Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals

The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In this paper, we modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian motion, and assume that the claim sizes and claim intervals are dependent. Using the properties of conditional expectation, we establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability.


Introduction
Various papers in ruin theory modify the classical Poisson risk model to describe the surplus process of an insurance portfolio.An extension of the classical model is that the risk process perturbed by a diffusion was first introduced by Gerber [1] and has been further studied by many authors during the last few years, e.g.Dufresne and Gerber [2], Gerber and Landry [3], Wang and Wu [4], Tsai and Willmot [5] [6], Chiu and Yin [7], and the references therein.
In the risk process that is perturbed by diffusion, the surplus process ( ) ( ) ( ) ( ) where 0 u ≥ is the initial surplus, 0 c > is the positive constant premium income rate, ( ) ( ) ∑ is the aggregate claims process, in which is the claim number process (denoting the number of claims up to time t), and the in- and density function ( ) is a standard Brownian motion that is independent of the aggregate claims process It is explicitly assumed in these papers that the interarrival times { } and the claim sizes { } , 1 i X i ≥ are mutually independent.However, this assumption is often too restrictive in practice, and there is a need for more general models where the independence assumptions can be relaxed.Recently, various results have been obtained concerning the asymptotic behavior of the probability of ruin for dependent claims, see [8]- [14], as well as the references therein.Zhao [14] assumed that the distribution of the time between two claim occurrences depends on the previous claim size.Motivated by the results of Zhao [14], the main aim of this paper is to modify the risk model (Equation (1)), and establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability in the new risk model.

Improved Risk Model
In this paper, it is assumed that the claim occurrence process to be of the following type: If a claim i X is larger than a random variable i Y , then the time until the next claim which is the net profit condition.
In the daily operation of insurance company, in addition to the premium income and claim to the operation of spending has a great influence on the outside, and there is also a factor that interest rates should not be neglected.As in [15], this paper assume that the risk model Equation ( 1) is invested in a stochastic interest process which is assumed to be a geometric Brownian motion , where r and σ 2 are positive constants, and

Let ( )
X t denote the surplus of the insurer at time t under this investment assumption.Thus, Denote T to be the ruin time (the first time that the surplus becomes negative), i.e., This article is interested in the expected discounted penalty (Gerber-Shiu) function: where ( ) > and satisfies ( ) Furthermore, let 1 T be the time when the first claim occurs, and random variable 1i T being exponentially distributed with rate

Integro-Differential Equations for
In this section, a system of integro-differential equations with initial value conditions satisfied by the Gerber-Shiu function , and ( ) ( ) Proof : acting on functions satisfying the reflecting boundary condition ( ) Here Using the initial condition ( ) Applying the Optional Stopping Theorem, it follows that and thus This ends the proof of Lemma 3.1.
Similarly, the following lemma can also be obtained.
with the initial value conditions Proof Let 11 T be the time when the first claim occurs which exponentially distributed with rate 1 0 λ > .Consider the risk process defined by Equation (2) in an infinitesimal time interval ( ) 0, t .There are three possible cases in ( ) 0, t as follows.
1) There are no claims in ( ) 0, t with probability 2) There is exactly one claim in ( ) 0, t with probability 1 t λ .According to different of the claim amount, there are three possible cases in this case as follows.
a) The amount of the claim ( ) , i.e., ruin does not occur, and thus ( ) ( ) b) The amount of the claim ( ) , i.e., ruin occurs due to the claim; c) The amount of the claim ( ) , i.e., ruin occurs due to oscillation (observe that the probability that this case occurs is zero).
3) There is more than one claim in ( ) 0, t with probability ( ) t ο .Thus, considering the three cases above and noting that is a strong Markov process, we have ) ) ( ) ( ) ( ) ( ) Then, by Itô's formula we have Therefore, by dividing t on both sides of Equation ( 12), letting 0 t → , using Equation (13), we obtain Equation ( 9), and similarly we can obtain Equation (10).
The condition follows from the oscillating nature of the sample paths of ( ) { } X t .Now, we prove Then, by the strong property of ( ) { } X t , it can be concluded that t of an insurance portfolio is given by How to cite this paper: Gao, H.L. (2016) Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals.
proof of Theorem 3.1.