1.Introduction YanWang, AiminSong, Cheng-DeZheng, andEnminFeng Nonzero-SumStochasticDifferentialGamebetweenControllerandStopperforJumpDiffusions ResearchArticle

(1)

Volume 2013, Article ID 761306,7pages http://dx.doi.org/10.1155/2013/761306

Research Article

Nonzero-Sum Stochastic Differential Game between Controller and Stopper for Jump Diffusions

Yan Wang,

^1,2

Aimin Song,

²

Cheng-De Zheng,

²

and Enmin Feng

¹

1School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China

2School of Science, Dalian Jiaotong University, Dalian 116028, China

Correspondence should be addressed to Yan Wang; [email protected] Received 5 February 2013; Accepted 7 May 2013

Academic Editor: Ryan Loxton

Copyright © 2013 Yan Wang et al. 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.

We consider a nonzero-sum stochastic differential game which involves two players, a controller and a stopper. The controller chooses a control process, and the stopper selects the stopping rule which halts the game. This game is studied in a jump diffusions setting within Markov control limit. By a dynamic programming approach, we give a verification theorem in terms of variational inequality-Hamilton-Jacobi-Bellman (VIHJB) equations for the solutions of the game. Furthermore, we apply the verification theorem to characterize Nash equilibrium of the game in a specific example.

1. Introduction

In this paper we study a nonzero-sum stochastic differential game with two players: a controller and a stopper. The state 𝑋(⋅)in this game evolves according to a stochastic differential equation driven by jump diffusions. The controller affects the control process𝑢(⋅)in the drift and volatility of𝑋(⋅)at time𝑡, and the stopper decides the duration of the game, in the form of a stopping rule𝜏for the process𝑋(⋅). The objectives of the two players are to maximize their own expected payoff.

In order to illustrate the motivation and the background of application for this game, we show a model in finance.

Example 1. Let (Ω,F, {F_𝑡}_𝑡≥0, 𝑃) be a filtered probability space, let𝐵(𝑡)be a𝑘-dimensional Brownian Motion, and let 𝑁(𝑑𝑡, 𝑑𝑧) = ( ̃̃ 𝑁₁(𝑑𝑡, 𝑑𝑧), . . . , ̃𝑁_𝑘(𝑑𝑡, 𝑑𝑧))be𝑘-independent compensated Poisson random measures independent of𝐵(𝑡), 𝑡 ∈ [0, ∞). For𝑖 = 1, . . . , 𝑘,𝑁̃_𝑖(𝑑𝑡, 𝑑𝑧) = 𝑁_𝑖(𝑑𝑡, 𝑑𝑧)−]_𝑖(𝑑𝑧)𝑑𝑡, where ]_𝑖 is the Lévy measure of a Lévy process 𝜂_𝑖(𝑡) with jump measure𝑁_𝑖 such that𝐸[𝜂²_𝑖(𝑡)] < ∞for all𝑡.{F_𝑡}_𝑡≥0 is the filtration generated by 𝐵(𝑡) and 𝑁(𝑑𝑡, 𝑑𝑧)̃ (as usual augmented with all the𝑃-null sets). We refer to [1,2] for more information about Lévy processes.

We firstly define a financial market model as follows.

Suppose that there are two investment possibilities:

(1) a risk-free asset (e.g., a bond), with unit price𝑆₀(𝑡)at time𝑡given by

𝑑𝑆₀(𝑡) = 𝑟 (𝑡) 𝑆₀(𝑡) 𝑑𝑡, 𝑆₀(0) = 1, (1) (2) a risky asset (e.g., a stock), with unit price𝑆₁(𝑡)at time

𝑡given by 𝑑𝑆₁(𝑡)

= 𝑆₁(𝑡−) [𝑏 (𝑡) 𝑑𝑡 + 𝜎 (𝑡) 𝑑𝐵 (𝑡) + ∫

R𝛾 (𝑡, 𝑧) ̃𝑁 (𝑑𝑡, 𝑑𝑧)] , 𝑆₁(0) > 0,

(2) where𝑟(𝑡)isF_𝑡-adapted with∫₀^𝑇|𝑟(𝑡)|𝑑𝑡 < ∞a.s.,𝑇 > 0 is a fixed given constant,𝑏,𝜎,𝛾isF_𝑡-predictable processes satisfying𝛾(𝑡, 𝑧) > −1for a.a.𝑡, 𝑧, a.s. and

∫^𝑇

0 {|𝑏 (𝑡)| + 𝜎²(𝑡) + ∫

R𝛾²(𝑡, 𝑧)](𝑑𝑧)} 𝑑𝑡 < ∞, ∀𝑇 < ∞.

(3)

(2)

Assume that an investor hires a portfolio manager to manage his wealth 𝑋(𝑡) from investments. The manager (controller) can choose a portfolio𝑢(𝑡), which represents the proportion of the total wealth𝑋(𝑡)invested in the stock at time𝑡. And the investor (stopper) can halt the wealth process 𝑋(𝑡)by selecting a stop-rule𝜏 : 𝐶[0, 𝑇] → [0, 𝑇]. Then the dynamics of the corresponding wealth process𝑋(𝑡) = 𝑋^𝑢(𝑡) is

𝑑𝑋^𝑢(𝑡) = 𝑋^𝑢(𝑡−) { [(1 − 𝑢 (𝑡)) 𝑟 (𝑡) + 𝑢 (𝑡) 𝑏 (𝑡)] 𝑑𝑡 + 𝑢 (𝑡) 𝜎 (𝑡) 𝑑𝐵 (𝑡)

+𝑢 (𝑡−) ∫

R𝛾 (𝑡, 𝑧) ̃𝑁 (𝑑𝑡, 𝑑𝑧)} , 𝑋^𝑢(0) = 𝑥 > 0

(4)

(see, e.g., [3–5]). We require that𝑢(𝑡−)𝛾(𝑡, 𝑧) > −1for a.a.𝑡, 𝑧, a.s. and that

∫^𝑇

0 { |(1 − 𝑢 (𝑡)) 𝑟 (𝑡)| + |𝑢 (𝑡) 𝑏 (𝑡)| + 𝑢²(𝑡) 𝜎²(𝑡) +𝑢²(𝑡) ∫

R𝛾²(𝑡, 𝑧)](𝑑𝑧)} 𝑑𝑡 < ∞ a.s.

(5)

At terminal time 𝜏, the stopper gives the controller a payoff𝐶(𝑋(𝜏)), where𝐶 : 𝐶[0, 𝑇] → Ris a deterministic mapping. Therefore, the controller aims to maximize his utility of the following form:

J^𝑥₁(𝑢, 𝜏)=𝐸^𝑥[𝑒^−𝛿𝜏𝑈₁(𝐶 (𝑋 (𝜏)))−∫^𝜏

0 𝑒^−𝛿𝑡ℎ (𝑡, 𝑋 (𝑡) , 𝑢_𝑡) 𝑑𝑡] , (6) where𝛿 > 0is the discounting rate,ℎis a cost function, and 𝑈₁ is the controller’s utility. We denote𝐸^𝑥 the expectation with respect to𝑃^𝑥and𝑃^𝑥the probability laws of𝑋(𝑡)starting at𝑥.

Meanwhile it is the stopper’s objective to choose the stopping time𝜏such that his own utility

J^𝑥₂(𝑢, 𝜏) = 𝐸^𝑥[𝑒^−𝛿𝜏𝑈₂{𝑋 (𝜏) − 𝐶 (𝑋 (𝜏))}] (7) is maximized, where𝑈₂is the stopper’s utility.

As this game is typically a nonzero sum, we seek a Nash equilibrium, namely, a pair(𝑢^∗, 𝜏^∗)such that

J^𝑥₁(𝑢^∗, 𝜏^∗) ≥J^𝑥₁(𝑢, 𝜏^∗) , ∀ 𝑢,

J^𝑥₂(𝑢^∗, 𝜏^∗) ≥J^𝑥₂(𝑢^∗, 𝜏) , ∀ 𝜏. (8) This means that the choice𝑢^∗ is optimal for the controller when the stopper uses𝜏^∗and vice verse.

The game (4) and (8) are a nonzero-sum stochastic differential game between a controller and a stopper. The existence of Nash equilibrium shows that, by an appropriate stopping rule design𝜏^∗, the stopper can induce the controller to choose the best portfolio he can. Similarly, by applying a suitable portfolio𝑢^∗, the controller can force the stopper to stop the employment relationship at a time of the controller’s choosing.

There have been significant advances in the research of stochastic differential games of control and stopping.

For example, in [6–9] the authors considered the zero-sum stochastic differential games of mixed type with both controls and stopping between two players. In these games, each of the players chooses an optimal strategy, which is composed by a control𝑢(⋅)and a stopping𝜏. Under appropriate conditions, they constructed a saddle pair of optimal strategies. For the nonsum case, the games of mixed type were discussed in [6, 10]. The authors presented Nash equilibria, rather than saddle pairs, of strategies [10]. Moreover, the papers [11–17]

considered a zero-sum stochastic differential game between controller and stopper, where one player (controller) chooses a control process 𝑢(⋅) and the other (stopper) chooses a stopping𝜏. One player tries to maximize the reward and the other to minimize it. They presented a saddle pair for the game.

In this paper, we study a nonzero-sum stochastic differential game between a controller and a stopper. The controller and the stopper have different payoffs. The objectives of them are to maximize their own payoffs. This game is considered in a jump diffusion context under the Markov control condition.

We prove a verification theorem in terms of VIHJB equations for the game to characterize Nash equilibrium.

Our setup and approach are related to [3,17]. However, their games are different from ours. In [3], the authors studied the stochastic differential games between two controllers.

The work in [17] was carried out for a zero-sum stochastic differential game between a controller and a stopper.

The paper is organized as follows: in the next section, we formulate the nonzero-sum stochastic differential game between a controller and a stopper and prove a general verification theorem. In Section 3, we apply the general results obtained inSection 2to characterize the solutions of a special game. Finally, we conclude this paper inSection 4.

2. A Verification Theorem for

Nonzero-Sum Stochastic Differential Game between Controller and Stopper

Suppose the state 𝑌(𝑡) = 𝑌^𝑢(𝑡) at time 𝑡 is given by the following stochastic differential equation:

𝑑𝑌 (𝑡) = 𝛼 (𝑌 (𝑡) , 𝑢₀(𝑡)) 𝑑𝑡 + 𝛽 (𝑌 (𝑡) , 𝑢₀(𝑡)) 𝑑𝐵 (𝑡) + ∫R^𝑘₀𝜃 (𝑌 (𝑡−) , 𝑢₁(𝑡−, 𝑧) , 𝑧) ̃𝑁 (𝑑𝑡, 𝑑𝑧) ,

𝑌 (0) = 𝑦 ∈R^𝑘, (9)

where𝛼 : R^𝑘×U → R^𝑘, 𝛽 : R^𝑘×U → R^𝑘×𝑘, 𝜃 : R^𝑘× U×R^𝑘 → R^𝑘×𝑘are given functions, andUdenotes a given subset ofR^𝑝.

We regard 𝑢₀(𝑡) = 𝑢₀(𝑡, 𝜔)and 𝑢₁(𝑡, 𝑧) = 𝑢₁(𝑡, 𝑧, 𝜔) as the control processes, assumed to be c`adl`ag,F_𝑡-adapted and with values in𝑢₀(𝑡) ∈ U,𝑢₁(𝑡, 𝑧) ∈ Ufor a.a.𝑡, 𝑧,𝜔.

And we put𝑢(𝑡) = (𝑢₀(𝑡), 𝑢₁(𝑡, 𝑧)). Then𝑌(𝑡) = 𝑌^(𝑢)(𝑡) is a controlled jump diffusion (see [18] for more information about stochastic control of jump diffusion).

(3)

Fix an open solvency setS⊂R^𝑘. Let

𝜏_S=inf{𝑡 > 0; 𝑌 (𝑡) ∉S} (10) be the bankruptcy time. 𝜏_S is the first time at which the stochastic process 𝑌(𝑡) exits the solvency set S. Similar optimal control problems in which the terminal time is governed by a stopping criterion are considered in [19–21] in the deterministic case.

Let 𝑓_𝑖 : R^𝑘 × 𝐾 → R and𝑔_𝑖 : R^𝑘 → Rbe given functions, for 𝑖 = 1, 2. Let A be a family of admissible controls, contained in the set of𝑢(⋅)such that (9) has a unique strong solution and

𝐸^𝑦[∫^𝜏^S

0 󵄨󵄨󵄨󵄨𝑓^𝑖(𝑌_𝑡, 𝑢_𝑡)󵄨󵄨󵄨󵄨 𝑑𝑡] < ∞, 𝑖 = 1, 2 (11) for all𝑦 ∈S, where𝐸^𝑦denotes expectation given that𝑌(0) = 𝑦 ∈ R^𝑘. Denote byΓthe set of all stopping times𝜏 ≤ 𝜏_S. Moreover, we assume that

the family {𝑔⁻_𝑖 (𝑌_𝜏) ; 𝜏 ∈ Γ} is uniformly integrable, for 𝑖 = 1, 2. (12) Then for𝑢 ∈ Aand𝜏 ∈ Γwe define the performance functionals as follows:

J^𝑦_𝑖 (𝑢, 𝜏) = 𝐸^𝑦[∫^𝜏

0 𝑓_𝑖(𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡 + 𝑔_𝑖(𝑌 (𝜏))] , 𝑖 = 1, 2.

(13)

We interpret𝑔_𝑖(𝑌(𝜏))as0if𝜏 = ∞. We may regardJ^𝑦₁(𝑢, 𝜏) as the payoff to the controller who controls𝑢andJ^𝑦₂(𝑢, 𝜏)as the payoff to the stopper who decides𝜏.

Definition 2(nash equilibrium). A pair(𝑢^∗, 𝜏^∗) ∈ A× Γis called a Nash equilibrium for the stochastic differential game (9) and (13), if the following holds:

J^𝑦₁(𝑢^∗, 𝜏^∗) ≥J^𝑦₁(𝑢, 𝜏^∗) , ∀ 𝑢 ∈U, 𝑦 ∈S, (14) J^𝑦₂(𝑢^∗, 𝜏^∗) ≥J^𝑦₂(𝑢^∗, 𝜏) , ∀ 𝜏 ∈ Γ, 𝑦 ∈S. (15) Condition (14) states that if the stopper chooses𝜏^∗, it is optimal for the controller to use the control 𝑢^∗. Similarly, condition (15) states that if the controller uses𝑢^∗, it is optimal for the stopper to decide𝜏^∗. Thus,(𝑢^∗, 𝜏^∗)is an equilibrium point in the sense that there is no reason for each individual player to deviate from it, as long as the other player does not.

We restrict ourselves to Markov controls; that is, we assume that 𝑢₀(𝑡) = ̃𝑢₀(𝑌(𝑡)), 𝑢₁(𝑡, 𝑧) = ̃𝑢₁(𝑌(𝑡), 𝑧). As customary we do not distinguish between𝑢₀and ̃𝑢₀,𝑢₁and

̃𝑢₁. Then the controls𝑢₀and𝑢₁can simply be identified with functions̃𝑢₀(𝑦)and̃𝑢₁(𝑦, 𝑧), where𝑦 ∈Sand𝑧 ∈R^𝑘.

When the control 𝑢 is Markovian, the corresponding process𝑌^(𝑢)(𝑡)becomes a Markov process with the generator 𝐴^𝑢of𝜙 ∈ 𝐶²(R^𝑘)given by

𝐴𝜙 (𝑦) =∑^𝑘

𝑖=1

𝛼_𝑖(𝑦, 𝑢₀(𝑦))𝜕𝜙

𝜕𝑦_𝑖(𝑦) +1

2

∑𝑘 𝑖,𝑗=1

(𝛽𝛽^𝑇)_𝑖,𝑗(𝑦, 𝑢₀(𝑦)) 𝜕²𝜙

𝜕𝑦_𝑖𝜕𝑦_𝑗 (𝑦) +∑^𝑘

𝑗=1

∫R{𝜙 (𝑦 + 𝜃^𝑗(𝑦, 𝑢₁(𝑦, 𝑧) , 𝑧)) − 𝜙 (𝑦)

−∇𝜙 (𝑦) 𝜃^𝑗(𝑦, 𝑢₁(𝑦, 𝑧) , 𝑧) }](𝑑𝑧) , (16) where∇𝜙 = (𝜕𝜙/𝜕𝑦₁, . . . , 𝜕𝜙/𝜕𝑦_𝑘)is the gradient of𝜙, and𝜃^𝑗 is the𝑗th column of the𝑘 × 𝑘matrix𝜃.

Now we can state the main result of this section.

Theorem 3 (verification theorem for game (9) and (13)).

Suppose there exist two functions𝜙_𝑖 : S → R; 𝑖 = 1, 2such that

(i)𝜙_𝑖∈ 𝐶¹(S⁰) ⋂ 𝐶(S),𝑖 = 1, 2, whereSis the closure of SandS⁰is the interior ofS;

(ii)𝜙_𝑖≥ 𝑔_𝑖onS,𝑖 = 1, 2.

Define the following continuation regions:

𝐷_𝑖= {𝑦 ∈S; 𝜙_𝑖(𝑦) > 𝑔_𝑖(𝑦)} , 𝑖 = 1, 2. (17) Suppose𝑌(𝑡) = 𝑌^𝑢(𝑡)spends0time on𝜕𝐷_𝑖 a.s.,𝑖 = 1, 2, that is,

(iii)𝐸^𝑦[∫₀^𝜏^𝑆𝜒_𝜕𝐷_𝑖(𝑌(𝑡))𝑑𝑡] = 0for all𝑦 ∈S,𝑢 ∈U,𝑖 = 1, 2, (iv)𝜕𝐷_𝑖is a Lipschitz surface,𝑖 = 1, 2,

(v)𝜙_𝑖∈ 𝐶²(S\𝜕𝐷_𝑖). The second-order derivatives of𝜙_𝑖are locally bounded near𝜕𝐷_𝑖, respectively,𝑖 = 1, 2, (vi)𝐷₁= 𝐷₂:= 𝐷,

(vii)there existŝ𝑢 ∈Asuch that, for𝑖 = 1, 2, 𝐴^̂𝑢𝜙_𝑖(𝑦) + 𝑓_𝑖(𝑦, ̂𝑢 (𝑦))

=sup

𝑢∈A{𝐴^𝑢𝜙_𝑖(𝑦) + 𝑓_𝑖(𝑦, 𝑢 (𝑦))} {= 0, 𝑦 ∈ 𝐷,

≤ 0, 𝑦 ∈S\ 𝐷, (18)

(viii)𝐸^𝑦[|𝜙_𝑖(𝑌_𝜏)| + ∫₀^𝜏|𝐴^𝑢𝜙_𝑖(𝑌^𝑢(𝑡))|𝑑𝑡] < ∞;𝑖 = 1, 2, for all 𝑢 ∈U,𝜏 ∈ Γ.

For𝑢 ∈Adefine

𝜏_𝐷= 𝜏_𝐷^𝑢 =inf{𝑡 > 0; 𝑌^𝑢(𝑡) ∉ 𝐷} < ∞, (19) and, in particular,

̂𝜏_𝐷= 𝜏_𝐷^̂𝑢 =inf{𝑡 > 0; 𝑌^̂𝑢(𝑡) ∉ 𝐷} < ∞. (20)

(4)

Suppose that

(ix)the family {𝜙_𝑖(𝑌(𝜏)); 𝜏 ∈ Γ, 𝜏 ≤ 𝜏_𝐷} is uniformly integrable, for all𝑢 ∈Aand𝑦 ∈S,𝑖 = 1, 2.

Then(̂𝜏_𝐷, ̂𝑢) ∈ Γ ×Ais a Nash equilibrium for game(9),(13), and

𝜙₁(𝑦) =sup

𝑢∈AJ^𝑢,̂𝜏₁ ^𝐷(𝑦) =J₁^{̂𝑢,̂𝜏}^𝐷(𝑦) , (21) 𝜙₂(𝑦) =sup

𝜏∈ΓJ₂^̂𝑢,𝜏(𝑦) =J₂^{̂𝑢,̂𝜏}^𝐷(𝑦) . (22) Proof. From (i), (iv), and (v) we may assume by an approx- imation theorem (see Theorem 2.1 in [18]) that𝜙_𝑖 ∈ 𝐶²(S), 𝑖 = 1, 2.

For a given𝑢 ∈A, we define, with𝑌(𝑡) = 𝑌^𝑢(𝑡),

𝜏_𝐷= 𝜏_𝐷^𝑢 =inf{𝑡 > 0; 𝑌^𝑢(𝑡) ∉ 𝐷} . (23) In particular, let̂𝑢be as in (vii). Then,

̂𝜏_𝐷= 𝜏_𝐷^̂𝑢 =inf{𝑡 > 0; 𝑌^̂𝑢(𝑡) ∉ 𝐷} . (24) We first prove that (21) holds. Let̂𝜏_𝐷∈ Γbe as in (24). For arbitrary𝑢 ∈ A, by (vii) and the Dynkin’s formula for jump diffusion (see Theorem 1.24 in [18]) we have

𝜙₁(𝑦) = 𝐸^𝑦[∫^̂𝜏^𝐷^∧𝑚

0 −𝐴^𝑢𝜙₁(𝑌 (𝑡)) 𝑑𝑡 + 𝜙₁(𝑌 (̂𝜏_𝐷∧ 𝑚))]

≥ 𝐸^𝑦[∫^̂𝜏^𝐷^∧𝑚

0 𝑓₁(𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡 + 𝜙₁(𝑌 (̂𝜏_𝐷∧ 𝑚))] , (25) where𝑚 = 1, 2, . . .. Therefore, by (11), (12), (i), (ii), (viii), and the Fatou lemma,

𝜙₁(𝑦)

≥lim inf_{𝑚 → ∞} 𝐸^𝑦[∫^̂𝜏^𝐷^∧𝑚

0 𝑓₁(𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡+𝜙₁(𝑌 (̂𝜏_𝐷∧ 𝑚))]

≥ 𝐸^𝑦[∫^̂𝜏^𝐷

0 𝑓₁(𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡 + 𝑔₁(𝑌 (̂𝜏_𝐷))]

=J^𝑦₁(𝑢, ̂𝜏_𝐷) .

(26) Since this holds for all𝑢 ∈A, we have

𝜙₁(𝑦) ≥sup

𝑢∈AJ^𝑦₁(𝑢, ̂𝜏_𝐷) . (27) In particular, applying the Dynkin’s formula to𝑢 = ̂𝑢we get an equality, that is,

𝜙₁(𝑦) = 𝐸^𝑦[∫^̂𝜏^𝐷^∧𝑚

0 −𝐴^̂𝑢𝜙₁(̂𝑌 (𝑡)) 𝑑𝑡 + 𝜙₁(̂𝑌 (̂𝜏_𝐷∧ 𝑚))]

= 𝐸^𝑦[∫^̂𝜏^𝐷^∧𝑚

0 𝑓₁(̂𝑌 (𝑡) , ̂𝑢 (𝑡)) 𝑑𝑡 + 𝜙₁(̂𝑌 (̂𝜏_𝐷∧ 𝑚))] , (28)

where ̂𝑌(𝑡) = 𝑌^̂𝑢(𝑡)and𝑚 = 1, 2, . . .. Hence we deduce that 𝜙₁(𝑦) =J^𝑦₁(̂𝑢, ̂𝜏_𝐷) . (29) Since we always have

J^𝑦₁(̂𝑢, ̂𝜏_𝐷) ≤sup

𝑢∈AJ^𝑦₁(𝑢, ̂𝜏_𝐷) , (30) we conclude by combining (27), (29), and (30) that

𝜙₁(𝑦) =J^𝑦₁(̂𝑢, ̂𝜏_𝐷) =sup

𝑢∈AJ^𝑦₁(𝑢, ̂𝜏_𝐷) , (31) which is (21).

Next we prove that (22) holds. Let̂𝑢 ∈ Abe as in (vii).

For𝜏 ∈ Γ, by the Dynkin’s formula and (vii), we have 𝐸^𝑦[𝜙₂(̂𝑌 (𝜏_𝑚))] = 𝜙₂(𝑦) + 𝐸^𝑦[∫^𝜏^𝑚

0 𝐴^̂𝑢𝜙₂(̂𝑌 (𝑡)) 𝑑𝑡]

≤ 𝜙₂(𝑦) − 𝐸^𝑦[∫^𝜏^𝑚

0 𝑓₂(̂𝑌 (𝑡) , ̂𝑢 (𝑡)) 𝑑𝑡] , (32) where𝜏_𝑚= 𝜏 ∧ 𝑚;𝑚 = 1, 2, . . ..

Letting𝑚 → ∞gives, by (11), (12), (i), (ii), (viii), and the Fatou Lemma,

𝜙₂(𝑦) ≥lim inf

𝑚 → ∞ 𝐸^𝑦[∫^𝜏∧𝑚

0 𝑓₂(̂𝑌 (𝑡) , ̂𝑢 (𝑡)) 𝑑𝑡 + 𝜙₂(̂𝑌 (𝜏_𝑚))]

≥ 𝐸^𝑦[∫^𝜏

0 𝑓₂(̂𝑌 (𝑡) , ̂𝑢 (𝑡)) 𝑑𝑡 + 𝑔₂(̂𝑌 (𝜏)) 𝜒_{𝜏<∞}]

=J^𝑦₂(̂𝑢, 𝜏) .

(33) The inequality (33) holds for all𝜏 ∈ Γ. Then we have

𝜙₂(𝑦) ≥sup

𝜏∈ΓJ^𝑦₂(̂𝑢, 𝜏) . (34) Similarly, applying the above argument to the pair(̂𝑢, ̂𝜏_𝐷)we get an equality in (34), that is,

𝜙₂(𝑦) =J^𝑦₂(̂𝑢, ̂𝜏_𝐷) . (35) We always have

J^𝑦₂(̂𝑢, ̂𝜏_𝐷) ≤sup

𝜏∈ΓJ^𝑦₂(̂𝑢, 𝜏) . (36) Therefore, combining (34), (35), and (36) we get

𝜙₂(𝑦) =J^𝑦₂(̂𝑢, ̂𝜏_𝐷) =sup

𝜏∈ΓJ^𝑦₂(̂𝑢, 𝜏) , (37) which is (22). The proof is completed.

(5)

3. An Example

In this section we come back toExample 1and useTheorem 3 to study the solutions of game (4) and (8). Here and in the following, all the processes are assumed to be one-dimension for simplicity.

To put game (4) and (8) into the framework ofSection 2, we define the process𝑌(𝑡) = (𝑌₀(𝑡), 𝑌₁(𝑡)); 𝑌(0) = 𝑦 = (𝑠, 𝑦₁) by

𝑑𝑌₀(𝑡) = 𝑑𝑡, 𝑌₀(0) = 𝑠 ∈R, 𝑑𝑌₁(𝑡) = 𝑑𝑋 (𝑡)

= 𝑌₁(𝑡) { [(1 − 𝑢 (𝑡)) 𝑟 (𝑡) + 𝑢 (𝑡) 𝑏 (𝑡)] 𝑑𝑡 + 𝑢 (𝑡) 𝜎 (𝑡) 𝑑𝐵 (𝑡)

+𝑢 (𝑡−) ∫

R𝛾 (𝑡, 𝑧) ̃𝑁 (𝑑𝑡, 𝑑𝑧)} , 𝑌₁(0) = 𝑥 = 𝑦₁.

(38)

Then the performance functionals to the controller (6) and the stopper (7) can be formulated as follows:

J^𝑦₁(𝑢, 𝜏) = 𝐸^𝑠,𝑦¹[𝑒^{−𝛿(𝑠+𝜏)}𝑈₁(𝐶 (𝑌₁(𝜏)))

− ∫^𝜏

0 𝑒^{−𝛿(𝑠+𝑡)}ℎ (𝑌₀(𝑡) , 𝑌₁(𝑡) , 𝑢 (𝑡)) 𝑑𝑡] , J^𝑦₂(𝑢, 𝜏) = 𝐸^𝑠,𝑦¹[𝑒^{−𝛿(𝑠+𝜏)}𝑈₂(𝑌₁(𝜏) − 𝐶 (𝑌₁(𝜏)))] .

(39) In this case the generator𝐴^𝑢in (16) has the form

𝐴^𝑢𝜙 (𝑠, 𝑥)

= 𝜕𝜙

𝜕𝑠 + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥]𝜕𝜙

𝜕𝑥+1

2𝑥²𝑢²𝜎²𝜕²𝜙

𝜕𝑥² + ∫R0

{𝜙 (𝑠, 𝑥 + 𝑥𝑢𝛾 (𝑧)) − 𝜙 (𝑠, 𝑥) − 𝑥𝑢𝛾 (𝑧)𝜕𝜙

𝜕𝑥}

×](𝑑𝑧) .

(40) To obtain a possible Nash equilibrium(̂𝑢, ̂𝜏) ∈A× Γfor game (4) and (8), according toTheorem 3, it is necessary to find a subset𝐷ofS = R²₊ := [0, ∞)²and𝜙_𝑖(𝑠, 𝑥); 𝑖 = 1, 2, such that

(i)𝜙₁(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝑈₁(𝐶(𝑥))and 𝜙₂(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝑈₂(𝑥 − 𝐶(𝑥)), for all(𝑠, 𝑥) ∈ 𝐷;

(ii)𝜙₁(𝑠, 𝑥) ≥ 𝑒^−𝛿𝑠𝑈₁(𝐶(𝑥))and 𝜙₂(𝑠, 𝑥) ≥ 𝑒^−𝛿𝑠𝑈₂(𝑥 − 𝐶(𝑥)), for all(𝑠, 𝑥) ∈S;

(iii)𝐴^𝑢𝜙₁(𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢) ≤ 0and𝐴^𝑢𝜙₂(𝑠, 𝑥) ≤ 0, for all (𝑠, 𝑥) ∈S\ 𝐷and𝑢;

(iv) there exists ̂𝑢 such that 𝐴^̂𝑢𝜙₁(𝑠, 𝑥) − ℎ(𝑠, 𝑥, ̂𝑢) = 𝐴^̂𝑢𝜙₂(𝑠, 𝑥) = 0, for all(𝑠, 𝑥) ∈ 𝐷.

Imposing the first-order condition on 𝐴^𝑢𝜙₁(𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢)and𝐴^𝑢𝜙₂(𝑠, 𝑥), we get the following equations for the optimal control processeŝ𝑢:

(𝑏𝑥 − 𝑟𝑥)𝜕𝜙₁

𝜕𝑥 (𝑠, 𝑥) + 𝑥²𝜎²̂𝑢𝜕²𝜙₁

𝜕𝑥² (𝑠, 𝑥) + ∫R𝑟𝛾 (𝑧) [𝜕𝜙₁

𝜕𝑥 (𝑠, 𝑥 + 𝑟̂𝑢𝛾 (𝑧)) − 𝜕𝜙₁

𝜕𝑥 (𝑠, 𝑥)]

−𝜕ℎ

𝜕𝑢(𝑠, 𝑥, ̂𝑢) = 0, (𝑏𝑥 − 𝑟𝑥)𝜕𝜙₂

𝜕𝑥 (𝑠, 𝑥) + 𝑥²𝜎²̂𝑢𝜕²𝜙₂

𝜕𝑥² (𝑠, 𝑥) + ∫R𝑟𝛾 (𝑧) [𝜕𝜙₂

𝜕𝑥 (𝑠, 𝑥 + 𝑟̂𝑢𝛾 (𝑧)) − 𝜕𝜙₂

𝜕𝑥 (𝑠, 𝑥)] = 0.

(41) Witĥ𝑢as in (41), we put

𝐴^̂𝑢𝜙_𝑖(𝑠, 𝑥)

=𝜕𝜙_𝑖

𝜕𝑠 + [(1 − ̂𝑢) 𝑟𝑥 + ̂𝑢𝑏𝑥]𝜕𝜙_𝑖

𝜕𝑥 +1

2𝑥²̂𝑢²𝜎²𝜕²𝜙_𝑖

𝜕𝑥² + ∫R0

{𝜙_𝑖(𝑠, 𝑥 + 𝑥̂𝑢𝛾 (𝑧)) − 𝜙_𝑖(𝑠, 𝑥) − 𝑥̂𝑢𝛾 (𝑧)𝜕𝜙_𝑖

𝜕𝑥}

×](𝑑𝑧) .

(42) Thus, we may reduce game (4) and (8) to the problem of solving a family of nonlinear variational-integro inequalities.

We summarize as follows.

Theorem 4. Suppose there exist̂𝑢satisfying(41)and two𝐶¹- functions𝜙_𝑖;𝑖 = 1, 2such that

(1)

𝐷 = {(𝑠, 𝑥) : 𝜙₂(𝑠, 𝑥) > 𝑒^−𝛿𝑠𝑈₂(𝑥 − 𝐶 (𝑥))}

= {(𝑠, 𝑥) : 𝜙₁(𝑠, 𝑥) > 𝑒^−𝛿𝑠𝑈₁(𝐶 (𝑥))} ; (43) (2)𝜙_𝑖∈ 𝐶²(𝐷),𝑖 = 1, 2;

(3)𝜙₂(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝑈₂(𝑥 − 𝐶(𝑥)) and 𝜙₁(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝑈₁(𝐶(𝑥))for all(𝑠, 𝑥) ∈S\ 𝐷;

(4)𝐴^𝑢𝜙₁(𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢) ≤ 0and𝐴^𝑢𝜙₂(𝑠, 𝑥) ≤ 0for all (𝑠, 𝑥) ∈S\ 𝐷and for all𝑢 ∈A;

(5)𝐴^̂𝑢𝜙₁(𝑠, 𝑥) − ℎ(𝑠, 𝑥, ̂𝑢) = 0for all(𝑠, 𝑥) ∈ 𝐷, where 𝐴^̂𝑢𝜙₁is given by(42);

(6)𝐴^̂𝑢𝜙₂(𝑠, 𝑥) = 0for all(𝑠, 𝑥) ∈ 𝐷, where𝐴^̂𝑢𝜙₂is given by(42).

Then the pair(̂𝑢, ̂𝜏) is a Nash equilibrium of the stochastic differential game(4)and(8), where

̂𝜏 =inf{𝑡 > 0; 𝑌^̂𝑢(𝑡) ∉ 𝐷} . (44)

(6)

Moreover, the corresponding equilibrium performances are 𝜙₁(𝑠, 𝑥) =J^(𝑠,𝑥)₁ (̂𝑢, ̂𝜏) ,

𝜙₂(𝑠, 𝑥) =J^(𝑠,𝑥)₂ (̂𝑢, ̂𝜏) . (45) In this paper we will not discuss general solutions of this family of nonlinear variational-integro inequalities. Instead we discuss a solution in special case when

𝛾 (𝑡, 𝑧) = 0, ℎ (𝑠, 𝑥, 𝑢) = 𝑢²

2. (46)

Let us try the functions𝜙_𝑖,𝑖 = 1, 2, of the form

𝜙_𝑖(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝜓_𝑖(𝑥) , (47) and a continuation region𝐷of the form

𝐷 = {(𝑠, 𝑥) ; 𝑥 < 𝑥₀} for some𝑥₀> 0. (48) Then we have

𝐴^𝑢𝜙_𝑖(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝐴^𝑢𝜓_𝑖(𝑥) , (49) where

𝐴^𝑢𝜓_𝑖(𝑥) = − 𝛿𝜓_𝑖(𝑥) + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥] 𝜓_𝑖^󸀠(𝑥) +1

2𝑥²𝑢²𝜎²𝜓_𝑖^󸀠󸀠(𝑥) . (50) By conditions (1) and (3) inTheorem 4, we get

𝜓₁(𝑥) = 𝑈₁(𝐶 (𝑥)) , 𝑥 ≥ 𝑥₀, 𝜓₁(𝑥) > 𝑈₁(𝐶 (𝑥)) , 0 < 𝑥 < 𝑥₀, 𝜓₂(𝑥) = 𝑈₂(𝑥 − 𝐶 (𝑥)) , 𝑥 ≥ 𝑥₀, 𝜓₂(𝑥) > 𝑈₂(𝑥 − 𝐶 (𝑥)) , 0 < 𝑥 < 𝑥₀.

(51)

From conditions (4), (5), and (6) ofTheorem 4, we get the candidatê𝑢for the optimal control as follows:

̂𝑢 =Argmax

𝑢∈A {𝐴^𝑢𝜓₁(𝑥) −𝑢² 2 }

=Argmax

𝑢∈A { − 𝛿𝜓₁(𝑥) + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥] 𝜓^󸀠₁(𝑥) +1

2𝑥²𝑢²𝜎²𝜓^󸀠󸀠₁ (𝑥) −𝑢² 2}

= (𝑏𝑥 − 𝑟𝑥) 𝜓₁^󸀠(𝑥) 1 − 𝑥²𝜎²𝜓^󸀠󸀠₁ (𝑥),

(52)

̂𝑢 =Argmax

𝑢∈A {𝐴^𝑢𝜓₂(𝑥)}

=Argmax

𝑢∈A { − 𝛿𝜓₂(𝑥) + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥] 𝜓₂^󸀠(𝑥) +1

2𝑥²𝑢²𝜎²𝜓^󸀠󸀠₂ (𝑥)}

= − (𝑏𝑥 − 𝑟𝑥) 𝜓^󸀠₂(𝑥) 𝑥²𝜎²𝜓₂^󸀠󸀠(𝑥) .

(53)

Let𝜓_𝑖(𝑥) = ̃𝜓_𝑖(𝑥)on0 < 𝑥 < 𝑥₀,𝑖 = 1, 2. By condition (5) inTheorem 4, we have𝐴^̂𝑢̃𝜓₁(𝑥) − ̂𝑢²/2 = 0for0 < 𝑥 < 𝑥₀. Substituting (52) into𝐴^𝑢̃𝜓₁(𝑥) − 𝑢²/2 = 0, we obtain

− 𝛿 ̃𝜓₁(𝑥) + [𝑟𝑥 +(𝑏𝑥 − 𝑟𝑥)²̃𝜓₁^󸀠(𝑥) 1 − 𝑥²𝜎²̃𝜓^󸀠󸀠₁ (𝑥)] ̃𝜓₁^󸀠(𝑥) + [(𝑏𝑥 − 𝑟𝑥) ̃𝜓₁^󸀠(𝑥)]²

2(1 − 𝑥²𝜎²̃𝜓₁^󸀠󸀠(𝑥))²[𝑥²𝜎²̃𝜓^󸀠󸀠₁ (𝑥) − 1] = 0.

(54)

Similarly, we obtain𝐴^̂𝑢̃𝜓₂(𝑥) = 0for0 < 𝑥 < 𝑥₀by condition (6) inTheorem 4. And we substitute (53) in𝐴^𝑢̃𝜓₂(𝑥) = 0to get

− 𝛿 ̃𝜓₂(𝑥) + 𝑟𝑥 ̃𝜓₂^󸀠(𝑥) +[(𝑏𝑥 − 𝑟𝑥) ̃𝜓^󸀠₂(𝑥)]² 1 − 𝑥²𝜎²̃𝜓^󸀠󸀠₂ (𝑥) +[𝑥𝜎 (𝑏𝑥 − 𝑟𝑥) ̃𝜓₂^󸀠(𝑥)]²̃𝜓^󸀠󸀠₂ (𝑥)

2(1 − 𝑥²𝜎²̃𝜓₂^󸀠󸀠(𝑥))² = 0.

(55)

Therefore, we conclude that

𝜓₁(𝑥) = {𝑈₁(𝐶 (𝑥)) , 𝑥 ≥ 𝑥₀,

̃𝜓₁(𝑥) , 0 < 𝑥 < 𝑥₀, (56) 𝜓₂(𝑥) = {𝑈₂(𝑥 − 𝐶 (𝑥)) , 𝑥 ≥ 𝑥₀,

̃𝜓₂(𝑥) , 0 < 𝑥 < 𝑥₀, (57) where ̃𝜓₁(𝑥)and ̃𝜓₂(𝑥)are the solutions of (56) and (57), respectively.

According to Theorem 4, we use the continuity and differentiability of𝜓_𝑖at𝑥 = 𝑥₀to determine𝑥₀,𝑖 = 1, 2, that is,

𝜓₁(𝑥₀) = 𝑈₁(𝐶 (𝑥₀)) , 𝜓₁^󸀠(𝑥₀) = 𝑈₁^󸀠(𝐶 (𝑥₀)) 𝐶^󸀠(𝑥₀) ,

𝜓₂(𝑥₀) = 𝑈₂(𝑥₀− 𝐶 (𝑥₀)) , 𝜓₂^󸀠(𝑥₀) = 𝑈₂^󸀠(𝑥₀− 𝐶 (𝑥₀)) (1 − 𝐶^󸀠(𝑥₀)) .

(58)

At the end of this section, we summarize the above results in the following theorem.

Theorem 5. Let 𝜓_𝑖,𝑖 = 1, 2and let 𝑥₀ be the solutions of equations(56)–(58). Then the pair(̂𝑢, ̂𝜏)given by

̂𝑢 = (𝑏𝑥 − 𝑟𝑥) 𝜓₁^󸀠(𝑥)

1 − 𝑥²𝜎²𝜓₁^󸀠󸀠(𝑥) = − (𝑏𝑥 − 𝑟𝑥) 𝜓₂^󸀠(𝑥) 𝑥²𝜎²𝜓₂^󸀠󸀠(𝑥) ,

̂𝜏 =inf{𝑡 > 0; 𝑥 ≥ 𝑥₀}

(59)

is a Nash equilibrium of game(4)and(8). The corresponding equilibrium performances are

𝜙_𝑖(𝑠, 𝑥) = 𝑒^−𝛿𝑠𝜓_𝑖(𝑥) , 𝑖 = 1, 2. (60)

(7)

4. Conclusion

A verification theorem is obtained for the general stochastic differential game between a controller and a stopper. In the special case of quadratic cost, we use this theorem to characterize the Nash equilibrium. However, the question of the existence and uniqueness of Nash equilibrium for the game remains open. It will be considered in our subsequent work.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant 61273022 and 11171050.

References

[1] J. Bertoin,L´evy Processes, Cambridge University Press, Cam- bridge, UK, 1996.

[2] D. Applebaum, L´evy Processes and Stochastic Calculus, Cam- bridge University Press, Cambridge, UK, 2003.

[3] S. Mataramvura and B. Øksendal, “Risk minimizing portfolios and HJBI equations for stochastic differential games,”Stochas- tics, vol. 80, no. 4, pp. 317–337, 2008.

[4] R. J. Elliott and T. K. Siu, “On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy,”Annals of Operations Research, vol. 176, pp. 271–291, 2010.

[5] G. Wang and Z. Wu, “Mean-variance hedging and forward- backward stochastic differential filtering equations,”Abstract and Applied Analysis, vol. 2011, Article ID 310910, 20 pages, 2011.

[6] I. Karatzas and W. Sudderth, “Stochastic games of control and stopping for a linear diffusion,” inRandom Walk, Sequential Analysis and Related Topics: A Festschrift in Honor of Y.S. Chow, A. Hsiung, Z. L. Ying, and C. H. Zhang, Eds., pp. 100–117, World Scientific Publishers, 2006.

[7] S. Hamad`ene, “Mixed zero-sum stochastic differential game and American game options,”SIAM Journal on Control and Optimization, vol. 45, no. 2, pp. 496–518, 2006.

[8] M. K. Ghosh, M. K. S. Rao, and D. Sheetal, “Differential games of mixed type with control and stopping times,” Nonlinear Differential Equations and Applications, vol. 16, no. 2, pp. 143–

158, 2009.

[9] M. K. Ghosh and K. S. Mallikarjuna Rao, “Existence of value in stochastic differential games of mixed type,”Stochastic Analysis and Applications, vol. 30, no. 5, pp. 895–905, 2012.

[10] I. Karatzas and Q. Li, “BSDE approach to non-zero-sum stochastic differential games of control and stopping,” in Stochastic Processes, Finance and Control, pp. 105–153, World Scientific Publishers, 2012.

[11] J.-P. Lepeltier, “On a general zero-sum stochastic game with stopping strategy for one player and continuous strategy for the other,”Probability and Mathematical Statistics, vol. 6, no. 1, pp.

43–50, 1985.

[12] I. Karatzas and W. D. Sudderth, “The controller-and-stopper game for a linear diffusion,”The Annals of Probability, vol. 29, no. 3, pp. 1111–1127, 2001.

[13] A. Weerasinghe, “A controller and a stopper game with degener- ate variance control,”Electronic Communications in Probability, vol. 11, pp. 89–99, 2006.

[14] I. Karatzas and I.-M. Zamfirescu, “Martingale approach to stochastic differential games of control and stopping,” The Annals of Probability, vol. 36, no. 4, pp. 1495–1527, 2008.

[15] E. Bayraktar, I. Karatzas, and S. Yao, “Optimal stopping for dynamic convex risk measures,”Illinois Journal of Mathematics, vol. 54, no. 3, pp. 1025–1067, 2010.

[16] E. Bayraktar and V. R. Young, “Proving regularity of the minimal probability of ruin via a game of stopping and control,”

Finance and Stochastics, vol. 15, no. 4, pp. 785–818, 2011.

[17] F. Bagherya, S. Haademb, B. Øksendal, and I. Turpina, “Optimal stopping and stochastic control differential games for jump diffusions,”Stochastics, vol. 85, no. 1, pp. 85–97, 2013.

[18] B. Øksendal and A. Sulem,Applied Stochastic Control of Jump Diffusions, Springer, Berlin, Germany, 2nd edition, 2007.

[19] Q. Lin, R. Loxton, K. L. Teo, and Y. H. Wu, “A new computa- tional method for a class of free terminal time optimal control problems,”Pacific Journal of Optimization, vol. 7, no. 1, pp. 63–

81, 2011.

[20] Q. Lin, R. Loxton, K. L. Teo, and Y. H. Wu, “Optimal control computation for nonlinear systems with state-dependent stopping criteria,”Automatica, vol. 48, no. 9, pp. 2116–2129, 2012.

[21] C. Jiang, Q. Lin, C. Yu, K. L. Teo, and G.-R. Duan, “An exact penalty method for free terminal time optimal control problem with continuous inequality constraints,”Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 30–53, 2012.

(8)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}