Ruin Probability in Compound Poisson Process with Investment

Yong Wu,Xiang Hu

doi:10.1155/2012/286792

Abstract

We consider that the surplus of an insurer follows compound Poisson process and the insurer would invest its surplus in risky assets, whose prices satisfy the Black‐Scholes model. In the risk process, we decompose the ruin probability into the sum of two ruin probabilities which are caused by the claim and the oscillation, respectively. We derive the integro‐differential equations for these ruin probabilities these ruin probabilities. When the claim sizes are exponentially distributed, third‐order differential equations of the ruin probabilities are derived from the integro‐differential equations and a lower bound is obtained.

Highlights

In classical risk models, the surplus process is defined as U t u ct − Nt i1 Xi, whereU 0 u ≥ 0 is the initial surplus, c > 0 is the premium rate, {N t, t ∈ R } is a homogeneousPoisson process with rate λ > 0, and {Xi, i ∈ N } is a sequence of independent and identically distributed i.i.d. nonnegative random variables with distribution F, denoting claim sizes.In this paper, we assume that the surplus can be invested in risky assets
We assume that the surplus can be invested in risky assets
The price of the nth risky asset follows dRn t μnRn t dt σnRn t dBn t, R 0 > 0, 1.1 where R 0 denotes the initial value of the risky asset, μn and σn are fixed constants, and {Bn t, t > 0} are standard Brownian motions

Summary

Introduction

U 0 u ≥ 0 is the initial surplus, c > 0 is the premium rate, {N t , t ∈ R } is a homogeneous. We assume that the surplus can be invested in risky assets. The probability of ruin from initial surplus u is defined as. We decompose the ruin probability into the sum of two ruin probabilities which are caused by the claim and the oscillation, respectively see 1–3. We denote by Ts inf{t | S t < 0, S h > 0, 0 < h < t} and Ts ∞ for all t ≥ 0; namely, Ts is the ruin time at which ruin is caused by a claim. We denote by Td inf{t | S t 0, S h > 0, 0 < h < t} and Td ∞ for all t ≥ 0; namely, Td is the ruin time at which ruin is caused by oscillation.

Integro-Differential Equation for Ruin Probability

Ruin with Exponential Claim Sizes