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

It is known that the slow motion $X^{\epsilon }$ in the time-scaled multidimensional averaging setup $\frac {dX^\varepsilon(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),,\xi(t/\varepsilon^2))+b(X^\varepsilon(t),,\xi(t/\ve^2)),, t\in [0,T]$convergesweaklyas$\varepsilon\to 0$ 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 $X^{\epsilon }$ and a family of diffusions ${\mathrm{\Xi }}^{\epsilon }$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-\Xi^\varepsilon(t)|^{2M}\leq C(M)\varepsilon^\del$forsome$C(M),\delta>0$andall$M\ge 1,,\varepsilon>0$,whereall$\Xi^\varepsilon,, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\epsilon$. 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.