ELECTRONICCOMMUNICATIONSinPROBABILITY A CONTROLLER AND A STOPPER GAME WITH DEGENERATE VARIANCE CONTROL

We consider a zero sum, stochastic diﬀerential game which involves two players, the controller and the stopper . The stopper selects the stopping rule which halts the game. The controller chooses the diﬀusion coeﬃcient of the corresponding state process which is allowed to degenerate. At the end of the game, the controller pays the stopper, the amount E R τ 0 e − αt C ( Z x ( t )) dt , where Z x ( · ) represents the state process with initial position x and α is a positive constant. Here C ( · ) is a reward function where the set { x : C ( x ) > 0 } is an open interval which contains the origin. Under some assumptions on the reward function C ( · ) and the drift coeﬃcient of the state process, we show that this game has a value. Furthermore, this value function is Lipschitz continuous, but it fails to be a C 1 function.


Introduction
We study a stochastic differential game in this article. This work is related to the controller and the stopper game studied by Karatzas and Sudderth in [4] and we use their formulation of the problem. Consider a weak solution to the one dimensional stochastic differential equation where x is a real number, µ is a twice differentiable function defined on R , {W (t) : t ≥ 0} is a standard Brownian motion adapted to a right continuous filtration {F t : t ≥ 0} on a probability space (Ω, F, P ). The σ-algebra F contains all the null sets in Ω and the Brownian increments W (t + s) − W (t) are independent of F t for all t ≥ 0 and s ≥ 0. The control process u(·) is progressively measurable with respect to the filtration {F t } and satisfies where σ 0 is a constant.
The quadruple ((Ω, F, P ), (F t ), W, u(·)) is an admissible control system if the corresponding state process X x satisfies (1.1) together with the above assumptions. For convenience, we let A(x) be the class of state processes X x available with the initial position x. Here, the reward function C(·) is a twice continuously differentiable defined on R and C(0) > 0.
In this zero sum stochastic differential game, there are two players, the controller and the stopper. Given any initial point x, the controller selects a state process X x from A(x) . The second player, the stopper selects a stopping rule τ to halt the evolution of the state process X x . As described in [4], such a stopping rule τ is considered a mapping from the space of continuous functions where B is the Borel σ-algebra generated by the open sets in C[0, ∞) and ϕ t : Let S be the collection of all such stopping rules. If the stopper decides to end the game at τ (X x ), then the controller pays the stopper an amount of τ (Xx) 0 e −αt C(X x (t))dt at that time. Here α is a positive constant. Thus, the controller would like to choose the control process u(·) to minimize the expected value E τ 0 e −αt C(X x (t))dt, while the stopper would like to select the stopping rule τ to maximize it. The positive discount factor α is useful, so that the expected pay-off will remain finite even for any degenerate diffusion process with state space contained in the set [C > 0].
We define the upper and lower value functions of this game by respectively.
If V (x) = V (x), then this game has a value, and in that case, we denote this common value function by V (x). The discrete time controller and stopper game has been studied in [5]. They showed that the game has a value when the reward function is Borel-measurable and the state space is a Polish space. Our problem is closely related to the continuous time controller and stopper game studied by [4].
Throughout the article, we make the following assumptions (i), (ii) and (iii) about the drift coefficient µ(·) and the reward function C(·). Here, µ and µ denote the first and second derivatives of the function µ. C denotes the first derivative of C.
(i) The function µ is twice continuously differentiable and it satisfies the following conditions: The function is decreasing on R. (1.10) Remarks.
Next, as in [4], we introduce the definition of a saddle point of the game.
for every τ in S and every X(·) in A(x).
The existence of a saddle point clearly implies that the game has a value and in this case, We intend to characterize a saddle point and to derive the explicit form of the value function for this game.
For early work on stochastic differential games, we refer to chapter 17 of [2]. There, stochastic differential games are discussed for non-degenerate diffusion processes in which the control variables occur in the drift coefficient. In [1], two player, zero sum stochastic differential games in finite time horizon are considered in the viscosity solutions framework. There, both players control the drift and diffusion coefficients in the presence of a running cost function and a terminal cost function. They assume that the drift and diffusion coefficients of the state process as well as both cost functions are bounded. They show the existence of the value function and characterize it as the unique viscosity solution to the corresponding "Bellman-Isaacs" partial differential equation.
In our problem, the cost function is unbounded and the drift coefficient of the state process also can be unbounded. Here, the game has a value and this value function is Lipschitz continuous, but it fails to be a C 1 function.
This paper is organized as follows. In section 2, we introduce a class of degenerate diffusion processes. Then we assume the existence of a function Q 0 which is a solution to a second order differential equation with overdetermined boundary data. Using this function Q 0 , we prove the main theorem of the article. There, we obtain a saddle point and the corresponding optimal state process which belongs to the above described class of degenerate diffusion processes. We discuss the properties of the value function of an optimal stopping problem in section 3. This will be used in the construction of Q 0 . Section 4 is devoted to the construction of the function Q 0 with the desired properties as required in section 2. In the case of linear drift coefficient µ(x) = −θx where θ is a positive number, the computations related to the construction of Q 0 in Theorem 4.1 simplify a great deal.

A Class of Degenerate Diffusion Processes and a Saddle Point
We begin with a class of degenerate diffusion processes used in a stochastic control problem in [6]. The construction of such processes is described in page 5 of [6], but it is an immediate application of the proof of Theorem 7.2 in chapter 4 of Ikeda and Watanabe (p.208-214, [3]). For each open interval (a, b) with a < 0 < b, consider a weak solution to where x is a real number, µ is a twice differentiable function on R which satisfies (1.6), {W (t) : t ≥ 0} is a standard Brownian motion adapted to a right continuous filtration {F t : t ≥ 0} on a probability space (Ω, F, P ). For a discussion on the existence of a weak solution and the uniqueness in law for such processes, we refer to [6].
To describe the saddle point, first we assume the existence of a function Q 0 and an open interval (a * , b * ) so that a * < 0 < b * and the following conditions are satisfied: It will be shown in section 5 that such a function Q 0 and the interval (a * , b * ) satisfying a * < 0 < b * exist. In the following lemma, we obtain two properties of the function Q 0 under the assumptions (2.2) and (2.3). They will be used in the derivation of a saddle point in the proof of Theorem 2.2.
Lemma 2.1. Assume the existence of a function Q 0 and a finite interval (a * , b * ) so that a * < 0 < b * and satisfying (2.2) and (2.3). Then Proof. Since the coefficients of the differential equation (2.2) are C 1 functions on R, the function Q 0 which satisfy (2.2) and (2.3) can be extended to R so that it satisfies (2.2) everywhere. Consider this extended function Q 0 and evaluate (2.2) at a * and b * . Using (2.3), then we obtain µ(a * )Q 0 (a * ) + C(a * ) = 0 and Consequently, C(a * ) ≤ 0 and C(b * ) ≤ 0. Therefore, using (1.9), we obtain a * ≤ r < 0 < s ≤ b * and part (i) follows.
To prove part (ii), we intend to use the maximum principle in differential equations (page 7, [8]). By differentiating the differential equation in (2.2) when a * < x < b * , we obtain By differentiating (2.6) again and rearranging the terms, we derive for a * < x < b * . Next, we use (2.6) to replace Q 0 (x) in the above equation and obtain for a * < x < b * . This can be rewritten as ]. Therefore, using (1.7) and (1.10), we can conclude that H(x) ≤ 0 on (a * , b * ). Since, P (a * ) = P (b * ) = 0, now we can apply the maximum principle for differential equations (page 7, [8]) to conclude P (x) ≤ 0 on [a * , b * ]. Hence, part (ii) follows and this completes the proof of the lemma.
To describe our candidate for a saddle point, first we introduce the process Z * x , which is a weak solution to where I A represents the indicator function of the set A. Notice that the corresponding control process u * (t) = I (a * ,b * ) (Z * x (t)) is a feed-back type control. Next, we introduce a stopping rule = +∞, if the above set is empty. (2.8) Now, we are ready to show that the above pair (τ * , Z * x ) is a saddle point.
Theorem 2.2. Assume the existence of a finite interval (a * , b * ) and a function Q 0 satisfying (2.2) and (2.3). Then the pair (τ * , Z * x ) is a saddle point. Furthermore, the stochastic game has a value function, is given by Proof. We intend to verify the two inequalities in (1.11) in the following proof. Notice that the domain of the infinitesimal generator of the Z * x process consists of all C 2 functions f which satisfy f (a * ) = f (b * ) = 0 (for details, see [6].) Let x be in [a * , b * ]. Then,we can apply Itô's lemma to Q 0 (Z * x (t))e −αt and use (2.2) and (2.3) to obtain Next, we take any stopping time τ and apply Itô's lemma to obtain Hence, This verifies the first inequality of (1.11) when the initial point x is deterministic until it enters the interval [a * , b * ] and we let T 0 be the first entrance time to the interval [a * , b * ]. Then e −αt C(Z * (t))dt.
By Lemma 2.1, the cost function C is negative outside the interval [a * , b * ]. Hence, the first term is less than or equal to zero. Similar to the proof of (2.11), we can show that the second term is also less than or equal to zero. Thus, This verifies the first inequality of (1.11) when x is outside [a * , b * ].
Therefore, we can conclude that the pair (τ * , Z * x ) is a saddle point and V (x) = V (x) for all x, where V (x) and V (x) are defined in (1.4) and (1.5) respectively. Consequently, this stochastic game has a value function and it is given by (2.9). This completes the proof.

An Optimal Stopping Problem
In this section, we consider an optimal stopping problem which is related to the construction of the function Q 0 . We begin with the process {Y x (t) : t ≥ 0} which satisfies for t ≥ 0. Here, {W (t) : t ≥ 0} is a Brownian motion adapted to a Brownian filtration {F t } in a probability space (Ω, F, P ). We introduce the infinitesimal generator G of (3.1) by and for α > 0, we define the differential operator G − α by Let D be the collection of all {F t } stopping times. We consider the optimal stopping problem with the value function U (x) given by Since C(·) is positive on the interval (r, s), U (·) is also strictly positive on (r, s). It is known that(see [7]) this value function U is a C 1 function and it satisfies the variational inequality for almost every x in R.
It is also known that, (see [7]) Then by Itô's lemma, we can conclude that 4 Construction of Q 0 Let G be the differential operator introduced in (3.2). We consider the collection of the functions Q ab along with their corresponding intervals [a, b] satisfying the following conditions: There is a finite interval [a * , b * ] and an associated function Q a * b * which satisfy (4.1), (4.2) and (4.3) above together with the following additional property: on the interval (a * , b * ).

Remark.
For convenience, we relabel this function Q a * b * as Q 0 and use it as required in Theorem 2.1.
Proof. We set First we show that [r, s] is in C. Since C(·) is strictly positive in (r, s), we can apply the maximum principle for differential equations (see [8], page 7) to Q rs to conclude Q rs (x) > 0 on (r, s). Also, by the boundary point lemma ( [8] page 7), Q rs (r) > 0 and Q rs (s) < 0. Hence, using the differential equation for Q rs it follows that Q rs (r) < 0 and Q rs (s) < 0. Therefore, the interval [r, s] is in C.
Now let ([a n , b n ]) be an increasing sequence of nested intervals in C. Then (a n ) is a decreasing sequence and (b n ) is an increasing sequence. We let l = lim n a n and m = lim Throughout the remaining arguments, we will use the following facts from the theory of differential equations. a) Two distinct solutions of the differential equation (G − α)Q(x) + C(x) = 0 cannot meet more than once.
b) The solution Q ab (x) and its first and second derivatives are jointly continuous in (a, b, x).
By elementary analysis, it can be shown that Q anbn is increasing to Q lm , lim n Q anbn (x) = Q lm (x) and lim n Q anbn (x) = Q lm (x) for all x in (l, m). It is easy to verify that Q lm satisfies Since [a * , b * ] is a maximal element in C, therefore it follows that Q a * b * (a * ) = Q a * b * (b * ) = 0. This completes the proof.