Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 761306, 7 pages
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 Feng1
1
2
School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China
School of Science, Dalian Jiaotong University, Dalian 116028, China
Correspondence should be addressed to Yan Wang; wymath@163.com
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 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, {F𝑡 }𝑡≥0 , 𝑃) be a filtered probability
space, let 𝐵(𝑡) be a 𝑘-dimensional Brownian Motion, and let
̃
̃1 (𝑑𝑡, 𝑑𝑧), . . . , 𝑁
̃𝑘 (𝑑𝑡, 𝑑𝑧)) be 𝑘-independent
𝑁(𝑑𝑡,
𝑑𝑧) = (𝑁
compensated Poisson random measures independent of 𝐵(𝑡),
̃𝑖 (𝑑𝑡, 𝑑𝑧) = 𝑁𝑖 (𝑑𝑡, 𝑑𝑧)−]𝑖 (𝑑𝑧)𝑑𝑡,
𝑡 ∈ [0, ∞). For 𝑖 = 1, . . . , 𝑘, 𝑁
where ]𝑖 is the Lévy measure of a Lévy process 𝜂𝑖 (𝑡) with
jump measure 𝑁𝑖 such that 𝐸[𝜂𝑖2 (𝑡)] < ∞ for all 𝑡. {F𝑡 }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)
̃ (𝑑𝑡, 𝑑𝑧)] ,
= 𝑆1 (𝑡−) [𝑏 (𝑡) 𝑑𝑡 + 𝜎 (𝑡) 𝑑𝐵 (𝑡) + ∫ 𝛾 (𝑡, 𝑧) 𝑁
R
𝑆1 (0) > 0,
(2)
𝑇
where 𝑟(𝑡) is F𝑡 -adapted with ∫0 |𝑟(𝑡)|𝑑𝑡 < ∞ a.s., 𝑇 > 0
is a fixed given constant, 𝑏, 𝜎, 𝛾 is F𝑡 -predictable processes
satisfying 𝛾(𝑡, 𝑧) > −1 for a.a. 𝑡, 𝑧, a.s. and
𝑇
∫ {|𝑏 (𝑡)| + 𝜎2 (𝑡) + ∫ 𝛾2 (𝑡, 𝑧) ] (𝑑𝑧)} 𝑑𝑡 < ∞,
0
R
∀𝑇 < ∞.
(3)
2
Abstract and Applied Analysis
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) = 𝑋𝑢 (𝑡)
is
𝑑𝑋𝑢 (𝑡) = 𝑋𝑢 (𝑡−) { [(1 − 𝑢 (𝑡)) 𝑟 (𝑡) + 𝑢 (𝑡) 𝑏 (𝑡)] 𝑑𝑡
+ 𝑢 (𝑡) 𝜎 (𝑡) 𝑑𝐵 (𝑡)
(4)
̃ (𝑑𝑡, 𝑑𝑧)} ,
+𝑢 (𝑡−) ∫ 𝛾 (𝑡, 𝑧) 𝑁
R
𝑋𝑢 (0) = 𝑥 > 0
(see, e.g., [3–5]). We require that 𝑢(𝑡−)𝛾(𝑡, 𝑧) > −1 for a.a. 𝑡, 𝑧,
a.s. and that
𝑇
∫ { |(1 − 𝑢 (𝑡)) 𝑟 (𝑡)| + |𝑢 (𝑡) 𝑏 (𝑡)| + 𝑢2 (𝑡) 𝜎2 (𝑡)
0
(5)
2
2
+𝑢 (𝑡) ∫ 𝛾 (𝑡, 𝑧) ] (𝑑𝑧)} 𝑑𝑡 < ∞ a.s.
R
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:
𝜏
J𝑥1 (𝑢, 𝜏) = 𝐸𝑥 [𝑒−𝛿𝜏 𝑈1 (𝐶 (𝑋 (𝜏)))−∫ 𝑒−𝛿𝑡 ℎ (𝑡, 𝑋 (𝑡) , 𝑢𝑡 ) 𝑑𝑡] ,
0
(6)
where 𝛿 > 0 is the discounting rate, ℎ is a cost function, and
𝑈1 is the controller’s utility. We denote 𝐸𝑥 the expectation
with respect to 𝑃𝑥 and 𝑃𝑥 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
J𝑥2 (𝑢, 𝜏) = 𝐸𝑥 [𝑒−𝛿𝜏 𝑈2 {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
J𝑥1 (𝑢∗ , 𝜏∗ ) ≥ J𝑥1 (𝑢, 𝜏∗ ) ,
∀𝑢,
J𝑥2 (𝑢∗ , 𝜏∗ ) ≥ J𝑥2 (𝑢∗ , 𝜏) ,
∀𝜏.
∗
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) = 𝑌𝑢 (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)
̃ (𝑑𝑡, 𝑑𝑧) ,
+ ∫ 𝜃 (𝑌 (𝑡−) , 𝑢1 (𝑡−, 𝑧) , 𝑧) 𝑁
(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.
R𝑘0
(9)
𝑌 (0) = 𝑦 ∈ R𝑘 ,
where 𝛼 : R𝑘 × U → R𝑘 , 𝛽 : R𝑘 × U → R𝑘×𝑘 , 𝜃 : R𝑘 ×
U × R𝑘 → R𝑘×𝑘 are given functions, and U denotes a given
subset of R𝑝 .
We regard 𝑢0 (𝑡) = 𝑢0 (𝑡, 𝜔) and 𝑢1 (𝑡, 𝑧) = 𝑢1 (𝑡, 𝑧, 𝜔)
as the control processes, assumed to be càdlàg, F𝑡 -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).
Abstract and Applied Analysis
3
Fix an open solvency set S ⊂ R𝑘 . Let
𝜏S = inf {𝑡 > 0; 𝑌 (𝑡) ∉ S}
(10)
be the bankruptcy time. 𝜏S 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 𝑓𝑖 : R𝑘 × 𝐾 → R and 𝑔𝑖 : R𝑘 → 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
𝜏S
󵄨
󵄨
𝐸 [∫ 󵄨󵄨󵄨𝑓𝑖 (𝑌𝑡 , 𝑢𝑡 )󵄨󵄨󵄨 𝑑𝑡] < ∞,
0
𝑦
𝑖 = 1, 2
the family
(𝑌𝜏 ) ; 𝜏 ∈ Γ} is uniformly integrable,
for 𝑖 = 1, 2.
(12)
Then for 𝑢 ∈ A and 𝜏 ∈ Γ we define the performance
functionals as follows:
𝑦
J𝑖
𝑘
𝐴𝜙 (𝑦) = ∑𝛼𝑖 (𝑦, 𝑢0 (𝑦))
𝑖=1
+
𝜕𝜙
(𝑦)
𝜕𝑦𝑖
𝜕2 𝜙
1 𝑘
(𝑦)
∑ (𝛽𝛽𝑇 )𝑖,𝑗 (𝑦, 𝑢0 (𝑦))
2 𝑖,𝑗=1
𝜕𝑦𝑖 𝜕𝑦𝑗
𝑘
+ ∑ ∫ {𝜙 (𝑦 + 𝜃𝑗 (𝑦, 𝑢1 (𝑦, 𝑧) , 𝑧)) − 𝜙 (𝑦)
𝑗=1 R
−∇𝜙 (𝑦) 𝜃𝑗 (𝑦, 𝑢1 (𝑦, 𝑧) , 𝑧) } ] (𝑑𝑧) ,
(16)
(11)
for all 𝑦 ∈ S, where 𝐸𝑦 denotes expectation given that 𝑌(0) =
𝑦 ∈ R𝑘 . Denote by Γ the set of all stopping times 𝜏 ≤ 𝜏S .
Moreover, we assume that
{𝑔𝑖−
When the control 𝑢 is Markovian, the corresponding
process 𝑌(𝑢) (𝑡) becomes a Markov process with the generator
𝐴𝑢 of 𝜙 ∈ 𝐶2 (R𝑘 ) given by
where ∇𝜙 = (𝜕𝜙/𝜕𝑦1 , . . . , 𝜕𝜙/𝜕𝑦𝑘 ) is the gradient of 𝜙, and 𝜃𝑗
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 𝜙𝑖 : S → R; 𝑖 = 1, 2 such
that
(i) 𝜙𝑖 ∈ 𝐶1 (S0 ) ⋂ 𝐶(S), 𝑖 = 1, 2, where S is the closure of
S and S0 is the interior of S;
(ii) 𝜙𝑖 ≥ 𝑔𝑖 on S, 𝑖 = 1, 2.
Define the following continuation regions:
𝜏
(𝑢, 𝜏) = 𝐸𝑦 [∫ 𝑓𝑖 (𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡 + 𝑔𝑖 (𝑌 (𝜏))] ,
0
(13)
𝑦
We interpret 𝑔𝑖 (𝑌(𝜏)) as 0 if 𝜏 = ∞. We may regard J1 (𝑢, 𝜏)
𝑦
as the payoff to the controller who controls 𝑢 and J2 (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:
𝑦
𝑦
∀𝑢 ∈ U, 𝑦 ∈ S,
(14)
𝑦
𝑦
∀𝜏 ∈ Γ, 𝑦 ∈ S.
(15)
J1 (𝑢∗ , 𝜏∗ ) ≥ J1 (𝑢, 𝜏∗ ) ,
J2 (𝑢∗ , 𝜏∗ ) ≥ J2 (𝑢∗ , 𝜏) ,
𝑖 = 1, 2.
(17)
Suppose 𝑌(𝑡) = 𝑌𝑢 (𝑡) spends 0 time on 𝜕𝐷𝑖 a.s., 𝑖 =
1, 2, that is,
𝑖 = 1, 2.
∗
𝐷𝑖 = {𝑦 ∈ S; 𝜙𝑖 (𝑦) > 𝑔𝑖 (𝑦)} ,
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) = 𝑢̃0 (𝑌(𝑡)), 𝑢1 (𝑡, 𝑧) = 𝑢̃1 (𝑌(𝑡), 𝑧). As
customary we do not distinguish between 𝑢0 and 𝑢̃0 , 𝑢1 and
𝑢̃1 . Then the controls 𝑢0 and 𝑢1 can simply be identified with
functions 𝑢̃0 (𝑦) and 𝑢̃1 (𝑦, 𝑧), where 𝑦 ∈ S and 𝑧 ∈ R𝑘 .
𝜏
(iii) 𝐸𝑦 [∫0 𝑆 𝜒𝜕𝐷𝑖 (𝑌(𝑡))𝑑𝑡] = 0 for all 𝑦 ∈ S, 𝑢 ∈ U, 𝑖 = 1, 2,
(iv) 𝜕𝐷𝑖 is a Lipschitz surface, 𝑖 = 1, 2,
(v) 𝜙𝑖 ∈ 𝐶2 (S\𝜕𝐷𝑖 ). The second-order derivatives of 𝜙𝑖 are
locally bounded near 𝜕𝐷𝑖 , respectively, 𝑖 = 1, 2,
(vi) 𝐷1 = 𝐷2 := 𝐷,
(vii) there exists 𝑢̂ ∈ A such that, for 𝑖 = 1, 2,
𝐴𝑢̂ 𝜙𝑖 (𝑦) + 𝑓𝑖 (𝑦, 𝑢̂ (𝑦))
= sup {𝐴𝑢 𝜙𝑖 (𝑦) + 𝑓𝑖 (𝑦, 𝑢 (𝑦))} {
𝑢∈A
(18)
= 0, 𝑦 ∈ 𝐷,
≤ 0, 𝑦 ∈ S \ 𝐷,
𝜏
(viii) 𝐸𝑦 [|𝜙𝑖 (𝑌𝜏 )| + ∫0 |𝐴𝑢 𝜙𝑖 (𝑌𝑢 (𝑡))|𝑑𝑡] < ∞; 𝑖 = 1, 2, for all
𝑢 ∈ U, 𝜏 ∈ Γ.
For 𝑢 ∈ A define
𝑢
= inf {𝑡 > 0; 𝑌𝑢 (𝑡) ∉ 𝐷} < ∞,
𝜏𝐷 = 𝜏𝐷
(19)
and, in particular,
𝑢̂
= inf {𝑡 > 0; 𝑌𝑢̂ (𝑡) ∉ 𝐷} < ∞.
𝜏̂𝐷 = 𝜏𝐷
(20)
4
Abstract and Applied Analysis
̂ = 𝑌𝑢̂ (𝑡) and 𝑚 = 1, 2, . . .. Hence we deduce that
where 𝑌(𝑡)
Suppose that
(ix) the family {𝜙𝑖 (𝑌(𝜏)); 𝜏 ∈ Γ, 𝜏 ≤ 𝜏𝐷} is uniformly
integrable, for all 𝑢 ∈ A and 𝑦 ∈ S, 𝑖 = 1, 2.
Then (̂
𝜏𝐷, 𝑢̂) ∈ Γ × A is a Nash equilibrium for game (9), (13),
and
𝑢,̂
𝜏𝐷
𝜙1 (𝑦) = supJ1
𝑢∈A
𝑢̂,̂
𝜏𝐷
(𝑦) = J1
𝑢̂,̂
𝜏𝐷
𝜙2 (𝑦) = supJ𝑢2̂,𝜏 (𝑦) = J2
𝜏∈Γ
(21)
(𝑦) .
(22)
Proof. From (i), (iv), and (v) we may assume by an approximation theorem (see Theorem 2.1 in [18]) that 𝜙𝑖 ∈ 𝐶2 (S),
𝑖 = 1, 2.
For a given 𝑢 ∈ A, we define, with 𝑌(𝑡) = 𝑌𝑢 (𝑡),
(23)
In particular, let 𝑢̂ be as in (vii). Then,
𝑢̂
= inf {𝑡 > 0; 𝑌𝑢̂ (𝑡) ∉ 𝐷} .
𝜏̂𝐷 = 𝜏𝐷
𝜏̂𝐷 ∧𝑚
0
≥ 𝐸𝑦 [∫
𝜏̂𝐷 ∧𝑚
0
𝑢∈A
we conclude by combining (27), (29), and (30) that
𝑦
𝑢∈A
𝑓1 (𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡 + 𝜙1 (𝑌 (̂
𝜏𝐷 ∧ 𝑚))] ,
0
𝜏𝑚
̂ (𝑡) , 𝑢̂ (𝑡)) 𝑑𝑡] ,
≤ 𝜙2 (𝑦) − 𝐸𝑦 [∫ 𝑓2 (𝑌
0
0
𝑓1 (𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡+𝜙1 (𝑌 (̂
𝜏𝐷 ∧ 𝑚))]
𝜙2 (𝑦) ≥ lim inf 𝐸𝑦 [∫
𝑚→∞
0
0
𝑦
J2
𝑢, 𝜏) .
(̂
(33)
The inequality (33) holds for all 𝜏 ∈ Γ. Then we have
𝑦
𝑢, 𝜏) .
𝜙2 (𝑦) ≥ supJ2 (̂
𝜏∈Γ
0
𝑦
(26)
(𝑢, 𝜏̂𝐷) .
𝑦
𝑢, 𝜏̂𝐷) .
𝜙2 (𝑦) = J2 (̂
(27)
In particular, applying the Dynkin’s formula to 𝑢 = 𝑢̂ we get
an equality, that is,
𝜙1 (𝑦) = 𝐸𝑦 [∫
𝜏̂𝐷 ∧𝑚
0
𝑦
= 𝐸 [∫
𝜏̂𝐷 ∧𝑚
0
̂ (𝑡)) 𝑑𝑡 + 𝜙1 (𝑌
̂ (̂
−𝐴𝑢̂ 𝜙1 (𝑌
𝜏𝐷 ∧ 𝑚))]
(28)
(35)
We always have
𝑦
𝑦
J2 (̂
𝑢, 𝜏̂𝐷) ≤ supJ2 (̂
𝑢, 𝜏) .
𝜏∈Γ
(36)
Therefore, combining (34), (35), and (36) we get
𝑦
̂ (𝑡) , 𝑢̂ (𝑡)) 𝑑𝑡 + 𝜙1 (𝑌
̂ (̂
𝑓1 (𝑌
𝜏𝐷 ∧ 𝑚))] ,
(34)
Similarly, applying the above argument to the pair (̂
𝑢, 𝜏̂𝐷) we
get an equality in (34), that is,
Since this holds for all 𝑢 ∈ A, we have
𝜙1 (𝑦) ≥
̂ (𝑡) , 𝑢̂ (𝑡)) 𝑑𝑡 + 𝜙2 (𝑌
̂ (𝜏𝑚 ))]
𝑓2 (𝑌
𝜏
𝜏̂𝐷
𝑦
supJ1
𝑢∈A
𝜏∧𝑚
̂ (𝑡) , 𝑢̂ (𝑡)) 𝑑𝑡 + 𝑔2 (𝑌
̂ (𝜏)) 𝜒{𝜏<∞} ]
≥ 𝐸𝑦 [∫ 𝑓2 (𝑌
≥ 𝐸𝑦 [∫ 𝑓1 (𝑌 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡 + 𝑔1 (𝑌 (̂
𝜏𝐷))]
= J1 (𝑢, 𝜏̂𝐷) .
(32)
where 𝜏𝑚 = 𝜏 ∧ 𝑚; 𝑚 = 1, 2, . . ..
Letting 𝑚 → ∞ gives, by (11), (12), (i), (ii), (viii), and the
Fatou Lemma,
=
𝜙1 (𝑦)
𝑚→∞
(31)
which is (21).
Next we prove that (22) holds. Let 𝑢̂ ∈ A be as in (vii).
For 𝜏 ∈ Γ, by the Dynkin’s formula and (vii), we have
(24)
where 𝑚 = 1, 2, . . .. Therefore, by (11), (12), (i), (ii), (viii), and
the Fatou lemma,
≥ lim inf 𝐸 [∫
𝑦
𝑢, 𝜏̂𝐷) = supJ1 (𝑢, 𝜏̂𝐷) ,
𝜙1 (𝑦) = J1 (̂
𝜏𝑚
−𝐴𝑢 𝜙1 (𝑌 (𝑡)) 𝑑𝑡 + 𝜙1 (𝑌 (̂
𝜏𝐷 ∧ 𝑚))]
𝜏̂𝐷 ∧𝑚
(30)
̂ (𝜏𝑚 ))] = 𝜙2 (𝑦) + 𝐸𝑦 [∫ 𝐴𝑢̂ 𝜙2 (𝑌
̂ (𝑡)) 𝑑𝑡]
𝐸𝑦 [𝜙2 (𝑌
(25)
𝑦
𝑦
𝑢, 𝜏̂𝐷) ≤ supJ1 (𝑢, 𝜏̂𝐷) ,
J1 (̂
We first prove that (21) holds. Let 𝜏̂𝐷 ∈ 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
𝜙1 (𝑦) = 𝐸𝑦 [∫
(29)
Since we always have
𝑦
(𝑦) ,
𝑢
𝜏𝐷 = 𝜏𝐷
= inf {𝑡 > 0; 𝑌𝑢 (𝑡) ∉ 𝐷} .
𝑦
𝜙1 (𝑦) = J1 (̂
𝑢, 𝜏̂𝐷) .
𝑦
𝑢, 𝜏̂𝐷) = supJ2 (̂
𝑢, 𝜏) ,
𝜙2 (𝑦) = J2 (̂
𝜏∈Γ
which is (22). The proof is completed.
(37)
Abstract and Applied Analysis
5
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,
Imposing the first-order condition on 𝐴𝑢 𝜙1 (𝑠, 𝑥) −
ℎ(𝑠, 𝑥, 𝑢) and 𝐴𝑢 𝜙2 (𝑠, 𝑥), we get the following equations for
the optimal control processes 𝑢̂:
(𝑏𝑥 − 𝑟𝑥)
+ ∫ 𝑟𝛾 (𝑧) [
R
𝑌0 (0) = 𝑠 ∈ R,
−
𝑑𝑌1 (𝑡) = 𝑑𝑋 (𝑡)
(𝑏𝑥 − 𝑟𝑥)
= 𝑌1 (𝑡) { [(1 − 𝑢 (𝑡)) 𝑟 (𝑡) + 𝑢 (𝑡) 𝑏 (𝑡)] 𝑑𝑡
+ 𝑢 (𝑡) 𝜎 (𝑡) 𝑑𝐵 (𝑡)
(38)
𝑌1 (0) = 𝑥 = 𝑦1 .
+ ∫ 𝑟𝛾 (𝑧) [
=
𝑦
J1 (𝑢, 𝜏) = 𝐸𝑠,𝑦1 [𝑒−𝛿(𝑠+𝜏) 𝑈1 (𝐶 (𝑌1 (𝜏)))
R0
− ∫ 𝑒−𝛿(𝑠+𝑡) ℎ (𝑌0 (𝑡) , 𝑌1 (𝑡) , 𝑢 (𝑡)) 𝑑𝑡] ,
𝑈2 (𝑌1 (𝜏) − 𝐶 (𝑌1 (𝜏)))] .
(39)
In this case the generator 𝐴𝑢 in (16) has the form
(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 𝑢̂ satisfying (41) and two 𝐶1 functions 𝜙𝑖 ; 𝑖 = 1, 2 such that
𝐴𝑢 𝜙 (𝑠, 𝑥)
=
𝜕𝜙𝑖
}
𝜕𝑥
× ] (𝑑𝑧) .
0
[𝑒
𝜕𝜙𝑖
𝜕2 𝜙
𝜕𝜙 1
+ [(1 − 𝑢̂) 𝑟𝑥 + 𝑢̂𝑏𝑥] 𝑖 + 𝑥2 𝑢̂2 𝜎2 2𝑖
𝜕𝑠
𝜕𝑥 2
𝜕𝑥
𝑢𝛾 (𝑧)
+ ∫ {𝜙𝑖 (𝑠, 𝑥 + 𝑥̂
𝑢𝛾 (𝑧)) − 𝜙𝑖 (𝑠, 𝑥) − 𝑥̂
𝜏
(𝑢, 𝜏) = 𝐸
𝜕𝜙2
𝜕𝜙
(𝑠, 𝑥 + 𝑟̂
𝑢𝛾 (𝑧)) − 2 (𝑠, 𝑥)] = 0.
𝜕𝑥
𝜕𝑥
(41)
𝐴𝑢̂ 𝜙𝑖 (𝑠, 𝑥)
Then the performance functionals to the controller (6) and
the stopper (7) can be formulated as follows:
−𝛿(𝑠+𝜏)
𝜕2 𝜙
𝜕𝜙2
(𝑠, 𝑥) + 𝑥2 𝜎2 𝑢̂ 22 (𝑠, 𝑥)
𝜕𝑥
𝜕𝑥
With 𝑢̂ as in (41), we put
R
𝑠,𝑦1
𝜕𝜙1
𝜕𝜙
(𝑠, 𝑥 + 𝑟̂
𝑢𝛾 (𝑧)) − 1 (𝑠, 𝑥)]
𝜕𝑥
𝜕𝑥
𝜕ℎ
(𝑠, 𝑥, 𝑢̂) = 0,
𝜕𝑢
R
̃ (𝑑𝑡, 𝑑𝑧)} ,
+𝑢 (𝑡−) ∫ 𝛾 (𝑡, 𝑧) 𝑁
𝑦
J2
𝜕2 𝜙
𝜕𝜙1
(𝑠, 𝑥) + 𝑥2 𝜎2 𝑢̂ 21 (𝑠, 𝑥)
𝜕𝑥
𝜕𝑥
𝜕𝜙
𝜕𝜙 1 2 2 2 𝜕2 𝜙
+ [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥]
+ 𝑥𝑢𝜎
𝜕𝑠
𝜕𝑥 2
𝜕𝑥2
(1)
𝜕𝜙
+ ∫ {𝜙 (𝑠, 𝑥 + 𝑥𝑢𝛾 (𝑧)) − 𝜙 (𝑠, 𝑥) − 𝑥𝑢𝛾 (𝑧) }
𝜕𝑥
R0
× ] (𝑑𝑧) .
(40)
𝐷 = {(𝑠, 𝑥) : 𝜙2 (𝑠, 𝑥) > 𝑒−𝛿𝑠 𝑈2 (𝑥 − 𝐶 (𝑥))}
= {(𝑠, 𝑥) : 𝜙1 (𝑠, 𝑥) > 𝑒−𝛿𝑠 𝑈1 (𝐶 (𝑥))} ;
(43)
(2) 𝜙𝑖 ∈ 𝐶2 (𝐷), 𝑖 = 1, 2;
(3) 𝜙2 (𝑠, 𝑥) = 𝑒−𝛿𝑠 𝑈2 (𝑥 − 𝐶(𝑥)) and 𝜙1 (𝑠, 𝑥)
𝑒−𝛿𝑠 𝑈1 (𝐶(𝑥)) for all (𝑠, 𝑥) ∈ S \ 𝐷;
=
To obtain a possible Nash equilibrium (̂
𝑢, 𝜏̂) ∈ 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 𝜙𝑖 (𝑠, 𝑥); 𝑖 = 1, 2,
such that
(4) 𝐴𝑢 𝜙1 (𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢) ≤ 0 and 𝐴𝑢 𝜙2 (𝑠, 𝑥) ≤ 0 for all
(𝑠, 𝑥) ∈ S \ 𝐷 and for all 𝑢 ∈ A;
(i) 𝜙1 (𝑠, 𝑥) = 𝑒−𝛿𝑠 𝑈1 (𝐶(𝑥)) and 𝜙2 (𝑠, 𝑥) = 𝑒−𝛿𝑠 𝑈2 (𝑥 −
𝐶(𝑥)), for all (𝑠, 𝑥) ∈ 𝐷;
(5) 𝐴𝑢̂ 𝜙1 (𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢̂) = 0 for all (𝑠, 𝑥) ∈ 𝐷, where
𝐴𝑢̂ 𝜙1 is given by (42);
(ii) 𝜙1 (𝑠, 𝑥) ≥ 𝑒−𝛿𝑠 𝑈1 (𝐶(𝑥)) and 𝜙2 (𝑠, 𝑥) ≥ 𝑒−𝛿𝑠 𝑈2 (𝑥 −
𝐶(𝑥)), for all (𝑠, 𝑥) ∈ S;
(6) 𝐴𝑢̂ 𝜙2 (𝑠, 𝑥) = 0 for all (𝑠, 𝑥) ∈ 𝐷, where 𝐴𝑢̂ 𝜙2 is given
by (42).
(iii) 𝐴𝑢 𝜙1 (𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢) ≤ 0 and 𝐴𝑢 𝜙2 (𝑠, 𝑥) ≤ 0, for all
(𝑠, 𝑥) ∈ S \ 𝐷 and 𝑢;
(iv) there exists 𝑢̂ such that 𝐴𝑢̂ 𝜙1 (𝑠, 𝑥) − ℎ(𝑠, 𝑥, 𝑢̂) =
𝐴𝑢̂ 𝜙2 (𝑠, 𝑥) = 0, for all (𝑠, 𝑥) ∈ 𝐷.
Then the pair (̂
𝑢, 𝜏̂) is a Nash equilibrium of the stochastic
differential game (4) and (8), where
𝜏̂ = inf {𝑡 > 0; 𝑌𝑢̂ (𝑡) ∉ 𝐷} .
(44)
6
Abstract and Applied Analysis
Moreover, the corresponding equilibrium performances are
𝜙1 (𝑠, 𝑥) =
J(𝑠,𝑥)
1
𝑢, 𝜏̂) ,
(̂
𝜙2 (𝑠, 𝑥) = J(𝑠,𝑥)
𝑢, 𝜏̂) .
(̂
2
(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
𝑢2
.
2
Let us try the functions 𝜙𝑖 , 𝑖 = 1, 2, of the form
𝛾 (𝑡, 𝑧) = 0,
ℎ (𝑠, 𝑥, 𝑢) =
𝜙𝑖 (𝑠, 𝑥) = 𝑒−𝛿𝑠 𝜓𝑖 (𝑥) ,
(46)
(47)
Let 𝜓𝑖 (𝑥) = 𝜓̃𝑖 (𝑥) on 0 < 𝑥 < 𝑥0 , 𝑖 = 1, 2. By condition (5)
in Theorem 4, we have 𝐴𝑢̂ 𝜓̃1 (𝑥) − 𝑢̂2 /2 = 0 for 0 < 𝑥 < 𝑥0 .
Substituting (52) into 𝐴𝑢 𝜓̃1 (𝑥) − 𝑢2 /2 = 0, we obtain
− 𝛿𝜓̃1 (𝑥) + [𝑟𝑥 +
+
(𝑏𝑥 − 𝑟𝑥)2 𝜓̃1󸀠 (𝑥)
] 𝜓̃1󸀠 (𝑥)
1 − 𝑥2 𝜎2 𝜓̃1󸀠󸀠 (𝑥)
[(𝑏𝑥 − 𝑟𝑥) 𝜓̃1󸀠 (𝑥)]
2
2(1 − 𝑥2 𝜎2 𝜓̃1󸀠󸀠 (𝑥))
for some 𝑥0 > 0.
(48)
− 𝛿𝜓̃2 (𝑥) + 𝑟𝑥𝜓̃2󸀠 (𝑥) +
Then we have
𝐴𝑢 𝜙𝑖 (𝑠, 𝑥) = 𝑒−𝛿𝑠 𝐴𝑢 𝜓𝑖 (𝑥) ,
+
where
1
+ 𝑥2 𝑢2 𝜎2 𝜓𝑖󸀠󸀠 (𝑥) .
2
By conditions (1) and (3) in Theorem 4, we get
𝜓1 (𝑥) = 𝑈1 (𝐶 (𝑥)) ,
𝜓1 (𝑥) > 𝑈1 (𝐶 (𝑥)) ,
0 < 𝑥 < 𝑥0 ,
𝑥 ≥ 𝑥0 ,
0 < 𝑥 < 𝑥0 .
From conditions (4), (5), and (6) of Theorem 4, we get the
candidate 𝑢̂ for the optimal control as follows:
𝑢̂ = Argmax {𝐴𝑢 𝜓1 (𝑥) −
𝑢∈A
=
(56)
𝑈2 (𝑥 − 𝐶 (𝑥)) , 𝑥 ≥ 𝑥0 ,
𝜓̃2 (𝑥) ,
0 < 𝑥 < 𝑥0 ,
(57)
𝜓2 (𝑥0 ) = 𝑈2 (𝑥0 − 𝐶 (𝑥0 )) ,
(52)
(58)
𝜓2󸀠 (𝑥0 ) = 𝑈2󸀠 (𝑥0 − 𝐶 (𝑥0 )) (1 − 𝐶󸀠 (𝑥0 )) .
At the end of this section, we summarize the above results
in the following theorem.
Theorem 5. Let 𝜓𝑖 , 𝑖 = 1, 2 and let 𝑥0 be the solutions of
equations (56)–(58). Then the pair (̂
𝑢, 𝜏̂) given by
𝑢∈A
𝑢̂ =
= Argmax { − 𝛿𝜓2 (𝑥) + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥] 𝜓2󸀠 (𝑥)
𝑢∈A
− (𝑏𝑥 − 𝑟𝑥) 𝜓2󸀠 (𝑥)
.
𝑥2 𝜎2 𝜓2󸀠󸀠 (𝑥)
𝑈1 (𝐶 (𝑥)) , 𝑥 ≥ 𝑥0 ,
0 < 𝑥 < 𝑥0 ,
𝜓̃1 (𝑥) ,
𝜓1󸀠 (𝑥0 ) = 𝑈1󸀠 (𝐶 (𝑥0 )) 𝐶󸀠 (𝑥0 ) ,
𝑢̂ = Argmax {𝐴𝑢 𝜓2 (𝑥)}
=
= 0.
𝜓1 (𝑥0 ) = 𝑈1 (𝐶 (𝑥0 )) ,
(𝑏𝑥 − 𝑟𝑥) 𝜓1󸀠 (𝑥)
,
1 − 𝑥2 𝜎2 𝜓1󸀠󸀠 (𝑥)
1
+ 𝑥2 𝑢2 𝜎2 𝜓2󸀠󸀠 (𝑥)}
2
2
(55)
where 𝜓̃1 (𝑥) and 𝜓̃2 (𝑥) are the solutions of (56) and (57),
respectively.
According to Theorem 4, we use the continuity and
differentiability of 𝜓𝑖 at 𝑥 = 𝑥0 to determine 𝑥0 , 𝑖 = 1, 2, that
is,
= Argmax { − 𝛿𝜓1 (𝑥) + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥] 𝜓1󸀠 (𝑥)
1
𝑢2
+ 𝑥2 𝑢2 𝜎2 𝜓1󸀠󸀠 (𝑥) − }
2
2
2
Therefore, we conclude that
𝑢2
}
2
𝑢∈A
[(𝑏𝑥 − 𝑟𝑥) 𝜓̃2󸀠 (𝑥)]
1 − 𝑥2 𝜎2 𝜓̃2󸀠󸀠 (𝑥)
2(1 − 𝑥2 𝜎2 𝜓̃2󸀠󸀠 (𝑥))
𝜓2 (𝑥) = {
(51)
(𝑥) − 1] = 0.
[𝑥𝜎 (𝑏𝑥 − 𝑟𝑥) 𝜓̃2󸀠 (𝑥)] 𝜓̃2󸀠󸀠 (𝑥)
𝜓1 (𝑥) = {
𝑥 ≥ 𝑥0 ,
𝜓2 (𝑥) = 𝑈2 (𝑥 − 𝐶 (𝑥)) ,
𝜓2 (𝑥) > 𝑈2 (𝑥 − 𝐶 (𝑥)) ,
(50)
[𝑥 𝜎
𝜓̃1󸀠󸀠
2
(49)
𝐴𝑢 𝜓𝑖 (𝑥) = − 𝛿𝜓𝑖 (𝑥) + [(1 − 𝑢) 𝑟𝑥 + 𝑢𝑏𝑥] 𝜓𝑖󸀠 (𝑥)
2 2
Similarly, we obtain 𝐴𝑢̂ 𝜓̃2 (𝑥) = 0 for 0 < 𝑥 < 𝑥0 by condition
(6) in Theorem 4. And we substitute (53) in 𝐴𝑢 𝜓̃2 (𝑥) = 0 to
get
and a continuation region 𝐷 of the form
𝐷 = {(𝑠, 𝑥) ; 𝑥 < 𝑥0 }
(54)
2
(53)
(𝑏𝑥 − 𝑟𝑥) 𝜓1󸀠 (𝑥) − (𝑏𝑥 − 𝑟𝑥) 𝜓2󸀠 (𝑥)
=
,
1 − 𝑥2 𝜎2 𝜓1󸀠󸀠 (𝑥)
𝑥2 𝜎2 𝜓2󸀠󸀠 (𝑥)
(59)
𝜏̂ = inf {𝑡 > 0; 𝑥 ≥ 𝑥0 }
is a Nash equilibrium of game (4) and (8). The corresponding
equilibrium performances are
𝜙𝑖 (𝑠, 𝑥) = 𝑒−𝛿𝑠 𝜓𝑖 (𝑥) ,
𝑖 = 1, 2.
(60)
Abstract and Applied Analysis
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.
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