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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

[1] Evans, L. C., Some min-max methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J., 33 (1984), 31–50.CrossRefGoogle Scholar
[2] Evans, L. C. and Souganidis, P. E., Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J., 33 (1984), 773–797.Google Scholar
[3] Fleming, W. F., The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Eq., 5 (1969), 515–530.Google Scholar
[4] Friedman, A., Differential games, Wiley, N. Y., 1971.Google Scholar
[5] Friedman, A., Stochastic differential games, J. Differential Eq., 11 (1972), 79–108.Google Scholar
[6] Fujita, Y. and Morimoto, H., On bang-bang solutions of stochastic differential games, preprint.Google Scholar
[7] Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Kodansha & North-Holland, 1981.Google Scholar
[8] Krylov, N. V., Controlled Diffusion Processes, Springer Verlag 1980. (England transl.)Google Scholar
[9] Krylov, N. V., Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR Izv., 20 (1983), 459–492.Google Scholar
[10] Lions, P. L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, part 1. The dynamic programming principle and applications, Comm. P. D. E., 8 (1983), 1101–1174.Google Scholar
[11] Lions, P. L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, part 2. Viscosity solutions and uniqueness, Comm. P. D. E., 8 (1983), 1229–1276.Google Scholar
[12] Lions, P. L., Some recent results in the optimal control of diffusion processes, Stochastic Analysis, ed. Ito, K., Kinokuniya, 1984.Google Scholar
[13] Lions, P. L. and Nisio, M., A uniqueness result for semigroup associated with the Hamilton-Jacobi-Bellman operator, Proc. Japan Acad., 58 (1982), 273–276.Google Scholar
[14] Nisio, M., On stochastic optimal controls and envelope of Markovian semigroups. Proc. Intern. Symp. SDE Kyoto 1976, ed. Ito, K., Kinokuniya, 297–325.Google Scholar
[15] Nisio, M., Stochastic Control Theory, ISI Lect. Notes 9, Macmillan India (1981).Google Scholar
[16] Skorokhod, V., Studies in the Theory of Random Processes, Scrip. Tech. 1965. (English transl.)Google Scholar
[17] Strook, D. W. and Varadhan, S. R. S., Multidimensional Diffusion Processes, Springer Verlag, 1979.Google Scholar