Numerical Approximations for Stochastic Differential Games
Numerical Approximations for Stochastic Differential Games
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DOI：
10.1137/s0363012901389457
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发表时间：
2002-02
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影响因子：
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通讯作者：
H. Kushner
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文献类型：
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作者：
H. Kushner
马尔可夫链近似方法是一种广泛使用、稳健、相对易于使用且高效的方法族，适用于反射跳跃扩散型模型的连续时间内的大量随机控制问题。它已被证明在广泛的条件下是收敛的，并且如果维数不太高，有很好的算法可以解决数值问题。这些方法的各个版本长期以来一直被用于各种两人微分和随机动态博弈的应用中，并且在某些情况下可以提供收敛证明，主要使用偏微分方程（PDE）类型的技术。在本文中，针对一大类此类问题给出了纯概率收敛证明，其中两个参与者的控制在动力学和成本函数中是分开的，并且涵盖了先前作品中未处理的大量类别。考虑折扣和停止时间成本函数。有限视野问题和过程在第一次触及先验给定边界时停止的问题可以通过采用[H. J. Kushner 和 P. Dupuis，《随机控制问题的数值方法》，连续时间，第 2 版，Springer-Verlag，柏林，纽约，2001 年] 正如本文针对处理的问题所做的那样。本质条件是解的弱意义存在性和唯一性，“几乎处处”的连续性条件，以及弱局部一致性条件对于数值近似而言“几乎处处”成立，就像控制问题一样。受控方差和跳跃问题还有一些扩展。
The Markov chain approximation method is a widely used, robust, relatively easy to use, and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected-jump-diffusion-type models. It has been shown to converge under broad conditions, and there are good algorithms for solving the numerical problems if the dimension is not too high. Versions of these methods have been used in applications to various two-player differential and stochastic dynamic games for a long time, and proofs of convergence are available for some cases, mainly using PDE-type techniques. In this paper, purely probabilistic proofs of convergence are given for a broad class of such problems, where the controls for the two players are separated in the dynamics and cost function, and which cover a substantial class not dealt with in previous works. Discounted and stopping time cost functions are considered. Finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of [H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems, in Continuous Time, 2nd ed., Springer-Verlag, Berlin, New York, 2001] as done in this paper for the treated problems. The essential conditions are the weak-sense existence and uniqueness of solutions, an "almost everywhere" continuity condition, and that a weak local consistency condition holds "almost everywhere" for the numerical approximations, just as for the control problem. There are extensions to problems with controlled variance and jumps.