The finite-time ruin probability for an inhomogeneous renewal risk model

In the paper, we give an asymptotic formula for the finite-time ruin probability in a generalized renewal risk model. We consider the renewal risk model with independent strongly subexponential claim sizes and independent not necessarily identically distributed inter occurrence times having finite variances. We find out that the asymptotic formula for the finite-time ruin probability is insensitive to the homogeneity of inter-occurrence times.


1.
Introduction. The renewal risk model has been extensively investigated in the literature since it was introduced by Sparre Andersen half a century ago (see [1]). In this risk model, the claim sizes Z 1 , Z 2 , ... form a sequence of independent and identically distributed (i.i.d.) nonnegative random variables (r.v.s) with a common distribution function (d.f.) F Z (u) = P (Z 1 u) and a finite mean β = EZ 1 , while the inter occurence times θ 1 , θ 2 , . . . are i.i.d. nonnegative r.v.s with common finite positive mean Eθ 1 = 1/ λ. In addition, it is assumed that {Z 1 , Z 2 , . . .} and {θ 1 , θ 2 , . . .} are mutually independent. In this model, the number of accidents in the interval [0, t] is given by a renewal counting process Θ(t) = sup{n 1 : θ 1 + θ 2 + . . . + θ n t} (1) which has a mean function λ(t) = EΘ(t) with λ(t) ∼ λt as t → ∞. The surplus process of the insurance company is then expressed as where x 0 is the initial risk reserve, and c > 0 is the constant premium rate.
The probability of ruin within time t is a bivariate function ψ(x, t) := P inf Under the assumptions that µ = cEθ 1 − EZ 1 = c/ λ − β > 0 and the equilibrium d.f.
1 β x 0 F Z (u) du is subexponential, Embrechts and Veraverbeke (see [7] and [18]) established a celebrated asymptotic relation for the ultimate ruin probability: We recall that a d.f. F supported on [0, ∞) is subexponential ( F belongs to the class S ) if where F * 2 denotes the convolution of F with itself.
In 2004, Tang showed that a formula similar to (4) holds for the finite-time ruin probability as well. More exactly, the following statement was proved in paper [17].
If d.f. F Z has a consistent variation and E θ p 1 < ∞ for some p > 1 + J + F Z then uniformly for all t such that t ∈ Λ = {t : λ(t) > 0}.
Here and further denote the upper Matuszevska index of a d.f. F . Furthermore, we say that a d.f. F concentrated on [0, ∞) (or on R) has a consistent variation ( F belongs to the class C ) if If d.f. F ∈ C has a finite mean, then the equilibrium d.f. of F is subexponential (see, for instance, Proposition 1.4.4 in [6]). In addition, the upper Matuszevska index J + F is finite for each d.f. F ∈ C (see, for instance, Section 2.1 in [4]). In [10] and [12], it was proved that the asymptotic formula (5) holds uniformly for t ∈ [a(x), ∞) with an arbitrary unboundedly increasing function a(x) if d.f.
A d.f. F belongs to class S * ( F is strongly subexponential according to Korshunov (see [11] It follows from Lemma 4 of [11] that each d.f. F ∈ C with finite mean value is strongly subexponential. Wang et al. [19] generalized the above results. It was shown that the asymptotic formula (5) preserves its form in the case when the inter occurrence times θ 1 , θ 2 , . . . have certain dependence structure (without restriction that Eθ p 1 < ∞ for some p > 1 + J + F Z ). In this paper, we consider an inhomogeneous (in time) renewal risk model. We suppose that inter occurrence times θ 1 , θ 2 , . . . are independent but not necessarily identically distributed. We obtain that the asymptotic formula (5) preserves also its form for such inter occurrence times satisfying some additional requirements. In fact, we consider a renewal risk model defined by equations (1) and (2) under the following three main assumptions.
Assumption H 1 . The claim sizes {Z 1 , Z 2 , . . .} are i.i.d. nonnegative r.v.s with common distribution function F Z and finite positive mean β.
The model assumptions H 1 and H 3 are natural, while assumption H 2 needs some additional comments. Hypothesis H 21 requires that r.v.s {θ 1 , θ 2 , . . .} should be uniformly integrable. Such requirement is used sufficiently frequently in the study of non identically distributed r.v.s (see, for instance, [16] or Chapter II in [15]). We use assumption H 21 together with H 23 to obtain an asymptotic formula for the exponential moment tail of renewal process (see Lemma 3.5) and to obtain an exponential estimate for maxima of sums of uniformly integrable r.v.s (see Lemma 3.4). These both auxiliary results are crucial to get the upper bound of Theorem 2.2. Requirements H 22 and H 23 are sufficient in order that the sequence {θ 1 , θ 2 , . . .} satisfies the strong law of large numbers (see Lemma 3.3), which we use to obtain the lower bound for the finite time ruin probability (see Theorem 2.1). Below we present two sequences ! of r.v.s {θ 1 , θ 2 , . . .} satisfying assumption H 2 . Example 1. Let {θ 1 , θ 2 , . . .} be independent r.v.s, such that θ 1 , θ 4 , θ 7 , . . . be distributed according to the Poisson law with parameter 1/λ 1 , r.v.s θ 2 , θ 5 , θ 8 , . . . be distributed according to the Poisson law with parameter 1/λ 2 and θ 3 , θ 6 , θ 9 , . . . be distributed according to the Poisson law with parameter 1/λ 3 . If λ 1 = λ 2 = λ 3 then the renewal counting process Θ(t) is inhomogeneous but assumption H 2 holds with λ = 3λ Example 2. Let {θ 1 , θ 2 , . . .} be independent r.v.s distributed in the following way: The renewal process with such inter occurrence times is also inhomogeneous and assumption H 2 holds again with λ = 2 because: 2. Main results. In this section, we present exact formulations of our assertions. Before these formulations we recall the definition of long tailed distribution.
A d.f. F supported on [0, ∞) (or on R) belongs to class L (is long tailed) if for each positive y The following two theorems and the corollary are the main results of the paper.
According to Lemma 3.6 below λ(t) ∼ λt if t → ∞. Therefore, Corollary 1 implies more simple asymptotic formula for the finite-time ruin probability in the case when the horizon of time t is restricted to a smaller region. Namely, under conditions of Corollary 1, we can obtain that Possibly, the asymptotic formulas, presented in Theorems 2.1 and 2.2 (and so in Corollary 1), hold uniformly for all t ∈ Λ, not only for t ∈ [T, ∞) with T ∈ Λ. At the moment, we do not know how we can extend the region of uniformity without additional requirements.
The rest of the paper is organized as follows. In Section 3 we collected all auxiliary results which we need to prove our Theorems 2.1 and 2.2. In Section 4 we obtain lower estimate of the finite-time ruin probability, while in the next Section 5 we prove the upper estimate for the same probability.
3. Auxiliary results. In this section, we present lemmas which we use in the proof of our main results.
for an arbitrary positive .  4. Lower bound. In this section, it is dealt with the proof of Theorem 2.1. Essentially, we keep in our proof the way of [19]. Let, as usual, ε, δ ∈ (0, 1), L ∈ N and for all positive x and t.
Since d.f. F Z is long-tailed we obtain using Lemma 3.1 that P max Substituting the last estimate into (6) we get lim inf for all for ε, δ ∈ (0, 1), L ∈ N and T 1 > 0. It is obvious that Conditions of Theorem 2.1 imply that for each sufficiently large K = K(δ) and L 2 So, due to Lemma 3.3,

Upper bound.
In this section, we obtain the assertion of Theorem 2.2. The proof of the assertion consists of two parts. In the first part of the proof we use the way from [12]. In the second part of proof we mainly use the consideration from [19].
If b = 1 + ∆ is chosen for a = (1 + ε/2)λ according to Lemma 3.5, then the last inequality implies that The last inequality together with equality (17) It remains to estimate ψ(x, t) in the case when t ∈ [T, T 5 ]. Suppose that function 1 ϕ(x) √ x, x 1, satisfies property (13). If x 1 and t T , then due to (3) we have Applying Lemma 3.2 we obtain because of condition (13).