Simulation of Integro-Differential Equation and Application in Estimation of Ruin Probability with Mixed Fractional Brownian Motion

TL;DR

This paper develops a numerical method using probability theory to solve integro-differential equations and applies it to estimate ruin probabilities influenced by mixed fractional Brownian motion, a process with complex dependence structures.

Contribution

It introduces a novel probability-based numerical approach for integro-differential equations and applies it to ruin probability estimation with mixed fractional Brownian motion.

Findings

Successful simulation of ruin probability with mixed fractional Brownian motion

New numerical method for integro-differential equations based on martingale theory

Demonstrates applicability to complex stochastic processes

Abstract

In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical L\'evy process but by the mixed fractional Brownian motion.