Stochastic Control for Small Noise Intensities

Abstract

This paper is concerned with the approximate solution of stochastic optimal control problems which arise by perturbing the system equations in the deterministic Pontryagin control model, through an additive white noise term with small coefficient. The system states are assumed completely observable. Mathematically the problem becomes one of singular perturbation of the Hamilton–Jacobi equation by a small second order term. Our main results concern expansions of solutions of the perturbed equation in powers $\varepsilon ,\varepsilon ^2 ,\varepsilon ^3 , \cdots $ of the noise variance coefficients. The results obtained hold in regions where the corresponding solution of the Hamilton-Jacobi equation is sufficiently well-behaved.