Ergodic BSDE with an unbounded and multiplicative underlying diffusion and application to large time behavior of viscosity solution of HJB equation

TL;DR

This paper investigates ergodic backward stochastic differential equations with unbounded, multiplicative diffusions, establishing their large-time behavior and applying findings to the asymptotic analysis of viscosity solutions of HJB equations in control problems.

Contribution

It extends the theory of EBSDEs to cases with unbounded, multiplicative diffusions and derives explicit convergence rates for associated viscosity solutions of HJB equations.

Findings

Solutions of finite-horizon BSDEs behave linearly in time with explicit convergence rates.

Established the large-time asymptotics of viscosity solutions for HJB equations with unbounded multiplicative diffusions.

Applied EBSDE results to ergodic optimal control, demonstrating exponential convergence rates.

Abstract

In this paper, we study ergodic backward stochastic differential equations (EBSDEs for short), for which the underlying diffusion is assumed to be multiplicative and of at most linear growth. The fact that the forward process has an unbounded diffusion is balanced with an assumption of weak dissipativity for its drift. Moreover, the forward equation is assumed to be non-degenerate. Like in [HMR15], we show that the solution of a BSDE in finite horizon T behaves basically as a linear function of T, with a shift depending on the solution of the associated EBSDE, with an explicit rate of convergence. Finally, we apply our results to an ergodic optimal control problem. In particular, we show the large time behaviour of viscosity solution of Hamilton-Jacobi-Bellman equation with an exponential rate of convergence when the undelrying diffusion is multiplicative and unbounded.