Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations

In this paper we study zero-sum two-player stochastic differential games with the help of theory of Backward Stochastic Differential Equations (BSDEs). At the one hand we generalize the results of the pioneer work of Fleming and Souganidis by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game (which implies that the cost functionals become random variables), on the other hand the application of BSDE methods, in particular that of the notion of stochastic "backward semigroups" introduced by Peng allows to prove a dynamic programming principle for the upper and the lower value functions of the game in a straight-forward way, without passing by additional approximations. The upper and the lower value functions are proved to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method is translated from the framework of stochastic control theory into that of stochastic differential games.Comment: The results were presented by Rainer Buckdahn at the "12th International Symposium on Dynamic Games and Applications" in Sophia-Antipolis (France) in June 2006; They were also reported by Juan Li at 2nd Workshop on "Stochastic Equations and Related Topics" in Jena (Germany) in July 2006 and at one seminar in the ETH of Zurich in November 200

Stochastic control problems for systems driven by normal martingales

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener--Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.Comment: Published in at http://dx.doi.org/10.1214/07-AAP467 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Regularity properties for general HJB equations. A BSDE method

In this work we investigate regularity properties of a large class of Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in form of a stochastic control system which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE (RBSDE). More precisely, we prove that, firstly, the unique viscosity solution $V(t,x)$ of such a HJB equation over the time interval $[0,T],$ with or without an obstacle, and with terminal condition at time $T$, is jointly Lipschitz in $(t,x)$, for $t$ running any compact subinterval of $[0,T)$. Secondly, for the case that $V$ solves a HJB equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that $V(t,x)$ is jointly semiconcave in $(t,x)$. These results extend earlier ones by Buckdahn, Cannarsa and Quincampoix [1]. Our approach embeds their idea of time change into a BSDE analysis. We also provide an elementary counter-example which shows that, in general, for the case that $V$ solves a HJB equation with a lower obstacle the semi-concavity doesn't hold true.Comment: 30 page

Differential games with asymmetric information and without Isaacs condition

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs condition. The dynamics is an ordinary differential equation parametrised by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. Moreover, the value function can be characterised in term of the unique viscosity solution in some dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the Isaacs condition which is usually assumed in differential games