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

Abstract

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 process 𝑢(⋅) in the drift and volatility of 𝑋(⋅) at time 𝑡, and the stopper decides the duration of the game, in the form of a stopping rule 𝜏 for the process 𝑋(⋅). 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, let 𝐵(𝑡) be a 𝑘-dimensional Brownian Motion, and let 𝑁(𝑑𝑡,𝑑𝑧)=(𝑁₁(𝑑𝑡,𝑑𝑧),…,𝑁_𝑘(𝑑𝑡,𝑑𝑧)) be 𝑘-independent compensated Poisson random measures independent of 𝐵(𝑡), 𝑡∈[0,∞). For 𝑖=1,…,𝑘, 𝑁_𝑖(𝑑𝑡,𝑑𝑧)=𝑁_𝑖(𝑑𝑡,𝑑𝑧)−𝜈_𝑖(𝑑𝑧)𝑑𝑡, where 𝜈_𝑖 is the Lévy measure of a Lévy process 𝜂_𝑖(𝑡) with jump measure 𝑁_𝑖 such that 𝐸[𝜂²_𝑖(𝑡)]<∞ for all 𝑡. {ℱ_𝑡}_𝑡≥0 is the filtration generated by 𝐵(𝑡) and 𝑁(𝑑𝑡,𝑑𝑧) (as usual augmented with all the 𝑃-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 price 𝑆₀(𝑡) at time 𝑡 given by 𝑑𝑆₀(𝑡)=𝑟(𝑡)𝑆₀(𝑡)𝑑𝑡,𝑆₀(0)=1, (1)(2)a risky asset (e.g., a stock), with unit price 𝑆₁(𝑡) at time 𝑡 given by 𝑑𝑆₁(𝑡)=𝑆₁(𝑡−)𝑏(𝑡)𝑑𝑡+𝜎(𝑡)𝑑𝐵(𝑡)+_ℝ𝛾()𝑁(,𝑆𝑡,𝑧𝑑𝑡,𝑑𝑧)₁(0)>0, (2)where 𝑟(𝑡) is ℱ_𝑡-adapted with ∫^𝑇₀|𝑟(𝑡)|𝑑𝑡<∞ a.s., 𝑇>0 is a fixed given constant, 𝑏, 𝜎, 𝛾 is ℱ_𝑡-predictable processes satisfying 𝛾(𝑡,𝑧)>−1 for a.a. 𝑡,𝑧, a.s. and |𝑏^𝑇₀(𝑡)|+𝜎²(𝑡)+_ℝ𝛾²(𝑡,𝑧)𝜈(𝑑𝑧)𝑑𝑡<∞,∀𝑇<∞. (3)
Assume that an investor hires a portfolio manager to manage his wealth 𝑋(𝑡) from investments. The manager (controller) can choose a portfolio 𝑢(𝑡), which represents the proportion of the total wealth 𝑋(𝑡) invested in the stock at time 𝑡. And the investor (stopper) can halt the wealth process 𝑋(𝑡) by selecting a stop-rule 𝜏∶𝐶[0,𝑇]→[0,𝑇]. Then the dynamics of the corresponding wealth process 𝑋(𝑡)=𝑋^𝑢(𝑡) is 𝑑𝑋^𝑢(𝑡)=𝑋^𝑢((𝑡𝑟(𝑡)(𝑡)𝑏(𝑡(𝑡)𝜎(𝑡)(𝑡)()𝑡−)[(1−𝑢))+𝑢)]𝑑𝑡+𝑢𝑑𝐵+𝑢𝑡−_ℝ𝛾()𝑁(,𝑋𝑡,𝑧𝑑𝑡,𝑑𝑧)_𝑢(0)=𝑥>0 (4) (see, e.g., [3–5]). We require that 𝑢(𝑡−)𝛾(𝑡,𝑧)>−1 for a.a. 𝑡,𝑧, a.s. and that |(1−𝑢(^𝑇₀𝑡))𝑟(𝑡)|+|𝑢(𝑡)𝑏(𝑡)|+𝑢²(𝑡)𝜎²(𝑡)+𝑢²(𝑡)_ℝ𝛾²(𝑡,𝑧)𝜈(𝑑𝑧)𝑑𝑡<∞a.s. (5)
At terminal time 𝜏, the stopper gives the controller a payoff 𝐶(𝑋(𝜏)), where 𝐶∶𝐶[0,𝑇]→ℝ is a deterministic mapping. Therefore, the controller aims to maximize his utility of the following form: 𝒥^𝑥₁(𝑢,𝜏)=𝐸^𝑥𝑒^−𝛿𝜏𝑈₁(𝐶(𝑋(𝜏)))−^𝜏₀𝑒^−𝛿𝑡ℎ𝑡,𝑋(𝑡),𝑢_𝑡𝑑𝑡, (6) where 𝛿>0 is the discounting rate, ℎ is a cost function, and 𝑈₁ is the controller’s utility. We denote 𝐸^𝑥 the expectation with respect to 𝑃^𝑥 and 𝑃^𝑥 the probability laws of 𝑋(𝑡) starting at 𝑥.
Meanwhile it is the stopper’s objective to choose the stopping time 𝜏 such that his own utility 𝒥^𝑥₂(𝑢,𝜏)=𝐸^𝑥𝑒^−𝛿𝜏𝑈₂{𝑋(𝜏)−𝐶(𝑋(𝜏))} (7) is maximized, where 𝑈₂ is 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^𝑥₁𝑢^∗,𝜏^∗≥𝒥^𝑥₁𝑢,𝜏^∗𝒥,∀𝑢,_𝑥2𝑢^∗,𝜏^∗≥𝒥^𝑥₂𝑢^∗,𝜏,∀𝜏. (8) This means that the choice 𝑢^∗ is optimal for the controller when the stopper uses 𝜏^∗ 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 design 𝜏^∗, the stopper can induce the controller to choose the best portfolio he can. Similarly, by applying a suitable portfolio 𝑢^∗, 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 control 𝑢(⋅) and a stopping 𝜏. 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 stopping 𝜏. 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 in Section 2 to characterize the solutions of a special game. Finally, we conclude this paper in Section 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)=𝛼𝑌(𝑡),𝑢₀(𝑡)𝑑𝑡+𝛽𝑌(𝑡),𝑢₀(𝑡)(𝑡)+𝑑𝐵_ℝ𝑘0𝜃𝑌(𝑡−),𝑢₁()𝑁(),𝑌(0)𝑡−,𝑧,𝑧𝑑𝑡,𝑑𝑧=𝑦∈ℝ^𝑘, (9) where 𝛼∶ℝ^𝑘×𝔘→ℝ^𝑘, 𝛽∶ℝ^𝑘×𝔘→ℝ^𝑘×𝑘, 𝜃∶ℝ^𝑘×𝔘×ℝ^𝑘→ℝ^𝑘×𝑘 are given functions, and 𝔘 denotes a given subset of ℝ^𝑝.

We regard 𝑢₀(𝑡)=𝑢₀(𝑡,𝜔) and 𝑢₁(𝑡,𝑧)=𝑢₁(𝑡,𝑧,𝜔) as the control processes, assumed to be càdlàg, ℱ_𝑡-adapted and with values in 𝑢₀(𝑡)∈𝔘, 𝑢₁(𝑡,𝑧)∈𝔘 for a.a. 𝑡, 𝑧, 𝜔. And we put 𝑢(𝑡)=(𝑢₀(𝑡),𝑢₁(𝑡,𝑧)). Then 𝑌(𝑡)=𝑌^(𝑢)(𝑡) is a controlled jump diffusion (see [18] for more information about stochastic control of jump diffusion).

Fix an open solvency set 𝒮⊂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 and 𝑔_𝑖∶ℝ^𝑘→ℝ be given functions, for 𝑖=1,2. Let 𝒜 be a family of admissible controls, contained in the set of 𝑢(⋅) such that (9) has a unique strong solution and 𝐸^𝑦𝜏𝒮₀𝑓_𝑖𝑌_𝑡,𝑢_𝑡𝑑𝑡<∞,𝑖=1,2 (11) for all 𝑦∈𝒮, where 𝐸^𝑦 denotes expectation given that 𝑌(0)=𝑦∈ℝ^𝑘. Denote by Γ the set of all stopping times 𝜏≤𝜏_𝒮. Moreover, we assume that thefamily𝑔⁻_𝑖𝑌_𝜏;𝜏∈Γisuniformlyintegrable,for𝑖=1,2. (12)

Then for 𝑢∈𝒜 and 𝜏∈Γ we define the performance functionals as follows: 𝒥^𝑦_𝑖(𝑢,𝜏)=𝐸^𝑦𝜏₀𝑓_𝑖(𝑌(𝑡),𝑢(𝑡))𝑑𝑡+𝑔_𝑖(𝑌(𝜏)),𝑖=1,2. (13) We interpret 𝑔_𝑖(𝑌(𝜏)) as 0 if 𝜏=∞. We may regard 𝒥^𝑦₁(𝑢,𝜏) as the payoff to the controller who controls 𝑢 and 𝒥^𝑦₂(𝑢,𝜏) as the payoff to the stopper who decides 𝜏.

Definition 2 (nash equilibrium). A pair (𝑢^∗,𝜏^∗)∈𝒜×Γ is called a Nash equilibrium for the stochastic differential game (9) and (13), if the following holds: 𝒥^𝑦₁𝑢^∗,𝜏^∗≥𝒥^𝑦₁𝑢,𝜏^∗𝒥,∀𝑢∈𝔘,𝑦∈𝒮,_𝑦2𝑢^∗,𝜏^∗≥𝒥^𝑦₂𝑢^∗,𝜏,∀𝜏∈Γ,𝑦∈𝒮. (14)(15)

Condition (14) states that if the stopper chooses 𝜏^∗, it is optimal for the controller to use the control 𝑢^∗. Similarly, condition (15) states that if the controller uses 𝑢^∗, it is optimal for the stopper to decide 𝜏^∗. 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 between 𝑢₀ and 𝑢₀, 𝑢₁ and 𝑢₁. Then the controls 𝑢₀ and 𝑢₁ can simply be identified with functions 𝑢₀(𝑦) and 𝑢₁(𝑦,𝑧), where 𝑦∈𝒮 and 𝑧∈ℝ^𝑘.

When the control 𝑢 is Markovian, the corresponding process 𝑌^(𝑢)(𝑡) becomes a Markov process with the generator 𝐴^𝑢 of 𝜙∈𝐶²(ℝ^𝑘) given by 𝐴𝜙𝑦=^𝑘_𝑖=1𝛼_𝑖𝑦,𝑢₀𝑦𝜕𝜙𝜕𝑦_𝑖𝑦+12^𝑘_𝑖,𝑗=1𝛽𝛽^𝑇_𝑖,𝑗𝑦,𝑢₀𝑦𝜕²𝜙𝜕𝑦_𝑖𝜕𝑦_𝑗𝑦+^𝑘_𝑗=1ℝ𝜙𝑦+𝜃^𝑗𝑦,𝑢₁𝑦𝑦𝜃𝑦,𝑧,𝑧−𝜙−∇𝜙^𝑗𝑦,𝑢₁𝑦,𝑧,𝑧𝜈(𝑑𝑧), (16) where ∇𝜙=(𝜕𝜙/𝜕𝑦₁,…,𝜕𝜙/𝜕𝑦_𝑘) is the gradient of 𝜙, and 𝜃^𝑗 is the 𝑗th column of the 𝑘×𝑘 matrix 𝜃.

Now we can state the main result of this section.

Theorem 3 (verification theorem for game (9) and (13)). Suppose there exist two functions 𝜙_𝑖∶𝒮→ℝ; 𝑖=1,2 such that(i)𝜙_𝑖∈𝐶¹(𝒮⁰)⋂𝐶(𝒮), 𝑖=1,2, where 𝒮 is the closure of 𝒮 and 𝒮⁰ is the interior of 𝒮; (ii)𝜙_𝑖≥𝑔_𝑖 on 𝒮, 𝑖=1,2. Define the following continuation regions: 𝐷_𝑖=𝑦∈𝒮;𝜙_𝑖𝑦>𝑔_𝑖𝑦,𝑖=1,2. (17) Suppose 𝑌(𝑡)=𝑌^𝑢(𝑡) spends 0 time on 𝜕𝐷_𝑖 a.s., 𝑖=1,2, that is, (iii)𝐸^𝑦[∫^𝜏𝑆₀𝜒_𝜕𝐷𝑖(𝑌(𝑡))𝑑𝑡]=0 for all 𝑦∈𝒮, 𝑢∈𝔘, 𝑖=1,2, (iv)𝜕𝐷_𝑖 is a Lipschitz surface, 𝑖=1,2, (v)𝜙_𝑖∈𝐶²(𝒮⧵𝜕𝐷_𝑖). The second-order derivatives of 𝜙_𝑖 are locally bounded near 𝜕𝐷_𝑖, respectively, 𝑖=1,2, (vi)𝐷₁=𝐷₂∶=𝐷, (vii) there exists 𝑢∈𝒜 such that, for 𝑖=1,2, 𝐴𝑢𝜙_𝑖𝑦+𝑓_𝑖𝑦,𝑢𝑦=sup_𝑢∈𝒜𝐴^𝑢𝜙_𝑖𝑦+𝑓_𝑖𝑦,𝑢𝑦=0,𝑦∈𝐷,≤0,𝑦∈𝒮⧵𝐷, (18)(viii)𝐸^𝑦[|𝜙_𝑖(𝑌_𝜏)|+∫^𝜏₀|𝐴^𝑢𝜙_𝑖(𝑌^𝑢(𝑡))|𝑑𝑡]<∞; 𝑖=1,2, for all 𝑢∈𝔘, 𝜏∈Γ. For 𝑢∈𝒜 define 𝜏_𝐷=𝜏^𝑢_𝐷=inf𝑡>0;𝑌^𝑢(𝑡)∉𝐷<∞, (19) and, in particular, 𝜏_𝐷=𝜏𝑢_𝐷=inf𝑡>0;𝑌𝑢(𝑡)∉𝐷<∞. (20) Suppose that (ix)the family {𝜙_𝑖(𝑌(𝜏));𝜏∈Γ,𝜏≤𝜏_𝐷} is uniformly integrable, for all 𝑢∈𝒜 and 𝑦∈𝒮, 𝑖=1,2. Then (𝜏_𝐷,𝑢)∈Γ×𝒜 is a Nash equilibrium for game (9), (13), and 𝜙₁𝑦=sup_𝑢∈𝒜𝒥^𝑢,_𝐷𝜏₁𝑦𝑢=𝒥^,_𝐷𝜏₁𝑦,𝜙₂𝑦=sup_𝜏∈Γ𝒥𝑢_2,𝜏𝑦𝑢=𝒥^,_𝐷𝜏₂𝑦. (21)(22)

Proof. From (i), (iv), and (v) we may assume by an approximation theorem (see Theorem  2.1 in [18]) that 𝜙_𝑖∈𝐶²(𝒮), 𝑖=1,2.
For a given 𝑢∈𝒜, we define, with 𝑌(𝑡)=𝑌^𝑢(𝑡), 𝜏_𝐷=𝜏^𝑢_𝐷=inf𝑡>0;𝑌^𝑢(𝑡)∉𝐷. (23) In particular, let 𝑢 be as in (vii). Then, 𝜏_𝐷=𝜏𝑢_𝐷=inf𝑡>0;𝑌𝑢(𝑡)∉𝐷. (24)
We first prove that (21) holds. Let 𝜏_𝐷∈Γ be as in (24). For arbitrary 𝑢∈𝒜, by (vii) and the Dynkin’s formula for jump diffusion (see Theorem  1.24 in [18]) we have 𝜙₁𝑦=𝐸^𝑦_𝐷𝜏_0∧𝑚−𝐴^𝑢𝜙₁(𝑌(𝑡))𝑑𝑡+𝜙₁𝑌𝜏_𝐷∧𝑚≥𝐸^𝑦_𝐷𝜏_0∧𝑚𝑓₁(𝑌(𝑡),𝑢(𝑡))𝑑𝑡+𝜙₁𝑌𝜏_𝐷∧𝑚, (25) where 𝑚=1,2,…. Therefore, by (11), (12), (i), (ii), (viii), and the Fatou lemma, 𝜙₁𝑦≥liminf_𝑚→∞𝐸^𝑦_𝐷𝜏_0∧𝑚𝑓₁(𝑌(𝑡),𝑢(𝑡))𝑑𝑡+𝜙₁𝑌𝜏_𝐷∧𝑚≥𝐸^𝑦_𝐷𝜏₀𝑓₁(𝑌(𝑡),𝑢(𝑡))𝑑𝑡+𝑔₁𝑌𝜏_𝐷=𝒥^𝑦₁𝑢,𝜏_𝐷. (26) Since this holds for all 𝑢∈𝒜, we have 𝜙₁𝑦≥sup_𝑢∈𝒜𝒥^𝑦₁𝑢,𝜏_𝐷. (27) In particular, applying the Dynkin’s formula to 𝑢=𝑢 we get an equality, that is, 𝜙₁𝑦=𝐸^𝑦_𝐷𝜏_0∧𝑚−𝐴𝑢𝜙₁𝑌(𝑡)𝑑𝑡+𝜙₁𝑌𝜏_𝐷∧𝑚=𝐸^𝑦_𝐷𝜏_0∧𝑚𝑓₁𝑌(𝑡),𝑢(𝑡)𝑑𝑡+𝜙₁𝑌𝜏_𝐷∧𝑚, (28) where 𝑌(𝑡)=𝑌𝑢(𝑡) and 𝑚=1,2,…. Hence we deduce that 𝜙₁𝑦=𝒥^𝑦₁𝑢,𝜏_𝐷. (29) Since we always have 𝒥^𝑦₁𝑢,𝜏_𝐷≤sup_𝑢∈𝒜𝒥^𝑦₁𝑢,𝜏_𝐷, (30) we conclude by combining (27), (29), and (30) that 𝜙₁𝑦=𝒥^𝑦₁𝑢,𝜏_𝐷=sup_𝑢∈𝒜𝒥^𝑦₁𝑢,𝜏_𝐷, (31) which is (21).
Next we prove that (22) holds. Let 𝑢∈𝒜 be as in (vii). For 𝜏∈Γ, by the Dynkin’s formula and (vii), we have 𝐸^𝑦𝜙₂𝑌𝜏_𝑚=𝜙₂𝑦+𝐸^𝑦𝜏𝑚₀𝐴𝑢𝜙₂𝑌(𝑡)𝑑𝑡≤𝜙₂𝑦−𝐸^𝑦𝜏𝑚₀𝑓₂𝑌(𝑡),𝑢(𝑡)𝑑𝑡, (32) where 𝜏_𝑚=𝜏∧𝑚; 𝑚=1,2,….
Letting 𝑚→∞ gives, by (11), (12), (i), (ii), (viii), and the Fatou Lemma, 𝜙₂𝑦≥liminf_𝑚→∞𝐸^𝑦₀^𝜏∧𝑚𝑓₂𝑌(𝑡),𝑢(𝑡)𝑑𝑡+𝜙₂𝑌𝜏_𝑚≥𝐸^𝑦𝜏₀𝑓₂𝑌(𝑡),𝑢(𝑡)𝑑𝑡+𝑔₂𝑌(𝜏)𝜒_{𝜏<∞}=𝒥^𝑦₂(𝑢,𝜏). (33) The inequality (33) holds for all 𝜏∈Γ. Then we have 𝜙₂𝑦≥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, 𝜙₂𝑦=𝒥^𝑦₂𝑢,𝜏_𝐷. (35) We always have 𝒥^𝑦₂𝑢,𝜏_𝐷≤sup_𝜏∈Γ𝒥^𝑦₂(𝑢,𝜏). (36) Therefore, combining (34), (35), and (36) we get 𝜙₂𝑦=𝒥^𝑦₂𝑢,𝜏_𝐷=sup_𝜏∈Γ𝒥^𝑦₂(𝑢,𝜏), (37) which is (22). The proof is completed.

3. An Example

In this section we come back to Example 1 and use Theorem 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 of Section 2, we define the process 𝑌(𝑡)=(𝑌₀(𝑡),𝑌₁(𝑡)); 𝑌(0)=𝑦=(𝑠,𝑦₁) by 𝑑𝑌₀(𝑡)=𝑑𝑡,𝑌₀(0)=𝑠∈ℝ,𝑑𝑌₁(𝑡)=𝑑𝑋(𝑡)=𝑌₁(𝑡(𝑡𝑟(𝑡)(𝑡)𝑏(𝑡(𝑡)𝜎(𝑡)(𝑡)())[(1−𝑢))+𝑢)]𝑑𝑡+𝑢𝑑𝐵+𝑢𝑡−_ℝ𝛾()𝑁(,𝑌𝑡,𝑧𝑑𝑡,𝑑𝑧)₁(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,𝑦1𝑒^{−𝛿(𝑠+𝜏)}𝑈₁𝐶𝑌₁(𝜏)−^𝜏₀𝑒^{−𝛿(𝑠+𝑡)}ℎ𝑌₀(𝑡),𝑌₁(𝑡)(𝑡),𝒥,𝑢𝑑𝑡_𝑦2(𝑢,𝜏)=𝐸^𝑠,𝑦1𝑒^{−𝛿(𝑠+𝜏)}𝑈₂𝑌₁(𝜏)−𝐶𝑌₁(𝜏). (39) In this case the generator 𝐴^𝑢 in (16) has the form 𝐴^𝑢𝜙()=𝑠,𝑥𝜕𝜙+)]𝜕𝑠[(1−𝑢𝑟𝑥+𝑢𝑏𝑥𝜕𝜙+1𝜕𝑥2𝑥²𝑢²𝜎²𝜕²𝜙𝜕𝑥²+_ℝ0𝜙𝑠,𝑥+𝑥𝑢𝛾(𝑧)−𝜙(𝑠,𝑥)−𝑥𝑢𝛾(𝑧)𝜕𝜙().𝜕𝑥×𝜈𝑑𝑧 (40)

To obtain a possible Nash equilibrium (𝑢,𝜏)∈𝒜×Γ for game (4) and (8), according to Theorem 3, it is necessary to find a subset 𝐷 of 𝒮=ℝ²₊∶=[0,∞)² and 𝜙_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)≤0 and 𝐴^𝑢𝜙₂(𝑠,𝑥)≤0, for all (𝑠,𝑥)∈𝒮⧵𝐷 and 𝑢; (iv)there exists 𝑢 such that 𝐴𝑢𝜙₁(𝑠,𝑥)−ℎ(𝑠,𝑥,𝑢)=𝐴𝑢𝜙₂(𝑠,𝑥)=0, for all (𝑠,𝑥)∈𝐷.

Imposing the first-order condition on 𝐴^𝑢𝜙₁(𝑠,𝑥)−ℎ(𝑠,𝑥,𝑢) and 𝐴^𝑢𝜙₂(𝑠,𝑥), we get the following equations for the optimal control processes 𝑢: (𝑏𝑥−𝑟𝑥)𝜕𝜙₁()𝜕𝑥𝑠,𝑥+𝑥²𝜎²𝑢𝜕²𝜙₁𝜕𝑥²()+𝑠,𝑥_ℝ𝑟𝛾(𝑧)𝜕𝜙₁(𝑧)−𝜕𝑥𝑠,𝑥+𝑟𝑢𝛾𝜕𝜙₁(−𝜕𝑥𝑠,𝑥)𝜕ℎ(𝑢)()𝜕𝑢𝑠,𝑥,=0,𝑏𝑥−𝑟𝑥𝜕𝜙₂()𝜕𝑥𝑠,𝑥+𝑥²𝜎²𝑢𝜕²𝜙₂𝜕𝑥²()+𝑠,𝑥_ℝ𝑟𝛾(𝑧)𝜕𝜙₂(𝑧)−𝜕𝑥𝑠,𝑥+𝑟𝑢𝛾𝜕𝜙₂(𝜕𝑥𝑠,𝑥)=0. (41) With 𝑢 as in (41), we put 𝐴𝑢𝜙_𝑖()=𝑠,𝑥𝜕𝜙_𝑖+𝑢)]𝜕𝑠[(1−𝑟𝑥+𝑢𝑏𝑥𝜕𝜙_𝑖+1𝜕𝑥2𝑥²𝑢²𝜎²𝜕²𝜙_𝑖𝜕𝑥²+_ℝ0𝜙_𝑖𝑠,𝑥+𝑥𝑢𝛾(𝑧)−𝜙_𝑖(𝑠,𝑥)−𝑥𝑢𝛾(𝑧)𝜕𝜙_𝑖().𝜕𝑥×𝜈𝑑𝑧 (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 exist 𝑢 satisfying (41) and two 𝐶¹-functions 𝜙_𝑖; 𝑖=1,2 such that (1)  𝐷=(𝑠,𝑥)∶𝜙₂(𝑠,𝑥)>𝑒^−𝛿𝑠𝑈₂((𝑥=)𝑥−𝐶))(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)≤0 and 𝐴^𝑢𝜙₂(𝑠,𝑥)≤0 for all (𝑠,𝑥)∈𝒮⧵𝐷 and for all 𝑢∈𝒜; (5)𝐴𝑢𝜙₁(𝑠,𝑥)−ℎ(𝑠,𝑥,𝑢)=0 for all (𝑠,𝑥)∈𝐷, where 𝐴𝑢𝜙₁ is given by (42); (6)𝐴𝑢𝜙₂(𝑠,𝑥)=0 for all (𝑠,𝑥)∈𝐷, where 𝐴𝑢𝜙₂ is given by (42). Then the pair (𝑢,𝜏) is a Nash equilibrium of the stochastic differential game (4) and (8), where 𝜏=inf𝑡>0;𝑌𝑢(𝑡)∉𝐷. (44) Moreover, the corresponding equilibrium performances are 𝜙₁(𝑠,𝑥)=𝒥₁^(𝑠,𝑥)(𝜏),𝜙𝑢,₂(𝑠,𝑥)=𝒥₂^(𝑠,𝑥)(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()()=𝑢𝑡,𝑧=0,ℎ𝑠,𝑥,𝑢²2. (46)

Let us try the functions 𝜙_𝑖, 𝑖=1,2, of the form 𝜙_𝑖(𝑠,𝑥)=𝑒^−𝛿𝑠𝜓_𝑖(𝑥), (47) and a continuation region 𝐷 of the form 𝐷=(𝑠,𝑥);𝑥<𝑥₀forsome𝑥₀>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)+12𝑥²𝑢²𝜎²𝜓_𝑖^′′(𝑥). (50)

By conditions (1) and (3) in Theorem 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) of Theorem 4, we get the candidate 𝑢 for the optimal control as follows: 𝑢=Argmax_𝑢∈𝒜𝐴^𝑢𝜓₁(𝑥)−𝑢²2=Argmax_𝑢∈𝒜−𝛿𝜓₁(𝑥)+[(1−𝑢)𝑟𝑥+𝑢𝑏𝑥]𝜓^′₁(𝑥)+12𝑥²𝑢²𝜎²𝜓₁^′′(𝑥)−𝑢²2=(𝑏𝑥−𝑟𝑥)𝜓^′₁(𝑥)1−𝑥²𝜎²𝜓₁^′′(𝑥),𝑢=Argmax_𝑢∈𝒜𝐴^𝑢𝜓₂(𝑥)=Argmax_𝑢∈𝒜−𝛿𝜓₂(𝑥)+[(1−𝑢)𝑟𝑥+𝑢𝑏𝑥]𝜓^′₂(𝑥)+12𝑥²𝑢²𝜎²𝜓₂^′′(𝑥=−()𝜓)𝑏𝑥−𝑟𝑥^′₂(𝑥)𝑥²𝜎²𝜓₂^′′(𝑥). (52)(53)

Let 𝜓_𝑖(𝑥)=𝜓_𝑖(𝑥) on 0<𝑥<𝑥₀, 𝑖=1,2. By condition (5) in Theorem 4, we have 𝐴𝑢𝜓₁(𝑥)−𝑢²/2=0 for 0<𝑥<𝑥₀. Substituting (52) into 𝐴^𝑢𝜓₁(𝑥)−𝑢²/2=0, we obtain −𝛿𝜓₁(𝑥)+()𝑟𝑥+𝑏𝑥−𝑟𝑥²𝜓^′₁(𝑥)1−𝑥²𝜎²𝜓₁^′′(𝑥)𝜓^′₁(𝑥)+(𝑏𝑥−𝑟𝑥)𝜓^′₁(𝑥)²21−𝑥²𝜎²𝜓₁^′′(𝑥)²𝑥²𝜎²𝜓₁^′′(𝑥)−1=0. (54) Similarly, we obtain 𝐴𝑢𝜓₂(𝑥)=0 for 0<𝑥<𝑥₀ by condition (6) in Theorem 4. And we substitute (53) in 𝐴^𝑢𝜓₂(𝑥)=0 to get −𝛿𝜓₂(𝑥)+𝑟𝑥𝜓^′₂(𝑥)+(𝑏𝑥−𝑟𝑥)𝜓^′₂(𝑥)²1−𝑥²𝜎²𝜓₂^′′(𝑥)+𝑥𝜎(𝑏𝑥−𝑟𝑥)𝜓^′₂(𝑥)²𝜓₂^′′(𝑥)21−𝑥²𝜎²𝜓₂^′′(𝑥)²=0. (55)

Therefore, we conclude that 𝜓₁(𝑥)=𝑈₁(𝐶(𝑥)),𝑥≥𝑥₀,𝜓₁(𝑥),0<𝑥<𝑥₀,𝜓₂(𝑥)=𝑈₂(𝑥−𝐶(𝑥)),𝑥≥𝑥₀,𝜓₂(𝑥),0<𝑥<𝑥₀, (56)(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 of 𝜓_𝑖 at 𝑥=𝑥₀ to determine 𝑥₀, 𝑖=1,2, that is, 𝜓₁𝑥₀=𝑈₁𝐶𝑥₀,𝜓^′₁𝑥₀=𝑈^′₁𝐶𝑥₀𝐶^′𝑥₀,𝜓₂𝑥₀=𝑈₂𝑥₀−𝐶𝑥₀,𝜓^′₂𝑥₀=𝑈^′₂𝑥₀−𝐶𝑥₀1−𝐶^′𝑥₀. (58)

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

Theorem 5. Let 𝜓_𝑖, 𝑖=1,2 and let 𝑥₀ be the solutions of equations (56)–(58). Then the pair (𝑢,𝜏) given by ()𝜓𝑢=𝑏𝑥−𝑟𝑥^′₁(𝑥)1−𝑥²𝜎²𝜓₁^′′(𝑥)=−(𝑏𝑥−𝑟𝑥)𝜓^′₂(𝑥)𝑥²𝜎²𝜓₂^′′(𝑥),𝜏=inf𝑡>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)

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.