Paper: arXiv 2410.15824

Authors: Abhishek Pal Majumder

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers rigorous mathematical proofs for long-time limit behaviors of semi-Markov modulated perpetuities and related processes, solving several open theoretical problems. However, it is purely theoretical and lacks empirical backtesting or real-world dataset validations beyond motivating references.

Research Flowchart

  flowchart TD
    A["Research Question: Long-time behavior of semi-Markov modulated integral processes I(a,b)_t"] --> B["Inputs: Ergodic semi-Markov environment Y, sign of E_πa(·), sojourn time tail properties"]
    B --> C["Methodology: Analyze state space models under semi-Markovian regime switching"]
    C --> D["Process: Evaluate asymptotic limits and tail conditions for the integral functional"]
    D --> E["Findings: Mixture-type limit laws represented via Affine SRE (X ≝ AX + B, X ⫫ (A,B))"]
    E --> F["Applications: Explicit representations for pitchfork bifurcations & regime-switching diffusions"]