G-stochastic maximum principle for risk-sensitive control problem and its applications

This study advances the G-stochastic maximum principle (G-SMP) from a riskneutral framework to a risk-sensitive one. A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under G-Brownian motion (G-SDEs), where the control variable may influence all terms. We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost. Initially, we introduced an auxiliary process to address risk-sensitive performance costs within the G-expectation framework. Subsequently, we established and validated the correlation between the G-expected exponential utility and the G-quadratic backward stochastic differential equation. Furthermore, we simplified the G-adjoint process from a dual-component structure to a singular component. Moreover, we explained the necessary optimality conditions for this model by considering a convex set of admissible controls. To describe the main findings, we present two examples: the first addresses the linear-quadratic problem and the second examines a Merton-type problem characterized by power utility.