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

TL;DR

This paper investigates a stochastic control problem involving bounded population dynamics under ambiguity, analyzing the associated nonlinear PIDE and providing numerical solutions for ergodic cases.

Contribution

It introduces a novel mathematical framework for controlling population dynamics under ambiguity, characterizing solutions via viscosity and distribution methods, and offers numerical approaches for ergodic scenarios.

Findings

Characterization of solutions to the nonlinear PIDE under ambiguity

Development of numerical methods for ergodic control cases

Insights into bounded noise processes 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.