Mathematics

Laplace transform

Poisson distribution

Financial economics

Statistics

Fractional calculus

Mathematical analysis

Artificial intelligence

Economics

Class (philosophy)

Computer science

Differential equation

Applied mathematics

Mathematical finance

Finance and Stochastics

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 size distributions exhibit rational Laplace transforms.

An application of fractional differential equations to risk theory