Stochastic HJB Equations and Regular Singular Points

TL;DR

This paper demonstrates that certain stochastic Hamilton-Jacobi-Bellman equations have regular singular points, enabling power series solutions, and introduces new algebraic and differential Riccati equations for stochastic control problems.

Contribution

It extends the theory of regular singular points in HJB equations to stochastic cases and introduces the Stochastic Algebraic Riccati Equation (SARE) and a novel method for computing higher degree solutions.

Findings

HJB equations with regular singular points can be solved by power series.

Introduction of the Stochastic Algebraic Riccati Equation (SARE).

A new degree-by-degree computation method for nonlinear stochastic control problems.

Abstract

IIn this paper we show that some HJB equations arising from both finite and infinite horizon stochastic optimal control problems have a regular singular point at the origin. This makes them amenable to solution by power series techniques. This extends the work of Al'brecht who showed that the HJB equations of an infinite horizon deterministic optimal control problem can have a regular singular point at the origin, Al'brekht solved the HJB equations by power series, degree by degree. In particular, we show that the infinite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a new type of algebraic Riccati equation which we call the Stochastic Algebraic Riccati Equation (SARE). If SARE can be solved then one has a complete solution to this infinite horizon stochastic optimal control problem. We also show that a finite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a Stochastic Differential Riccati Equation (SDRE) that is well known. If these problems are the linear-quadratic-bilinear part of a nonlinear finite horizon stochastic optimal control problem then we show how the higher degree terms of the solutions can be computed degree by degree. To our knowledge this computation is new.