ERGODIC BELLMAN SYSTEMS FOR STOCHASTIC GAMES IN ARBITRARY DIMENSION

Stochastic games with cost functionals J(rho,x)((i))(v) = E0 integral( 0)(infinity) e(-rho t)l(i)(y, v) dt, i = 1,2 with controls v = v(1), v (2) and state y(t) with y(0) = x are considered. Each player wants to minimize his (her) cost functional. E denotes the expected value and t he state variables y are coupled with the controls v via a stochastic differential equation with initial value x. The corresponding Bellman system, which is used for the calculation of feedback controls v = v(y ) and the solvability of the game, leads to a class of diagonal second -order nonlinear elliptic systems, which also occur in other branches of analysis. Their behaviour concerning existence and regularity of so lutions is, despite many positive results, not yet well understood, ev en in the case where the l(i) are simple quadratic functions. The obje ctive of this paper is to give new insight to these questions for fixe d rho > 0, and, primarily, to analyse the limiting behaviour as the di scount rho --> 0. We find that the modified solutions of the stochasti c games converge, for subsequences, to the solution of the so-called e rgodic Bellman equation and that the average cost converges. A former restriction of the space dimension has been removed. A reasonable clas s of quadratic integrands may be treated. More specifically, we consid er the Bellman systems of equations -Delta z + lambda = H(x, Dz), wher e the space variable x belongs to a periodic cube (for the sake of sim plifying the presentation). They are shown to have smooth solutions. I f u(rho) is the solution of -Delta u(rho) + rho u(rho) = H(x, Du(rho)) then the convergence of u rho - (u) over bar(rho), to z, as rho tends to 0, is established. The conditions on H are such that some quadrati c growth in Du is allowed.