Nonzero-Sum Risk-Sensitive Stochastic Differential Games: A Multi-parameter Eigenvalue Problem Approach

. We study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criterion. Under certain conditions, using multi parameter eigenvalue approach, we establish the existence of a Nash equilibrium in the space of stationary Markov strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Exploiting the stochastic representation of the principal eigenfunctions we completely characterize Nash equilibrium points in the space of stationary Markov strategies.

[1]  Somnath Pradhan Risk-sensitive zero-sum stochastic differential game for jump-diffusions , 2021, Syst. Control. Lett..

[2]  Ari Arapostathis,et al.  On the policy improvement algorithm for ergodic risk-sensitive control , 2019, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  Mrinal K. Ghosh,et al.  A nonzero-sum risk-sensitive stochastic differential game in the orthant , 2021, Mathematical Control & Related Fields.

[4]  M. K. Ghosh,et al.  Ergodic risk-sensitive stochastic differential games with reflecting diffusions in a bounded domain , 2020 .

[5]  Ari Arapostathis,et al.  A Variational Formula for Risk-Sensitive Control of Diffusions in ℝd , 2020, SIAM J. Control. Optim..

[6]  Somnath Pradhan Risk-Sensitive Ergodic Control of Reflected Diffusion Processes in Orthant , 2019, Applied Mathematics & Optimization.

[7]  A. Arapostathis,et al.  Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control , 2017, Journal de Mathématiques Pures et Appliquées.

[8]  A. Arapostathis A counterexample to a nonlinear version of the Kreĭn–Rutman theorem by R. Mahadevan , 2018, Nonlinear Analysis.

[9]  A. Biswas,et al.  Zero-Sum Stochastic Differential Games with Risk-Sensitive Cost , 2017, 1704.02689.

[10]  A. Arapostathis,et al.  Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions , 2016, 1601.00258.

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

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

[13]  P. Schrimpf,et al.  Dynamic Programming , 2011 .

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

[15]  B. Sirakov,et al.  Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators , 2008 .

[16]  M. Kocan,et al.  On Viscosity Solutions of Fully Nonlinear Equations with Measurable Ingredients , 2007 .

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

[18]  A. Nowak Notes on Risk-Sensitive Nash Equilibria , 2005 .

[19]  Robust control of discrete-time hybrid systems with uncertain modal dynamics , 2004 .

[20]  Hideo Nagai,et al.  Optimal Strategies for Risk-Sensitive Portfolio Optimization Problems for General Factor Models , 2002, SIAM J. Control. Optim..

[21]  T. Başar Nash Equilibria of Risk-Sensitive Nonlinear Stochastic Differential Games , 1999 .

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

[23]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

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

[25]  Some results on risk-sensitive control with full observation , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

[27]  P. Whittle Risk-Sensitive Optimal Control , 1990 .

[28]  N. Okada On the Banach-Saks property , 1984 .

[29]  Robert J. Elliott,et al.  Optimal Play in a Stochastic Differential Game , 1981 .

[30]  P. Varaiya N-player stochastic differential games , 1976 .

[31]  Rhodes,et al.  Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games , 1973 .

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

[33]  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.