Explicit Rates of Exponential Convergence for Reflected Jump-Diffusions on the Half-Line

TL;DR

This paper derives explicit exponential convergence rates for reflected jump-diffusions on the half-line using Lyapunov functions, extending previous work and applying results to Levy particle systems with rank-dependent dynamics.

Contribution

It provides explicit convergence rate estimates for reflected jump-diffusions and applies these to systems of Levy particles with rank-dependent interactions.

Findings

Explicit exponential convergence rates derived.

Application to Levy particle systems with rank dependence.

Extension of Lyapunov function methods to jump-diffusions.

Abstract

Consider a reflected jump-diffusion on the positive half-line. Assume it is stochastically ordered. We apply the theory of Lyapunov functions and find explicit estimates for the rate of exponential convergence to the stationary distribution, as time goes to infinity. This continues the work of Lund, Meyn and Tweedie (1996). We apply these results to systems of two competing Levy particles with rank-dependent dynamics.