Analysis and computation of an optimality equation arising in an impulse control problem with discrete and costly observations

Abstract

The framework of discrete and costly observations provides an attractive mathematical tool for stochastic control of biological and ecological systems in natural environment. However, this framework is still germinating, and its applications in these research areas are still rare. In this framework, the system dynamics to be controlled is continuous-time, while the observations by the decision-maker are discrete-time. The control problem of the system dynamics has a time-hybridized type nature, by which the optimality equation has a different form from the conventional ones. The objectives as well as contributions of this paper are formulation of the optimality equation focusing on population management problems, its detailed mathematical analysis, numerical discretization, and a demonstrative application. We show that the optimality equation is solvable uniquely and its solution can be approximated recursively. A numerical method to discretize the optimality equation is established with a simple finite difference scheme and the recursive formula. Mathematical analysis results such as monotonicity, stability, and convergence of the numerical method are discussed as well. The present model is then applied to a population management problem. Finally, an advanced problem subject to model ambiguity is also analyzed.