Infinite horizon Stackelberg differential games with random coefficients under control input constraint

Abstract

In this paper, we study Stackelberg stochastic differential game driven by stochastic differential equation (SDE for short) with random coefficients on infinite horizon under convex control constraint. The control enters into the drift and diffusion terms simultaneously. By employing the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to announce Pontryagin's maximum principle for the leader's global Stackelberg solution, within adapted open-loop structure, where the term global means that the leader's domination over the entire game duration. Due to the follower's adjoint equation develops into a BSDE, the leader will be confronted with a control problem where the state equation is a kind of infinite horizon fully coupled forward-backward stochastic differential equation (FBSDE for short). We shed light on an important application in a class of infinite horizon linear-quadratic (LQ for short) Stackelberg games in which the control process of leader's is constrained in a closed convex subset Γ 1 of full space R m 1 . Whenever the control domain is postulated to be full space, we derive a new kind of infinite horizon backward stochastic Riccati equation (BSRE for short) with asymmetric coefficients.