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

Abstract

It is known since (Theory Probab. Appl. 11 (1966) 390–406) that the slow motion ${\mathit{X}}^{\mathit{\epsilon }}$ in the time-scaled multidimensional averaging setup

$$\frac{\mathit{d}{\mathit{X}}^{\mathit{\epsilon }}(\mathit{t})}{\mathit{d}\mathit{t}}=\frac{1}{\mathit{\epsilon }}\mathit{B}({\mathit{X}}^{\mathit{\epsilon }}(\mathit{t}),\mathit{\xi }(\mathit{t}/{\mathit{\epsilon }}^2\left)\right)\mathbf{+}\mathit{b}({\mathit{X}}^{\mathit{\epsilon }}(\mathit{t}),\mathit{\xi }(\mathit{t}/{\mathit{\epsilon }}^2\left)\right),\mathit{t}\in [0,\mathit{T}]$$

converges weakly as $\mathit{\epsilon }\to 0$ to a diffusion process provided $\mathit{E}\mathit{B}(\mathit{x},\mathit{\xi }(\mathit{s}))\equiv 0$ where ξ is a sufficiently fast mixing stochastic process. In this paper we show that both ${\mathit{X}}^{\mathit{\epsilon }}$ and a family of diffusions ${\mathrm{\Xi }}^{\mathit{\epsilon }}$ can be redefined on a common sufficiently rich probability space so that $\mathit{E}{sup}_{0\le \mathit{t}\le \mathit{T}}|{\mathit{X}}^{\mathit{\epsilon }}(\mathit{t})-{\mathrm{\Xi }}^{\mathit{\epsilon }}(\mathit{t}){|}^{2\mathit{M}}\le \mathit{C}(\mathit{M}){\mathit{\epsilon }}^{\mathit{\delta }}$ for some $\mathit{C}(\mathit{M}),\mathit{\delta }>0$ and all $\mathit{M}\ge 1$, $\mathit{\epsilon }>0$, where all ${\mathrm{\Xi }}^{\mathit{\epsilon }},\mathit{\epsilon }>0$ have the same diffusion coefficients but underlying Brownian motions may change with ε. 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.