The Gerber-Shiu discounted penalty function in the thinning risk model with refunding

Introduction

The thinning process has an important application in risk theory. For example, the thinning risk model described by () 1 ( ) ( )

is considered (see literature [7]), where u ≥ 0 is the initial capital of an insurance company. {𝑁(𝑡), 𝑡 = 1,2,3, ⋯ } is a binomial process with parameter θ(θ > 0) and N(t) denotes the total number of the policies sold by insurance company in [0,t]. And the number of insurance policy charged each time is constant 1. { ( ), 1, 2, 3, } p N t t  is called the p-thinning process of { ( ), 1, 2,3, } N t t  , that is, { ( ), 1, 2, 3, } p N t t  is binomial process with parameter pθ(0<p<1), () is a sequence of independent and identically distributed nonnegative random variables with common distribution function () fx ( (0) 0 f  ), which are independent of { ( ), 1, 2,3,

the size of the i -th claim with the mean  . Now, we generalize the thinning risk model by adding the refund as follows

is the q-thinning process of { ( ), 1, 2,3, }

is a sequence of independent and identically distributed nonnegative random variables with common distribution function ( () gy (0) 0 g  and let () gy = () fx ), which are independent of { ( ),

and the Gerber-Shiu discounted penalty function as

UT  shows the instantaneous surplus before ruin,

| ( ) | UT

expresses the deficit at the time of ruin,



denotes the discount factor and () IA represents the indicator function of a set A. The Gerber-Shiu discounted penalty function was investigated by many authors. J. Pan and G.J. Wang (2009) [3] studied a thinning risk model and derived the integral equation, the integral differential equation and recursive formula for the expected discounted penalty function. L.J. He, C.Y. Wang and K. Zhang (2016) [4] observed a compound Poisson-Geometric risk model with variable premium rate and obtained a defective renewal equation of Gerber-Shiu discounted penalty function by applying the differential argument method. G. Shija and M.J. Jacob (2016) [2] considered a Markov modulated delayed by-claim type risk model with random incomes and derived the system of integral equations satisfied by the Gerber-Shiu discounted penalty function. In section 2, the relevant lemma is given and proved. In section 3, the defective renewal equation and the asymptotic estimate satisfied by the Gerber-Shiu discounted penalty function are derived, then the asymptotic estimate of the ruin probability is also obtained.

Preliminary

We represent the generating function of () fx on 0,1, 2, N  as ()

Hypothesis 1 There exits () fz  as zz   ( z  may be  ) when there is

has an unique positive solution  (

The Gerber-Shiu discounted penalty function Theorem 1

With relatively safe load condition, the defective renewal equation for the Gerber-Shiu discounted penalty function is followed by

Proof In the time interval [0,1] , it is well known that the probability is 1   if there is no premium, the probability is 1 q  if a refund occurs, and the probability is 1 p  if a claim happens. Thus, by considering whether a premium, refund and claim occurs in the time interval [0,1] , the Gerber-Shiu discounted penalty function yields that (1 ) ( 1 ) ( )

Let () mz be the generating function of () mu with respect to u on the entire real line, and () z  is the corresponding generating function of () u  . Now, taking the generating function both sides of the equation ( 4) with respect to u , we get [

From Lemma 1, it is followed that

Then we substitute the equation (3) into the equation ( 5) and derive that

And it is obvious to know that (1 ) (1 )(1 ) ( ) ( ) 0, q p p q pq f

we substitute the equation ( 7) into the equation ( 6) and yield that

Therefore, the equation ( 2) can be further obtained. And because

it is easily known that the equation ( 2) is the defective renewal equation.

Theorem 2 For 01   , the asymptotic estimate for the Gerber-Shiu discounted penalty function is

first, we multiply by u R both sides of the equation ( 2) and obtain that 0

References

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