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

It is known that the slow motion $X^\varepsilon$ in the time-scaled multidimensional averaging setup $\frac {dX^\varepsilon(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\,\xi(t/\varepsilon^2))+b(X^\varepsilon(t),\,\xi(t/\ve^2)),\, t\in [0,T]$ converges weakly as $\varepsilon\to 0$ to a diffusion process provided $EB(x,\xi(s))\equiv 0$ where $\xi$ is a sufficiently fast mixing stochastic process. In this paper we show that both $X^\varepsilon$ and a family of diffusions $\Xi^\varepsilon$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-\Xi^\varepsilon(t)|^{2M}\leq C(M)\varepsilon^\del$ for some $C(M),\delta>0$ and all $M\ge 1,\,\varepsilon>0$, where all $\Xi^\varepsilon,\, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\varepsilon$. This is the first strong approximation result both in the above setup and at all when the limit is a nontrivial multidimensional diffusion. We obtain also a similar result for the corresponding discrete time averaging setup which was not considered before at all. 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.