Strong diffusion approximation in averaging and value computation in dynkin's games

It is known since (Theory Probab. Appl. 11 (1966) 390-406) that the slow motion X-epsilon in the time-scaled multidimensional averaging setup dX(epsilon)(t)/dt = 1/epsilon B(X-epsilon(t), xi(t/epsilon(2))) +b(X-epsilon(t), xi(t/epsilon(2))), t is an element of [0, T] converges weakly as epsilon -> 0 to a diffusion process provided EB(x, xi(s)) 0 where xi is a sufficiently fast mixing stochastic process. In this paper we show that both X-epsilon and a family of diffusions Xi(epsilon) can be redefined on a common sufficiently rich probability space so that E sup(0 <= t <= T) vertical bar X-epsilon (t) - Xi(epsilon)(t)vertical bar(2M) <= C(M)epsilon(delta) for some C (M), delta > 0 and all M >= 1, epsilon > 0, where all Xi(epsilon), epsilon > 0 have the same diffusion coefficients but underlying Brownian motions may change with epsilon. We obtain also a similar result for the corresponding discrete time averaging setup. As an application we consider Dynkin's games with path dependent payoffs involving a diffusion and obtain error estimates for computation of values of such games by means of such discrete time approximations which provides a more effective computational tool than the standard discretization of the diffusion itself.