Runge--Kutta numerical methods for ruin probabilities in classical risk model

TL;DR

This paper develops and analyzes Runge--Kutta methods combined with quadrature formulas to efficiently compute ruin probabilities in the classical risk model, including implementations for specific claim-size distributions.

Contribution

It introduces a novel framework integrating Runge--Kutta schemes with quadrature for Volterra equations in risk modeling, including reformulation as ODEs and practical implementations.

Findings

Methods effectively approximate ruin probabilities for Gamma and Pareto distributions.

Numerical results demonstrate the accuracy and efficiency of the proposed methods.

Reformulation as ODEs offers computational advantages.

Abstract

In this paper, we study Runge--Kutta methods for the computation of ruin probabilities in the classical risk model through the associated Volterra integro-differential equation. The proposed framework combines fourth-order one-step and two-step Runge--Kutta schemes with numerical quadrature formulas to approximate the convolution term. In particular, the convolution term is approximated using Newton--Cotes and Gaussian quadrature formulas, including Simpson's 1/3 rule and Pareto-adapted Gauss--Jacobi quadrature. An equivalent reformulation of the Volterra equation as a system of ordinary differential equations is also considered. Implementations for Gamma and Pareto claim-size distributions are developed. Numerical results are presented to illustrate the effectiveness of the proposed methods.