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

TL;DR

This paper introduces self-exciting random evolutions (SEREs), a new class of stochastic processes based on superpositions of Markov and Hawkes processes, with applications in traffic, risk, and finance modeling.

Contribution

It presents a novel class of self-exciting random evolutions (SEREs) and their limit theorems, along with new models like the Swish process and diverse applications.

Findings

Established limit theorems for SEREs, including averaging and diffusion approximations.

Developed new self-exciting models for traffic, risk, and stock prices.

Demonstrated clustering effects and self-excitation in various processes.

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.