜 is an optimal control of this linear quadratic control problem, then it satisfies (4.12) and (4.13).
We can prove that (𝑢(),𝜉()) obtained in (4.12) and (4.13) is indeed an optimal control of this linear quadratic optimization problem. Note that Theorem 4.1 does not hold now since 𝑈 is not convex in this example. In what follows, we use the same notations as those in the proof of Theorem 4.1. In fact, as in the proof of Theorem 4.1, we can still derive 𝐽(𝑣(),𝜂())𝐽(𝑢(),𝜉())Ξ+Θ. On the one hand, it follows from (4.13) that Ξ=0. On the other hand, we have Θ=𝔼𝑇0𝑣,𝜂(𝑡)𝐻𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑧𝑡𝑣,𝜂,𝑢𝑡,𝑝𝑡𝑢,𝜉,𝑞𝑡𝑢,𝜉,𝑘𝑡𝑢,𝜉+Φ𝑡𝑑𝑡,(4.14) where Φ𝑡=𝐻𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑧𝑡𝑣,𝜂,𝑢𝑡,𝑝𝑡𝑢,𝜉,𝑞𝑡𝑢,𝜉,𝑘𝑡𝑢,𝜉𝑢,𝜉(𝑡)𝑥𝑢,𝜉𝑥(𝑡)𝑡𝑣,𝜂𝑥𝑡𝑢,𝜉𝑦𝑢,𝜉𝑦(𝑡)𝑡𝑣,𝜂𝑦𝑡𝑢,𝜉𝑧𝑢,𝜉𝑧(𝑡)𝑡𝑣,𝜂𝑧𝑡𝑢,𝜉.(4.15) From (4.12) and the definition of 𝐻, it is easy to get 𝑣,𝜂(𝑡)𝐻𝑡,𝑥𝑡𝑣,𝜂,𝑦𝑡𝑣,𝜂,𝑧𝑡𝑣,𝜂,𝑢𝑡,𝑝𝑡𝑢,𝜉,𝑞𝑡𝑢,𝜉,𝑘𝑡𝑢,𝜉=𝐺𝑝𝑡+𝐵𝑞𝑡𝑣𝑡+12𝑄𝑣2𝑡𝐺𝑝𝑡+𝐵𝑞𝑡𝑢𝑡+12𝑄𝑢2𝑡0.(4.16) Since 𝑀, 𝑁0, 𝐻 is convex in (𝑥,𝑦,𝑧), and thus Φ𝑡0, so we obtain Θ0. Consequently, it follows that 𝐽(𝑣(),𝜂())𝐽(𝑢(),𝜉())0 and the optimality of (𝑢(),𝜉()) is proved.

Remark 4.3. For the classical linear quadratic optimal control problem, one can usually obtain an optimal control in a linear state feedback form by virtue of the so-called Riccati equation, and along this line the solvability of the Riccati equation leads to that of the linear quadratic problem. However, it is difficult to obtain a state feedback optimal control in terms of the Riccati equation in Example 4.2 mainly due to the particular form of the regular control domain and the appearance of the impulse control in the control system.

Acknowledgments

The authors would like to thank the referees for valuable suggestions which helped to improve the first version of this paper. Z. Wu acknowledges the financial support from the National Natural Science Foundation of China (10921101 and 61174092) and the Natural Science Fund for Distinguished Young Scholars of China (11125102). F. Zhang acknowledges the financial support from the Natural Science Foundation of Shandong Province, China (ZR2011AQ018), and the Foundation of Doctoral Research Program, Shandong University of Finance and Economics.

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