Minimizing Lundberg inequality for ruin probability under correlated risk model by investment and reinsurance

Lin Xu,Bin Zhang,Minghan Wang

doi:10.1186/s13660-018-1838-0

Lin Xu, Bin Zhang + Show 1 more

Open Access

https://doi.org/10.1186/s13660-018-1838-0

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Abstract

This paper investigates optimal investment and reinsurance policies for an insurance company under a correlated risk model with common Poisson shocks. The goal of the insurance company is to minimize the ultimate ruin probability. By the dynamic programming principle, the Hamilton–Jacobi–Bellman (HJB for short) equation associated with this control problem is obtained. Since there is no explicit solution to the HJB equation, this paper alternates to find the minimal exponential upper bound of the ruin probability. The exponential upper bound of ruin probability is also called Lundberg inequality. Minimizing Lundberg inequality is equal to finding the maximal Lundberg coefficient. It turns out that the optimal investment and reinsurance polices are constant policies. Some numerical examples are given to illustrate the impact of the dependent structure and the investment chance on the upper bound.

Highlights

The past two decades have witnessed huge attention on the risk model with dependent structure
There is a lot of literature concentrating on a multivariate risk process, where the components of the multivariate process specify different business of insurance company and they cannot be integrated into a univariate process
Hu and Zhang [22] studied optimal reinsurance for minimizing the upper bound of ultimate ruin probability in a correlated risk model with common shocks

Summary

Introduction

The past two decades have witnessed huge attention on the risk model with dependent structure. [16] studied optimal dynamic reinsurance with dependent risks and variance premium principle, where the goal of insurance company is to maximize exponential utility at terminal time. Hu and Zhang [22] studied optimal reinsurance for minimizing the upper bound of ultimate ruin probability in a correlated risk model with common shocks. Numerical examples show the following: under optimal constant investment policy, an upper bound of the ultimate ruin probability is less than the corresponding one in [22]; when the correlation coefficient of each component risk process increases, the impact of investment on the upper bound of ruin probability inequality is less significant; when the claims are heavy-tailed, reinsurance plays a more important role for the insurance company than the investment.

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