Zero-Sum Risk-Sensitive Stochastic Differential Games

We study zero-sum risk-sensitive stochastic differential games on the infinite horizon with discounted and ergodic payoff criteria. Under certain assumptions, we establish the existence of values and saddle-point equilibria. We obtain our results by studying the corresponding Hamilton--Jacobi--Isaacs equations. Finally, we show that the value of the ergodic payoff criterion is a constant multiple of the maximal eigenvalue of the generators of the associated nonlinear semigroups.

[1]  K. Fan Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[2]  V. Benes Existence of Optimal Strategies Based on Specified Information, for a Class of Stochastic Decision Problems , 1970 .

[3]  N. Portenko Diffusion Processes with Unbounded Drift Coefficient , 1975 .

[4]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[5]  A. Veretennikov ON STRONG SOLUTIONS AND EXPLICIT FORMULAS FOR SOLUTIONS OF STOCHASTIC INTEGRAL EQUATIONS , 1981 .

[6]  Vivek S. Borkar Optimal Control of Diffusion Processes , 1989 .

[7]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[8]  A. Bensoussan,et al.  Some Results on Risk-Sensitive Control with Full Observation , 1995 .

[9]  W. Fleming,et al.  Risk-Sensitive Control on an Infinite Time Horizon , 1995 .

[10]  M. K. Ghosh,et al.  Stochastic differential games: Occupation measure based approach , 1996 .

[11]  S. Marcus,et al.  Risk sensitive control of Markov processes in countable state space , 1996 .

[12]  W. Fleming,et al.  Risk-Sensitive Control of Finite State Machines on an Infinite Horizon I , 1997 .

[13]  A. Bensoussan,et al.  Min-Max Characterization of a Small Noise Limit on Risk-Sensitive Control , 1997 .

[14]  Lukasz Stettner,et al.  Risk-Sensitive Control of Discrete-Time Markov Processes with Infinite Horizon , 1999, SIAM J. Control. Optim..

[15]  Risk-sensitive control of stochastic hybrid systems on infinitetime horizon , 2000 .

[16]  Sean P. Meyn,et al.  Risk-Sensitive Optimal Control for Markov Decision Processes with Monotone Cost , 2002, Math. Oper. Res..

[17]  Hideo Nagai Optimal Strategies for Risk-Sensitive Portfolio Optimization Problems for General Factor Models , 2003, SIAM J. Control. Optim..

[18]  S. Hamadène,et al.  BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations , 2003 .

[19]  Harold J. Kushner Numerical Approximations for Stochastic Differential Games: The Ergodic Case , 2004, SIAM J. Control. Optim..

[20]  J. Menaldi,et al.  Remarks on Risk-Sensitive Control Problems , 2005 .

[21]  Lukasz Stettner,et al.  Infinite Horizon Risk Sensitive Control of Discrete Time Markov Processes under Minorization Property , 2007, SIAM J. Control. Optim..

[22]  H. Kushner Numerical Methods for Stochastic Differential Games: The Ergodic Cost Criterion , 2007 .

[23]  V. Borkar,et al.  Risk-Sensitive Control with Near Monotone Cost , 2010 .

[24]  Vivek S. Borkar,et al.  Uniform Recurrence Properties of Controlled Diffusions and Applications to Optimal Control , 2010, SIAM J. Control. Optim..

[25]  A. Biswas Risk Sensitive Control of Diffusions with Small Running Cost , 2011 .

[26]  W. Fleming,et al.  On the value of stochastic differential games , 2011 .

[27]  V. Borkar Ergodic Control of Diffusion Processes , 2012 .