# Phase-type Approximation of the Gerber-Shiu Function

## TL;DR

This paper introduces a phase-type approximation method for the Gerber-Shiu function, enabling more tractable risk measurement in insurance models by fitting surplus processes with phase-type Levy processes.

## Contribution

It presents a novel closed-form approximation of the Gerber-Shiu function using phase-type Levy processes, advancing computational methods in risk theory.

## Key findings

- Provides a closed-form approximation for the Gerber-Shiu function.
- Demonstrates the effectiveness of the approximation through numerical results.
- Enhances computational efficiency in risk assessment models.

## Abstract

The Gerber-Shiu function provides a way of measuring the risk of an insurance company. It is given by the expected value of a function that depends on the ruin time, the deficit at ruin, and the surplus prior to ruin. Its computation requires the evaluation of the overshoot/undershoot distributions of the surplus process at ruin. In this paper, we use the recent developments of the fluctuation theory and approximate it in a closed form by fitting the underlying process by phase-type Levy processes. A sequence of numerical results are given.