Long time behavior of semi-Markov modulated perpetuity and some related processes
Abhishek Pal Majumder
TL;DR
This work analyzes the long-time behavior of a class of semi-Markov modulated integral processes, I^{(a,b)}_t, driven by a countable state environment Y. The authors derive explicit limit laws, revealing mixture-type asymptotics governed by an affine stochastic recurrence equation, with distinct regimes determined by the mean drift E_{\pi}a(\cdot) and tail properties of the sojourn times. In the stable regime (E_{\pi}a(\cdot)>0) the joint limit is a mixture where I^{(a,b)}_t is driven by J-storehouse dynamics Z_j built from a Markov renewal structure; in divergent/critical regimes they obtain Gaussian or stable limits, including heavy-tailed, regularly varying cases, contingent on the heaviest tail among the sojourns. Applications to pitchfork bifurcation and regime-switching generalized OU processes illustrate how these limits yield explicit, parametric descriptions of stationary or limiting behavior under semi-Markov modulation, with potential utility in finance, physics, and engineering. The core technical contribution is a regenerative structure based lemma (Lemma 1) enabling asymptotic independence in the limit, together with thorough handling of both finite-variance and heavy-tailed regimes via Goldie–Kesten theory and stable limit theorems for stopped renewal processes. Overall, the paper provides exact, mixture-based characterizations of long-run behavior for a broad class of semi-Markov modulated functionals, highlighting universal features in the divergent cases and the persistence of stochastic effects in the stable domain.
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 $π\in\mathcal{P}(S)$. Under different assumptions on signs of $E_πa(\cdot):=\sum_{j\in S}π_{j}a(j)$ 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.