The Asymptotics of Recovery Probability in the Dual Renewal Risk Model with Constant Interest and Debit Force

The asymptotic behavior of the recovery probability for the dual renewal risk model with constant interest and debit force is studied. By means the idea of Markov Skeleton method, we studied the times that the random premium incomes happened and transformed the continuous time model into a discrete time model. By investigating the fluctuations of this discrete time model, we obtained the asymptotic behavior when the random premium income belongs to a kind of heavy-tailed distributions.


Introduction
The classical risk model is specified as where ≥ 0 is the initial surplus and is the constant rate at which the premiums are received. The aggregate claims process { ( )} is assumed to be a compound Poisson process, which denotes the total number of claims up to time . Denote the time of arrival of the th claim by and the size of the th claim by . More details about the surplus process can be found in Asmussen and Albrecher [1] and Rolski et al. [2]. As pointed out by Albrecher et al. [3], its dual process may also be relevant for companies whose inherent business involves a constant flow of expenses while revenues arrive occasionally due to some contingent events (e.g., discoveries and sales). For instance, pharmaceutical or petroleum companies are prime examples of companies for which it is reasonable to model their surplus process as Here, is again the initial surplus, but the constant is now the rate of expenses, assumed to be deterministic and fixed. The aggregate claims process ( ) = ∑ =1 is assumed to be a compound renewal process. The negative claim sequence { , = 1, 2, 3, . . .} is assumed to be a sequence of independent and identically distributed (i.i.d.) nonnegative random variables (r.v.s.) with common distribution function = 1 − ; we said is the tailed distribution. The interoccurrence times { , = 1, 2, 3, . . .} form another sequence of i.i.d. positive random variables. { , = 1, 2, 3, . . .} and { , = 1, 2, 3, . . .} are mutually independent. The occurrence times of the successive claims, = ∑ =1 , = 1, 2, 3, . . ., constitute a renewal counting process The past decade has witnessed an increasing attention on the research of dual risk model. For example, see Albrecher et al. [3] for optimal dividend problem, see Cheung and Drekic [4] for dividend approximation and dual risk model with perturbation, and see Yao et al. [5] for optimal dividend and equity issuance. Due to the positive safety loading, in dual risk model, when the operation time scale is infinite, the ruin probability of dual risk model is zero; thus, it is meaningless to discuss the ruin problem under dual risk model. However, there is a constant consumption in the dual risk model; thus, it has occasionally happened that the surplus of dual risk model is negative; in this case, the decision maker will care the probability that the surplus of the model became positive; we define this probability as the "recovery probability. " This conception is of both theoretical value and practical relevance. In this paper, we focus on the asymptotic behavior of the recovery probability under the dual renewal risk model, which covers the compound Poisson dual risk model. The rest of this paper is organized as follows. In Section 2, we present an introduction to the dual renewal risk model with constant interest force and debit interest force and the problem to be investigated. Section 3 provides the main results and the corresponding proofs.

Model and Problem
In this section, we consider the case that the research institution would like to invest his surplus in bond market or borrow money from the bank; both the investment interest force or the debit interest force are the same, say > 0. If there are no claims in the interval (0, Δ ), then the surplus up to time is given by letting Δ → 0, we can obtain It is easy to see, if ( ) > / , then ( ) > 0; the surplus is increased and is not less than / ; if 0 < ( ) < / , then ( ) < 0; the surplus is decreased and the insurance company may be ruin; if ( ) < 0, we know it is possible that the surplus ( ) can recover to / . Above all, we can define the absolute positive profit as In this paper, we will discuss the recovery probability which is the probability of the surplus recover to / when ( ) ≤ 0. Now we let > 0 be the constant force of interest so that after time a capital becomes . Then, the total surplus which is denoted by ( ) is given by Now, we can define the finite-time recovery probability as At occurrence time = ∑ =1 , we observe the value ( ) which represents the surplus immediately after paying the th claim, = 1, 2, . . .. By virtue of (7), we have Let = ( ) − , = 1, 2, . . . ; (10) it follows that Since recovery can happen only at the time of a claim occurrence, we rewrite the finite-time recovery probability in (8) as Since So we can study the recovery probability in (8) by the (13).
Similarly, for the infinite-time recovery probability defined in (13), we have Here and henceforth, all limit relationships are for → ∞ unless stated otherwise. For two positive functions ( ) and ( ), the relation ( )∼ ( ) amounts to the conjunction of the relations lim sup ( )/ ( ) ≤ 1 and lim inf ( )/ ( ) ≥ 1, which are denoted as ( ) ≲ ( ) and ( ) ≳ ( ), respectively. Next, we will define the convolution. For two distributions 1 and 2 on [0, ∞), we can say 1 * 2 is their convolution; that is, Furthermore, we write 1 * = and * = ( −1) * * for every = 2, 3, . . .. For the general renewal risk model, an asymptotic expression for the finite time ruin probability is presented in [6]; that is, International Scholarly Research Notices 3 In recent years, with the study of classical model and the development of the financial and insurance business, more and more people are interested in dual risk model; however, an analogue problem in insurance company is the dividend payment or high gain tax payment. Thus, the problem studied here sounds reasonable.

Main Results
In order to complete our results, we should have some definitions and lemmas.
for every real number and the limit exists and is finite; that is, ∈ S( ). A larger class, L( ), is defined by relation (18) alone.
A distribution function concentrated on (−∞, ∞) is still said to be subexponential to the right if + ( ) = ( )1 (0≤ <∞) is subexponential, and we usually denote by S the subexponential classes; see for example, [7]. Since it was introduced by [8-10], the subexponential class S has been extensively investigated by many researchers and applied to various fields. This class is often used to model claim-size distributions; see, for example, [11][12][13].

Main Results and Proof
for arbitrary > 0.

International Scholarly Research Notices
Proof. Starting with (13) and conditioning on , we have Since ∏ =1 = − ∈ [ − , 1] for every 1 ≤ ≤ , by Lemma 7, we have We know { ≥ } ⇔ ≤ ; then, since ∈ S( ), then So we have We present an asymptotic expression for the finite-time recovery probability in Theorem 9. In the following, we will show the last main result of the paper.

Theorem 10. In the dual renewal risk model with constant force of interest
where { = − , = 1, 2, . . .} is a sequence of i.i.d. positive random variables with common distribution function .