# An Application of Fractional Differential Equations to Risk Theory

## TL;DR

This paper introduces fractional differential operators and their application to risk models, deriving explicit ruin probability solutions for specific cases with rational Laplace transforms.

## Contribution

It develops a new class of fractional differential operators and applies them to risk theory, providing explicit solutions for ruin probabilities in fractional risk models.

## Key findings

- Explicit ruin probability solutions for gamma-time risk models.
- Explicit ruin probability solutions for fractional Poisson risk models.
- Applicable when claim size distributions have rational Laplace transforms.

## Abstract

This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions, when claim sizes distributions exhibit rational Laplace transforms.