论文标题
在Dynkin游戏中的平均和价值计算中的强扩散近似
Strong diffusion approximation in averaging and value computation in Dynkin's games
论文作者
论文摘要
众所周知，慢动作$ x^\ varepsilon $在时间尺度的多维平均设置中b（x^\ varepsilon（t），\，ξ（t/\ varepsilon^2））+b（x^\ varepsilon（t），\，ξ（t/\ ve^2）），\，\，t \ in [0，t]在[0，t] $弱收集为$ \ varepsilon \ varepsilon \ varepsil \ varepsil \ diffusion a diffusion a diffuse a diffuse $ eb（x，ξ（s））\ equiv 0 $，其中$ξ$是一个足够快的随机过程。在本文中，我们表明，$ x^\ varepsilon $和一个扩散家族$ξ^\ varepsilon $都可以在常见的足够丰富的概率空间上重新定义，以便$ e \ e \ sup_ {0 \ leq t \ leq t \ leq t \ leq t} | x^\ varepsilon（x^\ varepsilon（t） - c（m）\ varepsilon^\ del $用于某些$ c（m），δ> 0 $和所有$ m \ ge 1，\，\，\ varepsilon> 0 $，其中所有$ξ^\ varepsilon，\ \ \ varepsilon> 0 $都具有相同的扩散系数，但与$ $差异$差异。这是上述设置中的第一个强近似结果，并且在极限是非平凡的多维扩散时。我们还为相应的离散时间平均设置获得了类似的结果，该设置根本没有考虑。作为一个应用程序，我们考虑使用涉及扩散的路径收益的Dynkin的游戏，并通过此类离散时间近似值获得该游戏值的错误估计，该近似值比扩散本身的标准离散化提供了更有效的计算工具。
It is known that the slow motion $X^\varepsilon$ in the time-scaled multidimensional averaging setup $\frac {dX^\varepsilon(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\,ξ(t/\varepsilon^2))+b(X^\varepsilon(t),\,ξ(t/\ve^2)),\, t\in [0,T]$ converges weakly as $\varepsilon\to 0$ to a diffusion process provided $EB(x,ξ(s))\equiv 0$ where $ξ$ is a sufficiently fast mixing stochastic process. In this paper we show that both $X^\varepsilon$ and a family of diffusions $Ξ^\varepsilon$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-Ξ^\varepsilon(t)|^{2M}\leq C(M)\varepsilon^\del$ for some $C(M),δ>0$ and all $M\ge 1,\,\varepsilon>0$, where all $Ξ^\varepsilon,\, \varepsilon>0$ 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.
