Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes

Kenji Handa

doi:10.1214/ejp.v19-2928

Abstract

We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.

Highlights

The study of ergodic behaviors of a Markov process is of quite interest for various reasons
A key idea there is to exploit a special relationship with measure-valued α-CIR models, which enabled us to give an expression for stationary distributions of our generalized Fleming-Viot processes in terms of those of the measure-valued α-CIR models
As α ↑ 1, LαF (z) → L1F (z) for any z > 0 and ‘nice’ functions F on R+, we call a Markov process associated with aLα for some constant a > 0 an α-CIR model

Summary

Introduction

The study of ergodic behaviors of a Markov process is of quite interest for various reasons. As α ↑ 1, LαF (z) → L1F (z) for any z > 0 and ‘nice’ functions F on R+, we call a Markov process associated with aLα for some constant a > 0 an α-CIR model This class of models would be of interest in its own right especially in the mathematical finance context, our main motivation to study it is the analysis of ergodicity for a jump-type version of a Wright-Fisher diffusion model with mutation, which is obtained through normalization and random time-change from two independent processes with generators of the form (1.2), say L′α and L′α′ , with common α and b.

The measure-valued α-CIR models

Associated Dirichlet forms

Spectral gap for the α-CIR model

An application to generalized Fleming-Viot processes