Self-Exciting Random Evolutions (SEREs) and their Applications (Version 2)

TL;DR

The paper develops Self-Exciting Random Evolutions (SEREs) by coupling a Markov chain with a Hawkes process through a novel Swish Process $x(t)=x_{N(t)}$, yielding a rich class of operator-valued stochastic evolutions. It establishes two principal limit theorems—an averaging result (Theorem 1) and a diffusion approximation (Theorem 2)—under a comprehensive set of structural and ergodic conditions, derived via a martingale framework in Banach spaces. The work provides explicit averaged and diffusion limits for continuous and discrete SEREs, with concrete forms for the drift and diffusion coefficients, and demonstrates applications to self-exciting traffic/transport and summation on a Markov chain. The findings offer new modeling tools for self-exciting and clustering dynamics in finance, insurance, and stochastic transport, with clear paths to broader applications and future refinements such as rate-of-convergence analyses.

Abstract

This paper is devoted to the study of a new class of random evolutions (RE), so-called self-exciting random evolutions (SEREs), and their applications. We also introduce a new random process $x(t)$ such that it is based on a superposition of a Markov chain $x_n$ and a Hawkes process $N(t),$ i.e., $x(t):=x_{N(t)}.$ We call this process self-walking imbedded semi-Hawkes process (Swish Process or SwishP). Then the self-exciting REs (SEREs) can be constructed in similar way as, e.g., semi-Markov REs, but instead of semi-Markov process $x(t)$ we have SwishP. We give classifications and examples of self-exciting REs (SEREs). Then we consider two limit theorems for SEREs such as averaging (Theorem 1) and diffusion approximation (Theorem 2). Applications of SEREs are devoted to the so-called self-exciting traffic/transport process and self-exciting summation on a Markov chain, which are examples of continuous and discrete SERE. From these processes we can construct many other self-exciting processes, e.g., such as impulse traffic/transport process, self-exciting risk process, general compound Hawkes process for a stock price, etc. We present averaged and diffusion approximation of self-exciting processes. The novelty of the paper associated with new models, such as $x(t)$ and SERE, and also new features of SEREs and their many applications, namely, self-exciting and clustering effects.