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Stochastic differential games and viscosity solutions of Isaacs equations

Published online by Cambridge University Press:  22 January 2016

Makiko Nisio
Makiko Nisio*
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe 657, Japan
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Recently P. L. Lions has demonstrated the connection between the value function of stochastic optimal control and a viscosity solution of Hamilton-Jacobi-Bellman equation [cf. 10, 11, 12]. The purpose of this paper is to extend partially his results to stochastic differential games, where two players conflict each other. If the value function of stochatic differential game is smooth enough, then it satisfies a second order partial differential equation with max-min or min-max type nonlinearity, called Isaacs equation [cf. 5]. Since we can write a nonlinear function as min-max of appropriate affine functions, under some mild conditions, the stochastic differential game theory provides some convenient representation formulas for solutions of nonlinear partial differential equations [cf. 1, 2, 3].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 110 , June 1988 , pp. 163 - 184
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

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