Abstract
We consider a nonzero-sum stochastic differential game which involves two players, a controller and a stopper. The controller chooses a control process, and the stopper selects the stopping rule which halts the game. This game is studied in a jump diffusions setting within Markov control limit. By a dynamic programming approach, we give a verification theorem in terms of variational inequality-Hamilton-Jacobi-Bellman (VIHJB) equations for the solutions of the game. Furthermore, we apply the verification theorem to characterize Nash equilibrium of the game in a specific example.
Highlights
In this paper we study a nonzero-sum stochastic differential game with two players: a controller and a stopper
We study a nonzero-sum stochastic differential game between a controller and a stopper
We prove a verification theorem in terms of VIHJB equations for the game to characterize Nash equilibrium
Summary
In this paper we study a nonzero-sum stochastic differential game with two players: a controller and a stopper. In [6,7,8,9] the authors considered the zero-sum stochastic differential games of mixed type with both controls and stopping between two players. We study a nonzero-sum stochastic differential game between a controller and a stopper. The paper is organized as follows: we formulate the nonzero-sum stochastic differential game between a controller and a stopper and prove a general verification theorem. 2. A Verification Theorem for Nonzero-Sum Stochastic Differential Game between Controller and Stopper. A pair (u∗, τ∗) ∈ A × Γ is called a Nash equilibrium for the stochastic differential game (9) and (13), if the following holds: Jy1 (u∗, τ∗) ≥ Jy1 (u, τ∗) , ∀u ∈ U, y ∈ S, (14).
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