Relationship between Maximum Principle and Dynamic Programming in Stochastic Differential Games and Applications

Abstract

This paper is concerned with the relationship between maximum principle and dynamic programming in zero-sum stochastic differential games. Under the assumption that the value function is smooth enough, relations among the adjoint processes, the generalized Hamiltonian function and the value function are given. A portfolio optimization problem under model uncertainty in the financial market is discussed to show the applications of our result.

Leader–follower stochastic differential game with asymmetric information and applications

“…First, the game problem has the Stackelberg or leader-follower feature, which means the two players act as different roles during the game. Thus the usual approach to deal with game problems, such as Yong [25], Hamadène [7], Wu [23], An and Øksendal [1], Wang and Yu [21], Yu [29], Hui and Xiao [10,11], Shi [17] where the two players act as equivalent roles, does not apply. Second, the game problem has the asymmetric information between the two players, which was not considered in Yong [26], Øksendal et al [15] and Bensoussan et al [5].…”

“…See the monographs by Basar and Olsder [4] for more information about differential games. For some most recent developments for stochastic differential games and their applications, please refer to Yong [25], Hamadène [7], Wu [23], An and Øksendal [1], Buckdahn and Li [6], Wang and Yu [20,21], Yu [29], Hui and Xiao [10,11], Shi [17] and the references therein.…”