Stochastic representation for solutions of Isaacs’ type integral–partial differential equations

In this paper we study the integral–partial differential equations of Isaacs’ type by zero-sum two-player stochastic differential games (SDGs) with jump-diffusion. The results of Fleming and Souganidis (1989) [9] and those of Biswas (2009) [3] are extended, we investigate a controlled stochastic system with a Brownian motion and a Poisson random measure, and with nonlinear cost functionals defined by controlled backward stochastic differential equations (BSDEs). Furthermore, unlike the two papers cited above the admissible control processes of the two players are allowed to rely on all events from the past. This quite natural generalization permits the players to consider those earlier information, and it makes more convenient to get the dynamic programming principle (DPP). However, the cost functionals are not deterministic anymore and hence also the upper and the lower value functions become a priori random fields. We use a new method to prove that, indeed, the upper and the lower value functions are deterministic. On the other hand, thanks to BSDE methods (Peng, 1997) [18] we can directly prove a DPP for the upper and the lower value functions, and also that both these functions are the unique viscosity solutions of the upper and the lower integral–partial differential equations of Hamilton–Jacobi–Bellman–Isaacs’ type, respectively. Moreover, the existence of the value of the game is got in this more general setting under Isaacs’ condition.

[1]  Juan Li,et al.  Stochastic Differential Games and Viscosity Solutions of Hamilton--Jacobi--Bellman--Isaacs Equations , 2008, SIAM J. Control. Optim..

[2]  J. Lepeltier,et al.  Zero-sum stochastic differential games and backward equations , 1995 .

[3]  Imran H. Biswas.  On Zero-Sum Stochastic Differential Games with Jump-Diffusion Driven State: A Viscosity Solution Framework , 2010, SIAM J. Control. Optim..

[4]  Shanjian Tang,et al.  Switching Games of Stochastic Differential Systems , 2007, SIAM J. Control. Optim..

[5]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[6]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[7]  S. Shreve,et al.  Methods of Mathematical Finance , 2010 .

[8]  G. Barles,et al.  Second-order elliptic integro-differential equations: viscosity solutions' theory revisited , 2007, math/0702263.

[9]  Catherine Rainer,et al.  Nash Equilibrium Payoffs for Nonzero-Sum Stochastic Differential Games , 2004, SIAM J. Control. Optim..

[10]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[11]  Rainer Buckdahn,et al.  Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers , 2008 .

[12]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[13]  S. Peng.  A Generalized dynamic programming principle and hamilton-jacobi-bellman equation , 1992 .

[14]  Xunjing Li,et al.  Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps , 1994 .

[15]  B. Øksendal,et al.  Risk minimizing portfolios and HJBI equations for stochastic differential games , 2008 .

[16]  Shige Peng,et al.  Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations , 2009 .

[17]  G. Barles,et al.  Backward stochastic differential equations and integral-partial differential equations , 1997 .