Risk-sensitive singular control for stochastic recursive systems and Hamilton-Jacobi-Bellman inequality

This paper investigates the risk-sensitive recursive utility control problem, where the system is governed by both regular controls and singular controls. The main feature of this problem is that the cost functional is given by a backward stochastic differential equation (BSDE) with quadratic growth, driven by a discontinuous semimartingale. We examine the existence and uniqueness of solutions to the BSDE, as well as the comparison theorem and stability. From this, we derive the continuity of the value function with respect to the initial state. Additionally, using the dynamic programming principle (DPP), we demonstrate that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman (HJB) inequality. Finally, we establish the connection between the DPP and the maximum principle for the risk-sensitive recursive utility singular control problem.