c 2004 Society for Industrial and Applied Mathematics

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SIAM J. COTROL OPTIM. Vol. 43, o. 4, pp. 1222 1233 c 2004 Society for Industrial and Applied Mathematics OZERO-SUM STOCHASTIC DIFFERETIAL GAMES WITH DISCOTIUOUS FEEDBACK PAOLA MAUCCI Abstract. The existence of a ash equilibrium feedback is established for a two-player nonzerosum stochastic differential game with discontinuous feedback. This is obtained by studying a parabolic system strongly coupled by discontinuous terms. Key words. nonzero-sum stochastic games, ash point, strongly coupled parabolic systems, discontinuous feedbacks AMS subject classifications. 35K50, 49K30, 4935, 91A10, 91A15 DOI. 10.1137/S0363012903423715 1. Introduction. The aim of this paper is to study the existence of ash equilibrium points for a two-player nonzero-sum stochastic differential game. The game is governed by a stochastic differential equation with two controls and two payoffs. This problem can be found, for instance, in Friedman [7] and in a series of papers by Bensoussan and Frehse [2], [3], [4]. All these papers make the assumption that feedback is continuous. We are interested in studying the problem assuming that the controls take values in compact sets. In this case one cannot expect a ash equilibrium among continuous feedback, and the Hamiltonian functions associated with the game are nonsmooth. We consider a simple multidimensional model problem taking two players, affine dynamics, affine payoff functions, and compact control sets. The loss of continuity of the feedback, due to the hard constraints, leads us to consider a parabolic system strongly coupled by discontinuous terms. In fact, from the usual necessary condition satisfied by the value of the ash equilibrium feedback in terms of the Hamilton Jacobi theory, we reduce ourselves to studying the existence of a sufficiently regular solution to a system of nonlinear parabolic equations which contains the Heaviside graph. By this regularity result, we are able to construct ash equilibrium feedback whose optimality is proved by using the verification approach in the sense of [2], [3], [4]. The motivation for studying games in compact control sets comes from standard nonlinear control theory; this seems a natural assumption in many applications. In particular, ash equilibria for nonzero-sum deterministic differential games were recently studied by Olsder [12] and Cardaliaguet and Plaskacz [5]. 2. Statement of the problem. Let Ω be a bounded smooth domain in R. Let X be a process which satisfies the following stochastic differential equation (2.1 (2.2 dx(s =f(s, X(s,u 1 (s, X(s,u 2 (s, X(sds + σ(s, X(sdw, X(t =x, s [t, T ], x Ω R. Received by the editors February 26, 2003; accepted for publication (in revised form March 8, 2004; published electronically December 1, 2004. This work was partially supported by the Italian MURST national project Analisi e controllo di equazion di evoluzione deterministiche e stocastiche. http://www.siam.org/journals/sicon/43-4/42371.html Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, 35135 Padova, Italy (mannucci@math.unipd.it. 1222

STOCHASTIC GAMES WITH DISCOTIUOUS FEEDBACK 1223 For each s, X(s represents the state evolution of a system controlled by two players. The ith player acts by means of a feedback control function u i :(t, T R U i R ki, where i =1, 2. Let U i : {u i Borel-measurable applications (t, T R U i R ki }, i =1, 2, be the set of the control functions u i with values u i (s, X inu i. The term σ(s, X(sdw represents the noise, where w is an -dimensional standard Brownian motion and σ is an matrix. We assume that σ does not depend on the control variables u 1, u 2 and that σ and σ 1 are bounded and Lipschitz on X. The function f(s, X, u 1,u 2 is called the dynamic of the game (2.1. We refer to [7] for the definitions about stochastic processes, stochastic differential equations and functional spaces. A control function u i U i will be called admissible if it is adapted to the filtration defined on the probability space. It is possible to prove, using Girsanov s theorem, that, under convenient assumptions on f, for all (u 1,u 2 U 1 U 2 admissible controls, there exists a unique weak solution to the problem (2.1, (2.2 (see, for example, [4], [9], [6, Chapter 4]. For any choice of admissible controls u 1, u 2 we have the following payoff functions: { τ } J i (t, x, u 1,u 2 =E tx l i (s, X(s,u 1 (s, X(s,u 2 (s, X(sds + g i (T,X(T, (2.3 t i =1, 2, where τ T inf{s t, X(s / Ω}, E tx is the expectation under the probability P tx, l i and g i are prescribed functions (the assumptions will be specified later, and X = X(s is the unique weak solution of (2.1 (2.2 corresponding to (u 1,u 2 U 1 U 2 admissible controls. Each player wants to maximize his own payoff. Definition 2.1. A pair of admissible controls (u 1, u 2 U 1 U 2 is called the ash equilibrium point of the differential game (2.1 (2.2, with payoff (2.3, if (2.4 (2.5 J 1 (t, x, u 1, u 2 J 1 (t, x, u 1, u 2, J 2 (t, x, u 1, u 2 J 2 (t, x, u 1,u 2, for all (t, x (0,T Ω and for all (u 1,u 2 U 1 U 2 admissible controls. The functions (2.6 V 1 (t, x : J 1 (t, x, u 1, u 2, V 2 (t, x : J 2 (t, x, u 1, u 2 are a value of the ash equilibrium point (u 1, u 2. We define the pre-hamiltonians H i (t, x, p, u 1,u 2 : (0,T R R U 1 U 2 R, i =1, 2: (2.7 H 1 (t, x, p, u 1 (t, x,u 2 (t, x : p f(t, x, u 1 (t, x,u 2 (t, x + l 1 (t, x, u 1 (t, x,u 2 (t, x, H 2 (t, x, p, u 1 (t, x,u 2 (t, x : p f(t, x, u 1 (t, x,u 2 (t, x + l 2 (t, x, u 1 (t, x,u 2 (t, x. We set a = 1 2 σσ (σ is the transpose of σ to be the matrix with elements a h,k, h, k =1,...,.

1224 PAOLA MAUCCI If the value functions V 1, V 2 C 1,2, we can apply Itô s formula; changing the time variable (T t t, we get that V 1, V 2 solve, in Ω T : (0,T Ω, the following nonlinear parabolic system coupled by the ash equilibrium problem: (2.8 V 1 (t, x (t, x 2 V 1 (t, x = max {u1 U 1}H 1 (t, x, x V 1 (t, x,u 1 (t, x, u 2 (t, x (2.9 V 2 (t, x = H 1 (t, x, x V 1 (t, x, u 1 (t, x, u 2 (t, x, (t, x 2 V 2 (t, x = max {u2 U 2}H 2 (t, x, x V 2 (t, x, u 1 (t, x,u 2 (t, x = H 2 (t, x, x V 2 (t, x, u 1 (t, x, u 2 (t, x, (2.10 (2.11 u 1 (t, x argmax {u1 U 1}H 1 (t, x, x V 1 (t, x,u 1 (t, x, u 2 (t, x, u 2 (t, x argmax {u2 U 2}H 2 (t, x, x V 2 (t, x, u 1 (t, x,u 2 (t, x, (2.12 V 1 = g 1 (t, x, V 2 = g 2 (t, x on p Ω T, where p Ω T : ( (0,T Ω ( {t =0} Ω. (Here and in the following we write, for the sake of brevity, argmax ui U i H i, which means argmax ui(t,x U i H i. The functions H 1 (t, x, x V 1 (t, x, u 1 (t, x, u 2 (t, x = max {u1 U 1}H 1 (t, x, x V 1 (t, x,u 1 (t, x, u 2 (t, x, H 2 (t, x, x V 1 (t, x, u 1 (t, x, u 2 (t, x = max {u2 U 2}H 2 (t, x, x V 2 (t, x, u 1 (t, x,u 2 (t, x are called the Hamiltonian functions associated with the game (2.1 (2.3. We want to outline here the classical procedure used in Friedman s book [7] to prove the existence of a ash equilibrium point u 1, u 2. 1. Suppose that, for any fixed p R, there exist u 1, u 2 such that u 1(t, x, p argmax {u1 U 1}H 1 (t, x, p, u 1 (t, x,u 2(t, x, u 2(t, x, p argmax {u2 U 2}H 2 (t, x, p, u 1(t, x,u 2 (t, x (2.13 are measurable in (t, x Ω T and continuous in p. 2. Solve the parabolic system (2.14 V 1 (t, x (t, x 2 V 1 (t, x = H 1 (t, x, x V 1,u 1(t, x, x V 1,u 2(t, x, x V 2, V 2 (t, x (t, x 2 V 2 (t, x = H 2 (t, x, x V 2,u 1(t, x, x V 1,u 2(t, x, x V 2, V 1 = g 1 (t, x, V 2 = g 2 (t, x on p Ω T.

STOCHASTIC GAMES WITH DISCOTIUOUS FEEDBACK 1225 3. Prove that the pair of functions (u 1, u 2 with values u 1 (t, x : u 1(t, x, x V 1 (t, x, u 2 (t, x : u 2(t, x, x V 2 (t, x is a ash equilibrium point (see Definition 2.1. Therefore to obtain ash equilibrium points for the associated stochastic differential game, we look for a sufficiently regular solution of system (2.14. A similar procedure is used, in the elliptic case, by Bensoussan and J. Frehse [2], [3], [4] to study systems of Bellman equations. We want to emphasize that the results of Friedman and Bensoussan and J. Frehse on the existence of classical solutions and of ash equilibrium points are obtained under the assumption that there exist some feedback u 1 argmax {u1 U 1}H 1 (t, x, p, u 1,u 2, u 2 argmax {u2 U 2}H 2 (t, x, p, u 1,u 2 that are continuous in p (see, for example, assumption (D [7, section 17, p. 497]. If we assume that the sets U i, i =1, 2, are compact, the assumption on the continuity of the feedback can be too restrictive. Weaker assumptions on the regularity of the feedback can be found in [8] and [9]. In this paper we consider a model problem with U 1, U 2 compact sets in R, affine dynamics of the game, and affine payoff. Let us list the assumptions: (2.15 U 1 = U 2 =[0, 1], (2.16 f(x, u 1 (t, x,u 2 (t, x : Ω U 1 U 2 R, f(x, u 1 (t, x,u 2 (t, x = f 1 (xu 1 (t, x+f 2 (xu 2 (t, x, f i (x : Ω R,f i (x C 1 (Ω, i=1, 2, (2.17 l i (x, u 1 (t, x,u 2 (t, x : Ω U 1 U 2 R, l i (x, u 1 (t, x,u 2 (t, x = l i (xu i (t, x, i =1, 2, l i (x : Ω R,l i (x C 1 (Ω, i =1, 2, (2.18 (2.19 g i (t, x H 1+α (Ω T,α (0, 1, i =1, 2, (t, x C 2 (Ω T, ν ξ 2 (t, xξ h ξ k µ ξ 2, ν,µ > 0, for all (t, x Ω T and for all ξ R. Taking into account the affine structure of f and l in (2.16 (2.17, the functions H 1 and H 2 in (2.7 become (2.20 H 1 (x, p, u 1 (t, x,u 2 (t, x = ( p f 1 (x+l 1 (x u 1 (t, x+p f 2 (xu 2 (t, x, H 2 (x, p, u 1 (t, x,u 2 (t, x = ( p f 2 (x+l 2 (x u 2 (t, x+p f 1 (xu 1 (t, x.

1226 PAOLA MAUCCI From (2.15 and (2.20, for any fixed p, wehave u 1 (t, x argmax {u1 U 1}H 1 (x, p, u 1 (t, x, u 2 (t, x = argmax {u1(t,x [0,1]}( p f1 (x+l 1 (x u 1 (t, x+p f 2 (xu 2 (t, x (2.21 = Heav ( p f 1 (x+l 1 (x, where Heav(η is the Heaviside graph defined as Heav(η = 1 if η>0, Heav(η = 0 if η<0, and Heav(0 = [0, 1]. Analogously (2.22 u 2 (t, x Heav ( p f 2 (x+l 2 (x. From (2.20, (2.21, (2.22, the Hamiltonian functions assume the form H 1 (x, p, u 1 (t, x, u 2 (t, x max {u1 U 1}H 1 (x, p, u 1 (t, x, u 2 (t, x (2.23 (2.24 = ( p f 1 (x+l 1 (x + p f + 2(xu 2 (t, x, H 2 (x, p, u 1, u 2 max {u2 U 2}H 2 (x, p, u 1 (t, x,u 2 (t, x = ( p f 2 (x+l 2 (x + p f + 1(xu 1 (t, x, where we denoted by (h + the positive part of the function h. Taking, for any fixed p, (2.25 (2.26 u 1(t, x, p Heav ( p f 1 (x+l 1 (x, u 2(t, x, p Heav ( p f 2 (x+l 2 (x, we have that u 1, u 2 do not satisfy (2.13 because they are not continuous in p. Hence, in this case, we cannot use the results of [7]. From (2.25, (2.26, the parabolic system (2.8, (2.9 becomes (2.27 (2.28 V 1 V 2 2 ( V 1 x V 1 f 1 (x+l 1 (x Heav( x V 1 f 1 (x+l 1 (x + x V 1 f 2 (xheav( x V 2 f 2 (x+l 2 (x in Ω T, 2 ( V 2 x V 2 f 2 (x+l 2 (x Heav( x V 2 f 2 (x+l 2 (x + x V 2 f 1 (xheav( x V 1 f 1 (x+l 1 (x in Ω T, (2.29 V 1 (t, x =g 1 (t, x, V 2 (t, x =g 2 (t, x on p Ω T. This is a uniformly parabolic system strongly coupled by the Heaviside graph containing the first order derivatives of the unknown functions. Equations (2.27 and (2.28 are to be interpreted in the following way: V 1 2 V 1 = ( x V 1 f 1 + l 1 h 1 (t, x+ x V 1 f 2 h 2 (t, x, V 2 2 V 2 = ( x V 2 f 2 + l 2 h 2 (t, x+ x V 2 f 1 h 1 (t, x, h 1 (t, x Heav( x V 1 f 1 (x+l 1 (x, h 2 (t, x Heav( x V 2 f 2 (x+l 2 (x.

STOCHASTIC GAMES WITH DISCOTIUOUS FEEDBACK 1227 Following the previous scheme, first we investigate the existence of a solution V 1, V 2 of (2.27 (2.29. ext, if we find sufficient regularity, it will be possible to prove the existence of a ash equilibrium point. In section 3 we provide an existence result for a solution V 1, V 2 in H 1+α (Ω T (Ω T of the system (2.27 (2.29, and in section 4 we prove the existence of a ash equilibrium point. W 1,2 q 3. Existence of a solution to the parabolic system. We give the following definition. Definition 3.1. (V 1,V 2 is a strong solution of the system (2.27 (2.29 if (a V 1 (t, x, V 2 (t, x H 1+α (Ω T Wq 1,2 (Ω T for some α (0, 1, q > +2; (b equations (2.27 (2.28 hold almost everywhere and (2.29 holds. Theorem 3.2. Under assumptions (2.15 (2.19, taking Heav(η =1if η>0, Heav(η =0if η<0, and Heav(0 = [0, 1], there exists at least a strong solution (V 1, V 2 of the parabolic system (2.27 (2.29. Proof. Let us consider the approximating problems obtained by replacing the Heaviside graph Heav(η with smooth functions H n : H n (η C (R, H n (η L (R, H n (η =0ifη 0, H n (η =1ifη 1 n, (3.1 H n 0, H n (η Heav(η in L p (K, p>1, K R is any compact of R, H n (η Heav(η in C 0 outside a neighbourhood of η =0. We denote by V 1n, V 2n the solution of the problem V 1n 2 V 1n = ( x V 1n f 1 + l 1 H n ( x V 1n f 1 + l 1 (3.2 + x V 1n f 2 H n ( x V 2n f 2 + l 2 in Ω T, V 2n 2 V 2n = ( x V 2n f 2 + l 2 H n ( x V 2n f 2 + l 2 (3.3 + x V 2n f 1 H n ( x V 1n f 1 + l 1 in Ω T, (3.4 V 1n = g 1,V 2n = g 2 in p Ω T. From [11, Theorem 7.1, p. 596] on quasi-linear parabolic systems with smooth coefficients, there exists an unique solution of problem (3.2 (3.4, V 1n (t, x, V 2n (t, x H 2+α (Ω T. At this point, regarding the terms H n ( x V 1n f 1 + l 1 +f 2 H n ( x V 2n f 2 + l 2 and H n ( x V 2n f 2 + l 2 +f 1 H n ( x V 1n f 1 + l 1 in (3.2 (3.3 as bounded uniformly on n, from [11, Theorem 9.1, p. 341], we find an uniform estimate in Wq 1,2 : (3.5 V 1n (2 q,ω T + V 2n (2 q,ω T C( H n q,ωt, g 1 (2 1/q q, pω T, g 2 (2 1/q q, pω T C, where C is independent of n and q>1. By means of an embedding theorem (see, for example, [11, Chapter 2, Lemma 3.3], taking q>+ 2, we obtain (3.6 V 1n (1+α Ω T + V 2n (1+α Ω T C, α =1 +2, q where C is independent of n.

1228 PAOLA MAUCCI We can now extract two subsequences, which we denote again by V 1n, V 2n such that (3.7 V in V i in C 0 (Ω T, i =1, 2, V in V i in C 0 (Ω T, i =1, 2, h=1,... x h x h From the weak precompactness of the unit ball of Wq 2,1,wehave (3.8 V in V i 2 V in weakly in L 2 (Ω T, i =1, 2, 2 V i weakly in L 2 (Ω T, i =1, 2,,... From (3.7, (3.8 (3.9 V 1 (t, x, V 2 (t, x H 1+α (Ω T Wq 1,2 (Ω T, with α =1 +2. q ow we have to prove that V 1, V 2 solve (2.27 (2.28 almost everywhere in Ω T. From assumptions (3.1 and (3.8, the two sequences H n ( x V 1n f 1 +l 1, H n ( x V 2n f 2 +l 2 are uniformly bounded in L 2 (Ω T, and hence we can extract two subsequences such that (3.10 H n ( x V 1n f 1 + l 1 h 1 (t, x weakly in L 2 (Ω T, H n ( x V 2n f 2 + l 2 h 2 (t, x weakly in L 2 (Ω T. We now show that h i (t, x Heav( x V i f i + l i almost everywhere i =1, 2. To do this let us consider the following sets: P i : {(t, x Ω T : x V i (t, x f i (x+l i (x > 0}, i : {(t, x Ω T : x V i (t, x f i (x+l i (x < 0}, Z i : {(t, x Ω T : x V i (t, x f i (x+l i (x =0}, i =1, 2. From (3.7, we have that, for a sufficiently large n, x V in (t, x f i (x+l i (x > 0 for all (t, x P i. Hence, from (3.1, we obtain that H n ( x V in (t, x f i (x+l i (x 1 = Heav( x V i (t, x f i (x+l i (x for all (t, x P i, i =1, 2. Analogously, in i, for a sufficiently large n, x V in (x, t f i (x+l i (x < 0, and hence H n ( x V in (x, t f i (x+l i (x 0 = Heav( x V i (x, t f i (x+l i (x for all (t, x i, i =1, 2.

STOCHASTIC GAMES WITH DISCOTIUOUS FEEDBACK 1229 In Z i, from the assumptions on H n (see (3.1, we have that 0 h i 1 almost everywhere, and hence (3.11 (3.12 h 1 (t, x Heav( x V 1 (t, x f 1 (x+l 1 (x almost everywhere in Ω T, h 2 (t, x Heav( x V 2 (t, x f 2 (x+l 2 (x almost everywhere in Ω T. At this point, from (3.7, (3.8, (3.10, (3.11, (3.12, we obtain that V 1, V 2 satisfy (2.27 (2.28 almost everywhere in Ω T and from the regularity of the functions V 1, V 2, we have that V 1 (t, x =g 1 (t, x, V 2 (t, x =g 2 (t, x for all (t, x p Ω T. Remark 3.1. If we choose W (η Heav(η, we are not able to solve the problem (3.13 (3.14 V 1 V 2 2 V 1 = ( x V 1 f 1 + l 1 W ( x V 1 f 1 + l 1 + x V 1 f 2 W ( x V 2 f 2 + l 2, 2 V 2 = ( x V 2 f 2 + l 2 W ( x V 2 f 2 + l 2 + x V 2 f 1 W ( x V 1 f 1 + l 1, because we cannot exclude that meas{(t, x Ω T : x V i f i (x+l i (x =0} > 0, i = 1, 2. Hence we cannot prove that but only that H n ( x V in f i + l i W( x V i f i + l i, i =1, 2, H n ( x V in f i + l i h i Heav( x V i f i + l i, i =1, 2. 4. Existence of a ash equilibrium point. We now prove the following. Theorem 4.1. Suppose that the assumptions of Theorem 3.2 hold. Let (V 1, V 2 be a strong solution of the parabolic system (2.27 (2.29; then any admissible control (u 1, u 2 such that (4.1 (4.2 u 1 (t, x Heav( x V 1 (t, x f 1 (x+l 1 (x, u 2 (t, x Heav( x V 2 (t, x f 2 (x+l 2 (x is a ash equilibrium point for the stochastic differential game (2.1 (2.2 with payoff (2.3. Proof. The existence of a strong solution (V 1,V 2 of the parabolic system (2.27 (2.29 is stated by Theorem 3.2. To prove that u i (t, x Heav( x V i (t, x f i (x+l i (x, i =1, 2, are the values of a ash equilibrium point, as in Definition 2.1, we have to show that (4.3 J 1 (t, x, u 1, u 2 J 1 (t, x, u 1, u 2, J 2 (t, x, u 1, u 2 J 2 (t, x, u 1,u 2 for all (u 1,u 2 U 1 U 2 admissible controls. Let us denote (4.4 v 1 (t, x : J 1 (t, x, u 1, u 2, v 2 (t, x : J 2 (t, x, u 1, u 2.

1230 PAOLA MAUCCI Using a generalization of Itô s formula applied to functions in Wq 1,2 (see, for example, [1, Theorem 4.1, p. 126], we have that the couple (v 1,v 2 solves the following parabolic system (here and in the following we omit, for the sake of brevity, the dependence on the variables (t, x: (4.5 (4.6 (4.7 v 1 v 2 2 v 1 = H 1 (x, t, x v 1, u 1, u 2 =( x v 1 f 1 + l 1 u 1 + x v 1 f 2 u 2 in Ω T, 2 v 2 = H 2 (x, t, x v 2, u 1, u 2 =( x v 2 f 2 + l 2 u 2 + x v 2 f 1 u 1 in Ω T, v 1 = g 1 (t, x, v 2 = g 2 (t, x on p Ω T. From (4.1, (4.2, we have (4.8 (4.9 v 1 v 2 2 v 1 ( x v 1 f 1 + l 1 Heav( x V 1 f 1 + l 1 + x v 1 f 2 Heav( x V 2 f 2 + l 2 in Ω T, 2 v 2 ( x v 2 f 2 + l 2 Heav( x V 2 f 2 + l 2 + x v 2 f 1 Heav( x V 1 f 1 + l 1 in Ω T, v 1 = g 1 (t, x, v 2 = g 2 (t, x on p Ω T. Let us now fix (u 1,u 2 U 1 U 2 admissible controls and denote (4.10 w 1 (t, x : J 1 (t, x, u 1, u 2, w 2 (t, x : J 2 (t, x, u 1,u 2. The couple (w 1, w 2 solves the following parabolic system: (4.11 w 1 2 w 1 = H 1 (x, t, x w 1,u 1, u 2 =( x w 1 f 1 + l 1 u 1 + x w 1 f 2 u 2 in Ω T, (4.12 w 2 2 w 2 = H 2 (x, t, x w 2, u 1,u 2 =( x w 2 f 2 + l 2 u 2 + x w 2 f 1 u 1 in Ω T, (4.13 w 1 = g 1 (t, x, w 2 = g 2 (t, x on p Ω T, w 1,w 2 Wq 1,2 (Ω T. From the expressions (2.20 of H 1, H 2, taking into account (2.10, (2.11, we have that, for any p fixed, (4.14 (pf 1 + l 1 u 1 (t, x (pf 1 + l 1 u 1 (t, x, (pf 2 + l 2 u 2 (t, x (pf 2 + l 2 u 2 (t, x.

STOCHASTIC GAMES WITH DISCOTIUOUS FEEDBACK 1231 Consider now the functions z 1 : v 1 w 1, z 2 : v 2 w 2. From systems (4.5 (4.7 and (4.11 (4.13 we have (4.15 z 1 z 2 2 z 1 =( x v 1 f 1 + l 1 u 1 + x v 1 f 2 u 2 ( x w 1 f 1 + l 1 u 1 x w 1 f 2 u 2 in Ω T, 2 z 2 =( x v 2 f 2 + l 2 u 2 + x v 2 f 1 u 1 ( x w 2 f 2 + l 2 u 2 x w 2 f 1 u 1 in Ω T, z 1 = z 2 =0 on p Ω T. Taking into account (4.14, we obtain z 1 2 z 1 ( x v 1 f 1 + l 1 u 1 + x v 1 f 2 u 2 (4.16 ( x w 1 f 1 + l 1 u 1 x w 1 f 2 u 2 = x z 1 (f 1 u 1 + f 2 u 2 in Ω T, z 2 2 z 2 ( x v 2 f 2 + l 2 u 2 + x v 2 f 1 u 1 (4.17 ( x w 2 f 2 + l 2 u 2 x w 2 f 1 u 1 = x z 2 (f 2 u 2 + f 1 u 1 in Ω T, z 1 = z 2 =0 on p Ω T. Equations (4.16, (4.17 are no longer coupled and the terms f 1 u 1 + f 2 u 2, f 2 u 2 + f 1 u 1 are known and bounded. Hence we can apply an extension of the maximum principle to parabolic equations whose coefficients are in L (see, for example, [10, Chapter 7], obtaining (4.18 z 1 (t, x 0, z 2 (t, x 0 in Ω T. Taking into account (4.4 and (4.10, from (4.18 we obtain (4.3, i.e., the result. Remark 4.1. The results proved in section 3 and 4 hold true even if we take M>2players and if we take the functions f and l linear and dependent explicitly on t, i.e., f(t, x, u 1,u 2 =f 1 (t, xu 1 + f 2 (t, xu 2 + f 3 (t, x, l 1 (t, x, u 1,u 2 =l 1 (t, xu 1 + h 1 (t, x, l 2 (t, x, u 1,u 2 =l 2 (t, xu 2 + h 2 (t, x. The only difference is the appearance in (2.27 (2.28, as source terms, of the functions f 3 (t, x+h 1 (t, x and f 3 (t, x+h 2 (t, x, respectively.

1232 PAOLA MAUCCI Remark 4.2. In [12], Olsder studied the ash equilibria of the following nonzerosum deterministic differential game with open-loop bang-bang controls: with payoff J 1 = T and controls subject to t ẋ =(1 xu 1 xu 2, (c 1 x u 1 ds, J 2 = T t 0 u i (t 1. (c 2 (1 x u 2 ds, Because of the hard constraints, also in this case, the optimal controls contain Heaviside functions. Remark 4.3. If we change the control sets taking U 1 = U 2 =[ 1, 1], as done in [5] in the deterministic case, the optimal feedback equilibria are u 1 sign ( p f 1 + l 1, u 2 sign ( p f 2 + l 2, where sign(η =1if η>0, sign(η = 1 if η<0, and sign(0 = [ 1, 1]. The Hamiltonian functions take the form H 1 (x, p, u 1, u 2 = p f1 (x+l 1 (x + p f2 (xu 2, H 2 (x, p, u 1, u 2 = p f2 (x+l 2 (x + p f1 (xu 1, and also in this case our method can be applied. Acknowledgments. The author would like to thank Martino Bardi, who stated the problem, for his helpful comments and suggestions. The author thanks the referees for their useful remarks. REFERECES [1] A. Bensoussan, Stochastic Control by Functional Analysis Methods, orth-holland, Amsterdam, ew York, 1982. [2] A. Bensoussan and J. Frehse, Regularity Results for onlinear Elliptic Systems and Applications, Appl. Math. Sci. 151, Springer-Verlag, Berlin, 2002. [3] A. Bensoussan and J. Frehse, Stochastic games for players, J. Optim. Theory Appl., 105 (2000, pp. 543 565. [4] A. Bensoussan and J. Frehse, onlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984, pp. 23 67. [5] P. Cardaliaguet and S. Plaskacz, Existence and uniqueness of a ash equilibrium feedback for a simple nonzero-sum differential game, Internat. J. Game Theory, 32 (2003, pp. 33 71. [6] W. H. Fleming and H. Mete Soner, Controlled Markov Processes and Viscosity Solutions Springer-Verlag, ew York, 1993. [7] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, ew York, 1976. [8] S. Hamedène and J. P. Lepeltier, Points d equilibre dans le jeux stochastiques de somme non nulle, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994, pp. 251 256. [9] S. Hamedène, J. P. Lepeltier, and S. Peng, BSDEs with continuous coefficients and stochastic differential games, in Backward Stochastic Differential Equations (Paris, 1995 1996, El Karoui et al., ed., Pitman Res. otes Math. Ser. 364, Longman, Harlow, 1997, pp. 115 128.

STOCHASTIC GAMES WITH DISCOTIUOUS FEEDBACK 1233 [10] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, J, 1996. [11] O. A. Ladyzhenskaya, V. A. Solonnikov, and.. Ural ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, AMS, Providence, RI, 1968. [12] G. J. Olsder, On open- and closed-loop bang-bang control in nonzero-sum differential games, SIAM J. Control Optim., 40 (2001, pp. 1087 1106.

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