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Long time behavior of semi-Markov modulated perpetuity and some related processes (2410.15824v2)

Published 21 Oct 2024 in math.PR, math.DS, q-fin.MF, and stat.CO

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.

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