A generalized stochastic control problem of bounded noise process under ambiguity arising in biological management

Abstract: The objectives and contributions of this paper are mathematical and numerical analyses of a stochastic control problem of bounded population dynamics under ambiguity, an important but not well-studied problem, focusing on the optimality equation as a nonlinear degenerate parabolic partial integro-differential equation (PIDE). The ambiguity comes from lack of knowledge on the continuous and jump noises in the dynamics, and its optimization appears as nonlinear and nonlocal terms in the PIDE. Assuming a strong dynamic programming principle for continuous value functions, we characterize its solutions from both viscosity and distribution viewpoints. Numerical computation focusing on an ergodic case are presented as well to complement the mathematical analysis.