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

Authors: Yan Wang; Aimin Song; Cheng-De Zheng; Enmin Feng
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

In this paper we study a nonzero-sum stochastic differential game with two players: a controller and a stopper. The stateX(·) 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 N~(dt,dz)=(N~1(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)=Ni(dt,dz)-νi(dz)dt, where νi 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 priceS0(t) at time t given by
(1)dS0(t)=r(t)S0(t)dt,S0(0)=1,(2)
a risky asset (e.g., a stock), with unit priceS1(t) at time t given by
(2)dS1(t)=S1(t-)[b(t)dt+σ(t)dB(t)+∫ℝ‍γ(t,z)N~(dt,dz)],S1(0)>0,where r(t) is ℱt-adapted with ∫0T‍|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
(3)∫0T‍{|b(t)|+σ2(t)+∫ℝ‍γ2(t,z)ν(dz)}dt<∞,∀T<∞.
Assume that an investor hires a portfolio manager to manage his wealthX(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
(4)dXu(t)=Xu(t-){∫ℝ‍γ(t,z)N~(dt,dz)[(1-u(t))r(t)+u(t)b(t)]dt+u(t)σ(t)dB(t)+u(t-)∫ℝ‍γ(t,z)N~(dt,dz)},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
(5)∫0T‍{∫ℝ‍γ2(t,z)ν(dz)|(1-u(t))r(t)|+|u(t)b(t)|+u2(t)σ2(t)+u2(t)∫ℝ‍γ2(t,z)ν(dz)}dt<∞a.s.
At terminal timeτ, 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:
(6)𝒥1x(u,τ)=Ex[e-δτU1(C(X(τ)))-∫0τ‍e-δth(t,X(t),ut)dt],
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 timeτ such that his own utility
(7)𝒥2x(u,τ)=Ex[e-δτU2{X(τ)-C(X(τ))}]
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
(8)𝒥1x(u*,τ*)≥𝒥1x(u,τ*),∀u,𝒥2x(u*,τ*)≥𝒥2x(u*,τ),∀τ.
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 Section3, 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 stateY(t)=Yu(t) at time t is given by the following stochastic differential equation:
(9)dY(t)=α(Y(t),u0(t))dt+β(Y(t),u0(t))dB(t)+∫ℝ0k‍θ(Y(t-),u1(t-,z),z)N~(dt,dz),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 regardu0(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).Fix an open solvency set𝒮⊂ℝk. Let
(10)τ𝒮=inf{t>0;Y(t)∉𝒮}
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.Letfi: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
(11)Ey[∫0τ𝒮‍|fi(Yt,ut)|dt]<∞,i=1,2
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
(12)thefamily{gi-(Yτ);τ∈Γ}isuniformlyintegrable,fori=1,2.Then foru∈𝒜 and τ∈Γ we define the performance functionals as follows:
(13)𝒥iy(u,τ)=Ey[∫0τ‍fi(Y(t),u(t))dt+gi(Y(τ))],i=1,2.
We interpret gi(Y(τ)) as 0 if τ=∞. We may regard 𝒥1y(u,τ) as the payoff to the controller who controls u and 𝒥2y(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:
(14)𝒥1y(u*,τ*)≥𝒥1y(u,τ*),∀u∈𝔘,y∈𝒮,(15)𝒥2y(u*,τ*)≥𝒥2y(u*,τ),∀τ∈Γ,y∈𝒮.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 thatu0(t)=u~0(Y(t)), u1(t,z)=u~1(Y(t),z). As customary we do not distinguish between u0 and u~0, u1 and u~1. Then the controls u0 and u1 can simply be identified with functions u~0(y) and u~1(y,z), where y∈𝒮 and z∈ℝk.When the controlu is Markovian, the corresponding process Y(u)(t) becomes a Markov process with the generator Au of ϕ∈C2(ℝk) given by
(16)Aϕ(y)=∑i=1k‍αi(y,u0(y))∂ϕ∂yi(y)+12∑i,j=1k‍(ββT)i,j(y,u0(y))∂2ϕ∂yi∂yj(y)+∑j=1k‍∫ℝ‍{ϕ(y+θj(y,u1(y,z),z))-ϕ(y)-∇ϕ(y)θj(y,u1(y,z),z)(y+θj(y,u1(y,z),z))}ν(dz),
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 functionsϕi: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:(17)Di={y∈𝒮;ϕi(y)>gi(y)},i=1,2.SupposeY(t)=Yu(t) spends 0 time on ∂Di a.s., i=1,2, that is,(iii)
Ey[∫0τS‍χ∂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 existsu^∈𝒜 such that, for i=1,2,
(18)Au^ϕi(y)+fi(y,u^(y))=supu∈𝒜{Auϕi(y)+fi(y,u(y))}{=0,y∈D,≤0,y∈𝒮∖D,(viii)
Ey[|ϕi(Yτ)|+∫0τ‍|Auϕi(Yu(t))|dt]<∞; i=1,2, for all u∈𝔘, τ∈Γ.Foru∈𝒜 define
(19)τD=τDu=inf{t>0;Yu(t)∉D}<∞,and, in particular,(20)τ^D=τDu^=inf{t>0;Yu^(t)∉D}<∞.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 (τ^D,u^)∈Γ×𝒜 is a Nash equilibrium for game (9), (13), and
(21)ϕ1(y)=supu∈𝒜𝒥1u,τ^D(y)=𝒥1u^,τ^D(y),(22)ϕ2(y)=supτ∈Γ𝒥2u^,τ(y)=𝒥2u^,τ^D(y).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 givenu∈𝒜, we define, with Y(t)=Yu(t),
(23)τD=τDu=inf{t>0;Yu(t)∉D}.
In particular, let u^ be as in (vii). Then,
(24)τ^D=τDu^=inf{t>0;Yu^(t)∉D}.
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
(25)ϕ1(y)=Ey[∫0τ^D∧m‍-Auϕ1(Y(t))dt+ϕ1(Y(τ^D∧m))]≥Ey[∫0τ^D∧m‍f1(Y(t),u(t))dt+ϕ1(Y(τ^D∧m))],
where m=1,2,…. Therefore, by (11), (12), (i), (ii), (viii), and the Fatou lemma,
(26)ϕ1(y)≥liminfm→∞Ey[∫0τ^D∧m‍f1(Y(t),u(t))dt+ϕ1(Y(τ^D∧m))]≥Ey[∫0τ^D‍f1(Y(t),u(t))dt+g1(Y(τ^D))]=𝒥1y(u,τ^D).
Since this holds for all u∈𝒜, we have
(27)ϕ1(y)≥supu∈𝒜𝒥1y(u,τ^D).
In particular, applying the Dynkin’s formula to u=u^ we get an equality, that is,
(28)ϕ1(y)=Ey[∫0τ^D∧m‍-Au^ϕ1(Y^(t))dt+ϕ1(Y^(τ^D∧m))]=Ey[∫0τ^D∧m‍f1(Y^(t),u^(t))dt+ϕ1(Y^(τ^D∧m))],
where Y^(t)=Yu^(t) and m=1,2,…. Hence we deduce that
(29)ϕ1(y)=𝒥1y(u^,τ^D).
Since we always have
(30)𝒥1y(u^,τ^D)≤supu∈𝒜𝒥1y(u,τ^D),
we conclude by combining (27), (29), and (30) that
(31)ϕ1(y)=𝒥1y(u^,τ^D)=supu∈𝒜𝒥1y(u,τ^D),
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
(32)Ey[ϕ2(Y^(τm))]=ϕ2(y)+Ey[∫0τm‍Au^ϕ2(Y^(t))dt]≤ϕ2(y)-Ey[∫0τm‍f2(Y^(t),u^(t))dt],
where τm=τ∧m; m=1,2,….
Lettingm→∞ gives, by (11), (12), (i), (ii), (viii), and the Fatou Lemma,
(33)ϕ2(y)≥liminfm→∞Ey[∫0τ∧m‍f2(Y^(t),u^(t))dt+ϕ2(Y^(τm))]≥Ey[∫0τ‍f2(Y^(t),u^(t))dt+g2(Y^(τ))χ{τ<∞}]=𝒥2y(u^,τ).
The inequality (33) holds for all τ∈Γ. Then we have
(34)ϕ2(y)≥supτ∈Γ𝒥2y(u^,τ).
Similarly, applying the above argument to the pair (u^,τ^D) we get an equality in (34), that is,
(35)ϕ2(y)=𝒥2y(u^,τ^D).
We always have
(36)𝒥2y(u^,τ^D)≤supτ∈Γ𝒥2y(u^,τ).
Therefore, combining (34), (35), and (36) we get
(37)ϕ2(y)=𝒥2y(u^,τ^D)=supτ∈Γ𝒥2y(u^,τ),
which is (22). The proof is completed.


## 3. An Example

In this section we come back to Example1 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
(38)dY0(t)=dt,Y0(0)=s∈ℝ,dY1(t)=dX(t)=Y1(t){∫ℝ‍[(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.
Then the performance functionals to the controller (6) and the stopper (7) can be formulated as follows:
(39)𝒥1y(u,τ)=Es,y1[∫0τ‍e-δ(s+τ)U1(C(Y1(τ)))-∫0τ‍e-δ(s+t)h(Y0(t),Y1(t),u(t))dt],𝒥2y(u,τ)=Es,y1[e-δ(s+τ)U2(Y1(τ)-C(Y1(τ)))].
In this case the generator Au in (16) has the form
(40)Auϕ(s,x)=∂ϕ∂s+[(1-u)rx+ubx]∂ϕ∂x+12x2u2σ2∂2ϕ∂x2+∫ℝ0‍{ϕ(s,x+xuγ(z))-ϕ(s,x)-xuγ(z)∂ϕ∂x}×ν(dz).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=ℝ+2:=[0,∞)2 and ϕi(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)
Auϕ1(s,x)-h(s,x,u)≤0 and Auϕ2(s,x)≤0, for all (s,x)∈𝒮∖D and u;(iv)
there existsu^ such that Au^ϕ1(s,x)-h(s,x,u^)=Au^ϕ2(s,x)=0, for all (s,x)∈D.Imposing the first-order condition onAuϕ1(s,x)-h(s,x,u) and Auϕ2(s,x), we get the following equations for the optimal control processes u^:
(41)(bx-rx)∂ϕ1∂x(s,x)+x2σ2u^∂2ϕ1∂x2(s,x)+∫ℝ‍rγ(z)[∂ϕ1∂x(s,x+ru^γ(z))-∂ϕ1∂x(s,x)]-∂h∂u(s,x,u^)=0,(bx-rx)∂ϕ2∂x(s,x)+x2σ2u^∂2ϕ2∂x2(s,x)+∫ℝ‍rγ(z)[∂ϕ2∂x(s,x+ru^γ(z))-∂ϕ2∂x(s,x)]=0.
With u^ as in (41), we put
(42)Au^ϕi(s,x)=∂ϕi∂s+[(1-u^)rx+u^bx]∂ϕi∂x+12x2u^2σ2∂2ϕi∂x2+∫ℝ0‍{ϕi(s,x+xu^γ(z))-ϕi(s,x)-xu^γ(z)∂ϕi∂x}×ν(dz).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 existu^ satisfying (41) and two C1-functions ϕi; i=1,2 such that (1)
(43)D={(s,x):ϕ2(s,x)>e-δsU2(x-C(x))}={(s,x):ϕ1(s,x)>e-δsU1(C(x))};(2)
ϕi∈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)
Auϕ1(s,x)-h(s,x,u)≤0 and Auϕ2(s,x)≤0 for all (s,x)∈𝒮∖D and for all u∈𝒜;(5)
Au^ϕ1(s,x)-h(s,x,u^)=0 for all (s,x)∈D, where Au^ϕ1 is given by (42);(6)
Au^ϕ2(s,x)=0 for all (s,x)∈D, where Au^ϕ2 is given by (42). Then the pair (u^,τ^) is a Nash equilibrium of the stochastic differential game (4) and (8), where
(44)τ^=inf{t>0;Yu^(t)∉D}.
Moreover, the corresponding equilibrium performances are
(45)ϕ1(s,x)=𝒥1(s,x)(u^,τ^),ϕ2(s,x)=𝒥2(s,x)(u^,τ^).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(46)γ(t,z)=0,h(s,x,u)=u22.Let us try the functionsϕi, i=1,2, of the form
(47)ϕi(s,x)=e-δsψi(x),
and a continuation region D of the form
(48)D={(s,x);x<x0}forsomex0>0.
Then we have
(49)Auϕi(s,x)=e-δsAuψi(x),
where
(50)Auψi(x)=-δψi(x)+[(1-u)rx+ubx]ψi′(x)+12x2u2σ2ψi′′(x).By conditions (1) and (3) in Theorem4, we get
(51)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.From conditions (4), (5), and (6) of Theorem4, we get the candidate u^ for the optimal control as follows:
(52)u^=Argmaxu∈𝒜{Auψ1(x)-u22}=Argmaxu∈𝒜{u22-δψ1(x)+[(1-u)rx+ubx]ψ1′(x)+12x2u2σ2ψ1′′(x)-u22}=(bx-rx)ψ1′(x)1-x2σ2ψ1′′(x),(53)u^=Argmaxu∈𝒜{Auψ2(x)}=Argmaxu∈𝒜{12-δψ2(x)+[(1-u)rx+ubx]ψ2′(x)+12x2u2σ2ψ2′′(x)}=-(bx-rx)ψ2′(x)x2σ2ψ2′′(x).Letψi(x)=ψ~i(x) on 0<x<x0, i=1,2. By condition (5) in Theorem 4, we have Au^ψ~1(x)-u^2/2=0 for 0<x<x0. Substituting (52) into Auψ~1(x)-u2/2=0, we obtain
(54)-δψ~1(x)+[rx+(bx-rx)2ψ~1′(x)1-x2σ2ψ~1′′(x)]ψ~1′(x)+[(bx-rx)ψ~1′(x)]22(1-x2σ2ψ~1′′(x))2[x2σ2ψ~1′′(x)-1]=0.
Similarly, we obtain Au^ψ~2(x)=0 for 0<x<x0 by condition (6) in Theorem 4. And we substitute (53) in Auψ~2(x)=0 to get
(55)-δψ~2(x)+rxψ~2′(x)+[(bx-rx)ψ~2′(x)]21-x2σ2ψ~2′′(x)+[xσ(bx-rx)ψ~2′(x)]2ψ~2′′(x)2(1-x2σ2ψ~2′′(x))2=0.Therefore, we conclude that(56)ψ1(x)={U1(C(x)),x≥x0,ψ~1(x),0<x<x0,(57)ψ2(x)={U2(x-C(x)),x≥x0,ψ~2(x),0<x<x0,
where ψ~1(x) and ψ~2(x) are the solutions of (56) and (57), respectively.According to Theorem4, we use the continuity and differentiability of ψi at x=x0 to determine x0, i=1,2, that is,
(58)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)).At the end of this section, we summarize the above results in the following theorem.Theorem 5.
Letψi, i=1,2 and let x0 be the solutions of equations (56)–(58). Then the pair (u^,τ^) given by
(59)u^=(bx-rx)ψ1′(x)1-x2σ2ψ1′′(x)=-(bx-rx)ψ2′(x)x2σ2ψ2′′(x),τ^=inf{t>0;x≥x0}
is a Nash equilibrium of game (4) and (8). The corresponding equilibrium performances are
(60)ϕi(s,x)=e-δsψi(x),i=1,2.


## 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.



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*Source: 761306-2013-05-16.xml*