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

Examples of stochastic processes whose state space representations involve functions of an integral type structure $$I_t^{(a,b)}:={\int }_0^{t}b(Y_s)e^{-{\int }_s^{t}a(Y_r)dr}ds,t\ge 0$$ 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 $I_{}^{(a,b)}:=(I_t^{(a,b)}:t\ge 0)\mathrm{.}$ In all cases mixture type of laws emerge which are naturally represented through an affine stochastic recurrence equation (SRE) $XAX+B,X\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp (A,B)$. Examples include explicit long-time representations of pitchfork bifurcation, and regime-switching diffusions under semi-Markov modulated environments, etc.