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

TL;DR

This paper establishes a strong approximation for slow motion processes in averaging setups, showing they can be closely coupled with diffusions on a common probability space, and applies this to improve computations in Dynkin's games.

Contribution

It provides the first strong approximation results for multidimensional diffusions in averaging, including discrete setups, with applications to Dynkin's game value computations.

Findings

Strong coupling of $X^ ext{varepsilon}$ and $\\Xi^ ext{varepsilon}$ with error bounds

Extension of strong approximation to discrete time averaging

Application to error estimation in Dynkin's games

Abstract

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.