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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.<\/jats:p>","DOI":"10.1137\/s0363012901389457","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"457-486","source":"Crossref","is-referenced-by-count":43,"title":["Numerical Approximations for Stochastic Differential Games"],"prefix":"10.1137","volume":"41","author":[{"given":"Harold J.","family":"Kushner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998333517"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009855"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(98)00164-2"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"MartinoBardi, SandraBottacin, MaurizioFalcone, Convergence of discrete schemes for discontinuous value functions of pursuit\u2010evasion games, Ann. 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