# Self-Exciting Multifractional Processes

## TL;DR

This paper introduces a novel self-exciting multifractional process extending multifractional Brownian motion, with proven existence, uniqueness, and convergence properties, applicable to modeling phenomena like earthquakes.

## Contribution

It develops a new stochastic process with self-exciting behavior, defined via a stochastic Volterra equation, and analyzes its mathematical properties and applications.

## Key findings

- Proved existence and uniqueness of the process
- Established convergence and rate of the Euler-Maruyama scheme
- Provided examples for modeling self-exciting phenomena

## Abstract

We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this through a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as give bounds on the p-order moments, for all p>=1. We show convergence of an Euler-Maruyama scheme for the process, and also give the rate of convergence, which is depending on the self-exciting dynamics of the process. Moreover, we discuss different applications of this process, and give examples of different functions to model self-exciting behavior.