XU HUAIAND TANG LING

(1.School of Mathematics,Anhui University,Hefei,230039)

(2.Department of mathematics,Anhui Institute of Architecture and Industry,Hefei,230601)

A Joint Density Function in the Renewal Risk Model*

XU HUAI1AND TANG LING2

(1.School of Mathematics,Anhui University,Hefei,230039)

(2.Department of mathematics,Anhui Institute of Architecture and Industry,Hefei,230601)

Communicated by Wang De-hui

In this paper,we consider a general expression for φ(u,x,y),the joint density function of the surplus prior to ruin and the def i cit at ruin when the initial surplus is u.In the renewal risk model,this density function is expressed in terms of the corresponding density function when the initial surplus is 0.In the compound Poisson risk process with phase-type claim size,we derive an explicit expression for φ(u,x,y).Finally,we give a numerical example to illustrate the application of these results.

def i cit at ruin,surplus prior to ruin,phase-type distribution,renewal risk model,maximal aggregate loss

1 Introduction

The renewal risk model{U(t)}t≥0is def i ned by where u is the initial surplus,c is the rate of premium income per unit time,is a sequence of independent and identically distributed(i.i.d.)random variables,where Xirepresents the amount of the ith claim,andis a counting process with N(t) denoting the number of claims up to time t.In addition,Xihas a density function θ(x)and a distribution function

where X is an arbitrary Xi.Let

The sequence of i.i.d.random variablesrepresents the claim inter-arrival times,with W1being the time until the fi rst claim.Wihas a density function k(t)and a distribution function

where W is an arbitrary Wi.Let

We assume that claim amounts are independent of claim inter-arrival times.Further,we assume that

De fi ne the time of ruin

where T=∞if U(t)≥0 for all t＞0.Denote the ruin probability by

and the survival probability by

It is well known that

where ρ=ψ(0),L is the well-known maximal aggregate loss in the renewal risk model,and

is the so-called ladder height distribution function,which can be interpreted as either the distribution function of the de fi cit at ruin when initial surplus u=0 or the distribution function of the amount of a drop in surplus,given that a drop below its initial level occurs.

is the distribution function of the n-fold convolution of F(y)with itself(see[1]).

Let

where U(T-)denotes the surplus prior to ruin,and U(T)denotes the de fi cit at ruin. Φ(u,x,y)may be interpreted as the probability that ruin occurs from initial surplus u with the de fi cit at ruin no greater than y and the surplus prior to ruin no greater than x. φ(u,r,s)denotes the joint density function.Let

where h(u,x)may be interpreted as the defective density function of the surplus prior to ruin from initial surplus u.Let

where g(u,y)may be interpreted as the defective density function of the de fi cit at ruin from initial surplus u.De fi ne the proper density function of the de fi cit at ruin when initial surplus u=0 by

Clearly,we have

The Sparre Andersen risk model is a well recognized risk model.As it was commented by Gerber and Shiu[2],although the model was proposed almost half a century ago,it remains an important area of research in actuarial science.A large number of researchers have studied this model on a variety of topics.Albrecher et al.[3]considered the threshold dividend strategies in the renewal risk model.Borovkov and Dickson[4]gave the distribution of ruin time in the renewal risk model.Yang and Zhang[5]studied the Gerber-Shiu function in a Sparre Andersen model with multi-layer dividend strategy.Landriault and Willmot[6]considered discounted penalty function in the renewal risk model with general inter-claim times.

The remainder of this paper is organized as follows.In Section 2,we provide a general solution for Φ(u,x,y),and consequently its joint density function φ(u,x,y).In Section 3, we consider a simpli fi cations in compound Poisson process with phase-type claim amount. In Section 4,we give a numerical example to illustrate the application of these results.

2 An Expression for φ(u,x,y)

In this section,we derive the explicit expression of φ(u,x,y).

First,we consider the case when u≥x.In order for the surplus immediately prior to ruin to be less than or equal to x,the surplus cannot fall below 0 on the fi rst occasion that it drops below its initial level u.Hence it follows that

Taking partial derivatives with respect to x and y yields

Secondly,in the case when 0≤u＜x,it is possible for ruin to occur at the time the surplus fi rst falls below its initial level u,and for the surplus prior to ruin to be less than or equal to x,and for the de fi cit at ruin to be less than or equal to y.The probability of thisevent is

as the event is equivalent to ruin occurring from initial surplus 0 with a surplus immediately prior to ruin less than or equal to x-u and a de fi cit at ruin between u and u+y.Hence, for 0≤u＜x,we have

and

Therefore,for u≥0,from(2.1)and(2.2),we have

where

with

Let

Taking Laplace transform for(2.3)with respects to u,by basic properties of Laplace transform,we obtain

From(1.1)we obtain

(see[1]),which implies that

Since the product of two transforms is the transform of a convolution,it immediately followsthat

Hence the above equation provides a means of fi nding φ(u,x,y)provided that we know both δ(z)and φ(0,x,y).

3 Simpli fi cations in the Classical Risk Process with Phase-type Claim

In this section,we derive the explicit expression in compound Poisson process with phasetype claim amount.

Assume that{N(t)}t≥0is a Poisson process with rate λ＞0.So the claim inter-arrival time Wihas distribution function

First,we introduce the phase-type distributions.Phase-type distributions have become an extremely popular tool for applied probabilists wishing to generalize beyond the exponential while retaining some of its key properties(see[7]–[8]).The phase-type family includes the exponential,mixture of exponentials,Erlangian and Coxian distributions as special cases. The class of phase-type distributions is dense in the space of probability distributions on [0,∞).We can always use phase-type distribution as the approximate distribution.Readers interested in fi nding a good approximating phase-type distributions may refer to[9]–[10].

Phase-type distributions were fi rst introduced by Neuts[11]in 1975.A shortened treatment can be stated as follows.Consider a Markov process with transient states{1,2,···,m} and absorbing state m+1,whose in fi nitesimal generator Q has the form

The diagonal entries Siiare necessarily negative,other entries are non-negative,and S0= -Se′(e′is an m×1 column vector of ones)represents the rates at which transitions occur from the individual transient states to the absorbing state.Let the process start in state i with probability ai(i=1,2,···,m,m+1),and a=(a1,a2,···,am)(in many practical problems,am+1=0).Under these assumptions,the time V until absorption has occurred has distribution function

and density function

where the matrix exponential is de fi ned by

As this distribution is completely determined by a and S,we say either that V has a phasetype distribution with representation(a,S),or write V～PH(a,S).Occasionally,we say that F(x)has PH representation(a,S).For a more detailed description of phase-type distributions,see[12].

Several well-known ruin-theoretic results can be summarized as follows(see[13]):

If the i.i.d.claim amount random variables Xi～PH(a,S),from Theorem 4.4 in [12]we know that the probability of ultimate ruin in the general renewal risk model with phase-distributed claim amounts is given by

where B=S+Se′a+,and the row vector a+is the unique solution of a fi xed-point problem, i.e.,a+satis fi es the equation

while

In the classical compound Poisson risk process,the claim inter-arrival times are exponentially distributed with

Note that

where Imrepresents the m×m identity matrix.Therefore,substituting(3.2)into(3.1),we obtain the following equation:

Based on Corollary 3.1 in[12],we try as the candidate solution

Then the left-hand side of(3.3)becomes

Thus the probability of ultimate ruin in the compound Poisson risk process with phase-type distribution claim amounts is given by

where

It is well known that in the compound Poisson risk process

(see[14]).Thus when X～PH(a,S),we have

i

Substituting(3.4)and(3.5)into(2.4),for 0≤u≤x,we obtain

and for u＞x we obtain

4 Example

In this section,we illustrate the application of the results of the previous section with an example.We comment that the computation of matrix exponentials is a simple task with the aid of software.The results in this section can be readily obtained using packages such as Mathematica.

We consider that individual claim amount Xi～PH(a,S)withandIn this case,the distribution is an equal mixture of two exponentials at rates 3 and 7,respectively,where{N(t)}t≥0is a Poisson process with rate λ=1 and the rate of premium income per unit timeFrom(3.4),we have

The matrix exponential exp{uB}can be calculated as

From(3.5)–(3.7),we have

and

Thus,we have the defective density function of the surplus prior to ruin h(u,x)and the defective density functions of the de fi cit at ruin g(u,y),namely,

and

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tion:60P05,60H10

A

1674-5647(2013)01-0088-09

*Received date:Jan.26,2011.

Tianyuan NSF(11126176)of China.