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Differential games for stochastic partial differential equations

Published online by Cambridge University Press:  22 January 2016

W.H. Fleming and
M. Nisio
W.H. Fleming
Affiliation:
Division of Applied Mathematics Brown University, Providence R. I,. 02912, USA
M. Nisio
Affiliation:
Department of Mathematics Kobe University, Rokko, Kobe 657, Japan, Department of Mathematics Osaka Electro-Communication University, Hatsue ho, Neyagawa 572, Japan
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In this paper we are concerned with zero-sum two-player finite horizon games for stochastic partial differential equations (SPDE in short). The main aim is to formulate the principle of dynamic programming for the upper (or lower) value function and investigate the relationship between upper (or lower) value function and viscocity solution of min-max (or max-min) equation on Hilbert space.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 131 , September 1993, pp. 75 - 107
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[1] Aronson, D.G., Non-negative solution of linear parabolic equation, Ann. della Scuola Norm. Sup. Pisa, 22 (1968), 607–694.Google Scholar
[2] Crandall, M.G. and Lions, P.L., Hamilton-Jacobi equations in infinite dimensions, Part 4, J. Funct. Anal., 90 (1990), 273–283, Part 5, 97 (1991), 417–465.Google Scholar
[3] Crandall, M.G., Ishii, H. and Lions, P.L., A user’s guide to viscosity solutions, Bull. Amer. Math. Soc., N. S., 27 (1992), 1–67.Google Scholar
[4] Fleming, W.H. and Souganidis, P.E., On the existence of value function of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J., 38, (1989), 293–314.CrossRefGoogle Scholar
[5] Fleming, W.H. and Soner, H.M., Controlled Markov Processes and Viscosity Solutions, Springer Verlag, 1992.Google Scholar
[6] Ishii, H. Viscosity solution of nonlinear second-order partial differential equation in Hilbert spaces, Preprint.Google Scholar
[7] Krylov, N.V. and Rozovskii, B.L.. On the Cauchy problem for linear stochatic partial differential equations, Math. USSR Izv., 11 (1977), 1267-1284.Google Scholar
[8] Lions, P.L., Viscosity solutions and optimal stochastic control in infinite dimensions, Part 1, Acta Math., 161 (1988), 243–278, Part 2 L. N. M., 1390 (1988), 147–170, Part 3, J. Funct. Anal., 86 (1989), 1–18.Google Scholar
[9] Nisio, M. Optimal control for stochastic differential equations and viscosity solutions of Bellman equations, Nagoya Math. J., 123 (1991), 13–37.CrossRefGoogle Scholar
[10] Pardoux, E. Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127–167.Google Scholar