Asymptotic Investment Behaviors under a Jump-Diffusion Risk Process

TL;DR

This paper analyzes the optimal investment strategies for an insurance company with a surplus modeled by a jump-diffusion process, focusing on minimizing ruin probability and exploring asymptotic behaviors at different surplus levels.

Contribution

It introduces new operators to solve the HJB equation for ruin probability and provides novel asymptotic results for low and high surplus levels under various claim distributions.

Findings

Minimal ruin probability function is a classical solution to the HJB equation.

Asymptotic behaviors are characterized for low surplus levels.

New results are obtained for large surplus levels with exponential claim distributions.

Abstract

We study an optimal investment control problem for an insurance company. The surplus process follows the Cramer-Lundberg process with perturbation of a Brownian motion. The company can invest its surplus into a risk free asset and a Black-Scholes risky asset. The optimization objective is to minimize the probability of ruin. We show by new operators that the minimal ruin probability function is a classical solution to the corresponding HJB equation. Asymptotic behaviors of the optimal investment control policy and the minimal ruin probability function are studied for low surplus levels with a general claim size distribution. Some new asymptotic results for large surplus levels in the case with exponential claim distributions are obtained. We consider two cases of investment control - unconstrained investment and investment with a limited amount.