Paper: arXiv 2412.10592

Authors: Anatoliy Swishchuk

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation of Self-Exciting Random Evolutions (SEREs) and the Swish Process, featuring advanced probability theory and operator-valued stochastic differential equations. However, it is entirely theoretical with zero empirical validation, real-world data backtesting, or released code, placing it firmly in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop Self-Exciting Random Evolutions SEREs & Applications] --> B[Core Modeling: Superpose Markov Chain x_n & Hawkes Process N_t]
    B --> C[New Process Construction: Swish Process x_t = x_N_t]
    C --> D[Construct SEREs using Swish Process]
    D --> E[Apply Limit Theorems: Averaging & Diffusion Approximation]
    E --> F[Applications: Traffic/Transport, Risk Processes & Compound Hawkes Stock Models]
    F --> G[Key Outcomes: Models capturing self-exciting & clustering effects]