Maximum Principle for Stochastic Differential Games with Partial Information

Abstract

In this paper we first deal with the problem of optimal control for zero-sum stochastic differential games. We give a necessary and sufficient maximum principle for that problem with partial information. Then we use the result to solve a problem in finance. Finally, we extend our approach to general stochastic games (nonzero-sum), and obtain an equilibrium point of such game.

Leader–follower stochastic differential game with asymmetric information and applications

“…Suppose that the state of the system is described by the SDE          dx u1,u2 (t) = b t, x u1,u2 (t), u 1 (t), u 2 (t) dt + σ t, x u1,u2 (t), u 1 (t), u 2 (t) dW (t) + σ t, x u1,u2 (t), u 1 (t), u 2 (t) d W (t), x u1,u2 (0) = x 0 , (1) where u 1 (·) and u 2 (·) are control processes taken by the two players in the game, labeled 1 (the follower) and 2 (the leader), with values in nonempty convex sets U 1 ⊆ R m1 , U 2 ⊆ R m2 , respectively. x u1,u2 (·), the solution to SDE (1) with values in R n , is the corresponding state process with initial state x 0 ∈ R n . Here b(t, x, u 1 , u 2 ) : Ω × [0, T ] × R n × U 1 × U 2 → R n , σ(t, x, u 1 , u 2 ) : Ω × [0, T ] × R n × U 1 × U 2 → R n×d1 , σ(t, x, u 1 , u 2 ) : Ω × [0, T ] × R n × U 1 × U 2 → R n×d2 are given F t -adapted processes, for each (x, u 1 , u 2 ).…”

“…subject to (1) and (4). Such a u * 1 (·) = u * 1 (·; u 2 (·)) is called a (partial information) optimal control, and the corresponding solution x u * 1 ,u2 (·) to (1) is called a (partial information) optimal state process for the follower.…”

“…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.…”

Relationship between maximum principle and dynamic programming for stochastic differential games of jump diffusions

“…For this problem, Matatamvura and Øksendal (2008) related its saddle point to some HJBI equation and obtained the stochastic verification theorem. An and Øksendal (2008) proved both sufficient and necessary maximum principles, which state some conditions of optimality via the Hamiltonian function and adjoint equation. The main contribution of this paper is * Email: shijingtao@sdu.edu.cn that we connect the maximum principle of An and Øksendal (2008) with the dynamic programming of Matatamvura and Øksendal (2008), and obtain relations among the adjoint processes, the generalised Hamiltonian function and the value function under the assumption that the value function is smooth enough.…”

“…In Section 2, we state our zero-sum stochastic differential game problem of jump diffusions. Under suitable assumptions, we reformulate the sufficient maximum principle of An and Øksendal (2008) by an adjoint equation and a Hamiltonian function, and the stochastic verification theorem of Matatamvura and Øksendal (2008) by an HJBI equation. In Section 3, we prove the relationship between maximum principle and dynamic programming for our zero-sum stochastic differential game problem of jump diffusions, under the assumption that the value function is smooth 694 Jingtao Shi enough.…”