Representation of asymptotic values for nonexpansive stochastic control systems

TL;DR

This paper investigates the asymptotic behavior of value functions in ergodic stochastic control problems under nonexpansivity assumptions, extending classical results to more general settings including second-order HJB equations and backward stochastic differential equations.

Contribution

It introduces a new analysis framework for nonexpansive stochastic control systems, providing explicit limit formulas for the scaled value functions as the discount factor approaches zero.

Findings

Limit of λV_λ is explicitly characterized under nonexpansivity.

Extends classical results to second-order HJB equations not necessarily linked to control problems.

Includes analysis of backward stochastic differential equations with infinite horizon.

Abstract

In ergodic stochastic problems the limit of the value function $V_\lambda$ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor $\lambda$ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function $\lambda V_\lambda$ converges uniformly to a constant as $\lambda\to 0$. The objective of this work consists in studying these problems under assumptions, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also $V_\lambda$ given by a Hamilton-Jacobi-Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of $\lambda V_\lambda$, as $\lambda\to 0$.