Value functions and the Dirichlet problem for Isaacs equation in a smooth domain
HTML articles powered by AMS MathViewer
- by Jay Kovats PDF
- Trans. Amer. Math. Soc. 361 (2009), 4045-4076 Request permission
Abstract:
In this paper, we investigate probabilistic solutions of the Dirichlet problem for the elliptic Isaacs equation in a smooth bounded domain in Euclidean space.References
- A. Bensoussan and J.-L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Méthodes Mathématiques de l’Informatique, No. 6, Dunod, Paris, 1978 (French). MR 0513618
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- Xavier Cabré and Luis A. Caffarelli, Interior $C^{2,\alpha }$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl. (9) 82 (2003), no. 5, 573–612 (English, with English and French summaries). MR 1995493, DOI 10.1016/S0021-7824(03)00029-1
- E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der mathematischen Wissenschaften, Band 121, vol. 122, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR 0193671, DOI 10.1007/978-3-662-00031-1
- L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J. 33 (1984), no. 5, 773–797. MR 756158, DOI 10.1512/iumj.1984.33.33040
- W. H. Fleming and M. Nisio, Differential games for stochastic partial differential equations, Nagoya Math. J. 131 (1993), 75–107. MR 1238634, DOI 10.1017/S0027763000004554
- W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38 (1989), no. 2, 293–314. MR 997385, DOI 10.1512/iumj.1989.38.38015
- Avner Friedman, Differential games, Pure and Applied Mathematics, Vol. XXV, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London, 1971. MR 0421700
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Rufus Isaacs, Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0210469
- H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), no. 1, 26–78. MR 1031377, DOI 10.1016/0022-0396(90)90068-Z
- Hitoshi Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42 (1989), no. 1, 15–45. MR 973743, DOI 10.1002/cpa.3160420103
- Markos A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations, Nonlinear Anal. 24 (1995), no. 2, 147–158. MR 1312585, DOI 10.1016/0362-546X(94)00170-M
- N. V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR 601776, DOI 10.1007/978-1-4612-6051-6
- N. V. Krylov, Smoothness of the payoff function for a controllable diffusion process in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 66–96 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 65–95. MR 992979, DOI 10.1070/IM1990v034n01ABEH000603
- —, On Controlled Diffusion Processes with Unbounded Coefficients, vol. 19, Izv. Acad. Nauk. SSSR Ser. Mat., 1982, pp. 41-64, English transl. in Math. USSR Izv..
- P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Comm. Partial Differential Equations 8 (1983), no. 10, 1101–1174. MR 709164, DOI 10.1080/03605308308820297
- Pierre-Louis Lions and José-Luis Menaldi, Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations. I, II, SIAM J. Control Optim. 20 (1982), no. 1, 58–81, 82–95. MR 642179, DOI 10.1137/0320006
- Makiko Nisio, Stochastic differential games and viscosity solutions of Isaacs equations, Nagoya Math. J. 110 (1988), 163–184. MR 945913, DOI 10.1017/S0027763000002932
- Makiko Nisio, Some remarks on stochastic optimal controls, Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975) Lecture Notes in Math., Vol. 550, Springer, Berlin, 1975, pp. 446–460. MR 0439373
- —, Stochastic Control Theory, ISI Lecture Notes 9, Macmillan, India, 1981.
- Andrzej Święch, Another approach to the existence of value functions of stochastic differential games, J. Math. Anal. Appl. 204 (1996), no. 3, 884–897. MR 1422779, DOI 10.1006/jmaa.1996.0474
- Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR 532498
Additional Information
- Jay Kovats
- Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901
- MR Author ID: 635359
- Email: jkovats@fit.edu
- Received by editor(s): May 7, 2007
- Published electronically: April 1, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 4045-4076
- MSC (2000): Primary 35B65, 35J60, 49N70, 91A05
- DOI: https://doi.org/10.1090/S0002-9947-09-04732-1
- MathSciNet review: 2500878