Long time behavior of semi-Markov modulated perpetuity and some related processes

TL;DR

This paper investigates the long-term behavior of certain stochastic processes influenced by semi-Markov environments, revealing mixture laws and explicit representations for processes like bifurcations and regime-switching diffusions.

Contribution

It introduces new limit results for semi-Markov modulated processes and connects them to affine stochastic recurrence equations, providing explicit long-term process representations.

Findings

Mixture laws emerge in long-term limits under various conditions.

Explicit representations of processes like bifurcations and regime-switching diffusions.

Different asymptotic behaviors depending on the signs of expected values and tail properties.

Abstract

Examples of stochastic processes whose state space representations involve functions of an integral type structure $$I_{t}^{(a,b)}:=\int_{0}^{t}b(Y_{s})e^{-\int_{s}^{t}a(Y_{r})dr}ds, \quad t\ge 0$$ are studied under an ergodic semi-Markovian environment described by an $S$ valued jump type process $Y:=(Y_{s}:s\in\mathbb{R}^{+})$ that is ergodic with a limiting distribution $\pi\in\mathcal{P}(S)$. Under different assumptions on signs of $E_{\pi}a(\cdot):=\sum_{j\in S}\pi_{j}a(j)$ and tail properties of the sojourn times of $Y$ we obtain different long time limit results for $I^{(a,b)}_{}:=(I^{(a,b)}_{t}:t\ge 0).$ In all cases mixture type of laws emerge which are naturally represented through an affine stochastic recurrence equation (SRE) $X\stackrel{d}{=}AX+B,\,\, X\perp\!\!\!\perp (A, B)$. Examples include explicit long-time representations of pitchfork bifurcation, and regime-switching diffusions under semi-Markov modulated environments, etc.