Backward Stackelberg Games with Delay and Related Forward-Backward Stochastic Differential Equations

In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear-quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward-backward stochastic differential delayed Equation (AFBSDDE), which is composed of anticipated stochastic differential equations (ASDEs) and BSDDEs. Moreover, we obtain the unique solvability of the AFBSDDE using the continuation method. As an application of the theoretical results, the pension fund problem with delay effect is considered.

Automated summary

In this paper, the authors study a Stackelberg game with controlled systems described by backward stochastic differential delayed equations (BSDDEs). They introduce a new adjoint equation and establish the sufficient verification theorem for the leader and follower's optimal strategies in a general case. They also focus on a linear-quadratic (LQ) backward Stackelberg game with delay and present the backward Stackelberg equilibrium using the generalized fully coupled anticipated forward-backward stochastic differential delayed equation (AFBSDDE). The unique solvability of the AFBSDDE is obtained using the continuation method. Theoretical results are applied to the pension fund problem with delay effect.