Global Conference on Mathematics and Applications (GCMA)

This paper investigates a class of risk-sensitive stochastic optimal control problems formulated on the Wasserstein space, where the control consists of both continuous and impulsive components. The controlled dynamics and cost functional depend nonlinearly on the state process and its probability distribution and are described by mean-field forward-backward stochastic differential equations (FBSDEs). The objective is given by an exponential integral cost functional, with the parameter θ characterizing the decision maker’s attitude toward risk: risk aversion for θ > 0, risk neutrality for θ = 0, and risk seeking for θ < 0. Under a convex control domain and suitable regularity and boundedness assumptions on the system coefficients, we establish variational equations and derive the corresponding convergence and variational inequality results. Based on these results, we formulate a risk-sensitive stochastic maximum principle of Pontryagin type, providing necessary optimality conditions in terms of an appropriate Hamiltonian and adjoint equations associated with both continuous and impulsive controls. Furthermore, under additional convexity assumptions on the running and terminal cost functions, we prove sufficient optimality conditions and establish a verification theorem characterizing optimal control pairs. The obtained results extend stochastic maximum principle techniques to a risk-sensitive mean-field control framework with impulse effects on the Wasserstein space. They provide a theoretical basis for analyzing stochastic control problems in which distributional dependence, risk sensitivity, and impulsive interventions are simultaneously present, with potential applications in finance, engineering, and economics.