# ASYMPTOTICS FOR THE RUIN PROBABILITIES OF A TWO-DIMENSIONAL RENEWAL RISK MODEL.

1. INTRODUCTION

The multi-dimensional risk models were initially investigated by Hult et al. (2005). Since then, many researchers have been devoted to the research of this field and have obtained many meaningful results, see, for instance, Yuen et al. (2006), Li et al. (2007), Chen et al. (2011), Zhang and Wang (2012), Chen et al. (2013), Gao and Yang (2014), Yang and Li (2014), Jiang et al. (2015), Lu and Zhang (2016) and Yang and Yuen (2016), and so on.

Among the above-mentioned results, Chen et al. (2011) considered a two- dimensional insurance model satisfying the following assumptions.

Assumption 1.1. The claims come in pairs and the claim amounts {[X.sub.in], n [greater than or equal to] 1}, i = 1, 2 form two sequences of independent, identically distributed (i.i.d.) and nonnegative random variables (r.v.s) with finite means and distributions [F.sub.1] and [F.sub.2] respectively.

Assumption 1.2. The claims of the two classes of insurance business share the same claim arrival process

(1.1) N(t) = inf {n: [n.summation over (i=1)] [[theta].sub.i] [less than or equal to] t}, t [greater than or equal to] 0,

where the claim inter-arrival times {[[theta].sub.n], n [greater than or equal to] 1} are i.i.d. r.v.s with a common finite positive mean [[lambda].sup.-1]. Namely, {N(t), t [greater than or equal to] 0} is a renewal counting process. We always assume that {[X.sub.1n], n [greater than or equal to] 1}, {[X.sub.2n], n [greater than or equal to] 1} and {N(t), t [greater than or equal to] 0} are mutually independent.

Assumption 1.3. The premium rates of the two classes of insurance business are two positive constants [c.sub.1] and [c.sub.2] such that the following safety loading conditions hold:

[[mu].sub.i] = [[lambda].sup.-1] [c.sub.i] - E[X.sub.i1] > 0, i = 1, 2.

With the initial surplus vector of the insurance company [mathematical expression not reproducible], the ruin probability by time t, t [greater than or equal to] 0 can be defined as

[mathematical expression not reproducible]

or

[mathematical expression not reproducible],

where

[mathematical expression not reproducible]

is the aggregate loss process of the insurance company. See Lu and Zhang (2016) for some other definitions of the ruin probabilities.

Before introducing a brief review and our main result on the two-dimensional ruin probabilities, we need some notions and notation. For a r.v. X with a distribution F, we denote its tail by [bar.F], namely, [bar.F](x) := P(X > x) for any - [infinity] < x < [infinity]. We say that X is bounded above if there exists some -[infinity] < C < [infinity] such that F(C) = 1, and unbounded above otherwise. Hereafter, we always suppose that X is unbounded above. We say that X (or F) is long-tailed, denoted by F [member of] L, if [lim.sub.x[right arrow][infinity]] [bar.F](x-1)/[bar.F](x) = 1; we say that X (or F) is dominatedly-varying tailed, denoted by F [member of] V, if lim [sup.sub.x[right arrow][infinity]] [bar.F](xy)/[bar.F](x) < [infinity] for some 0 < y < 1; we say that X (or F) is consistently-varying tailed, denoted by F [member of] C, if [lim.sub.y[up arrow]1] lim [sup.sub.x[right arrow][infinity]] [bar.F](xy)/[bar.F](x) = 1; we say that X (or F) is subexponential, denoted by F [member of] S, if F [member of] L and [lim.sub.x[right arrow][infinity]] [[bar.F.sup.*2]](x)/[bar.F](x) = 2, where [F.sup.*2] is the two-fold convolution of F; We say that F is strongly subexponential, denoted by F [member of] [S.sub.*], if [[integral].sup.[infinity].sub.0] [bar.F](x) dx < [infinity] and lim [sup.sub.x[right arrow][infinity]] [sup.sub.1[less than or equal to]u<[infinity]] [absolute value of ([bar.[F.sup.*2]](x)/[[bar.F].sub.u](x) - 1)] = 0, where

[mathematical expression not reproducible].

If [[integral].sup.[infinity].sub.0] [bar.F](x) dx < [infinity], then we have the following inclusion relationships:

C [subset or equal to] L [intersection] D [subset or equal to] [S.sub.*] [subset or equal to] S [subset or equal to] L,

see, for instance, Cline et al. (1994), Korshunov (2002) and Denisov et al. (2004), among many others.

Now we return to the main topic of this paper. Based on the one-dimensional results of Tang (2004), Chen et al. (2011) derived uniform asymptotics of [[psi].sub.a]([??], t) and [[psi].sub.b]([??], t) under the condition that the claim distributions belonged to the class C. Chen et al. (2013) generalized this model by enlarging the class of the claim distributions from the class C to the class L [intersection] D. Besides, they allowed the claim inter-arrival times to be dependent according to certain dependence structure. More recently, Lu and Zhang (2016) obtained uniform asymptotics for [[psi].sub.a]([??], t), [[psi].sub.b]([??], t) and ruin probabilities of some other forms by assuming that the claim distributions belonged to the class [S.sub.*]. However, they needed some extra conditions.

This paper aims to remove the limitations in Lu and Zhang (2016). More concretely, we will establish the following uniform asymptotics for [[psi].sub.a]([??], t), [[psi].sub.b]([??], t) for general strongly subexponential claims. For notational convenience, for two functions a(x, t) and b(x, t), we write a(x, t) [??] b(x, t), if lim [sup.sub.(x,t)[right arrow]([infinity],[infinity])] [a.sup.-1](x,t) b(x,t) [less than or equal to] 1; a(x, t) [??] b(x, t), if lim [inf.sub.(x,t)[right arrow]([infinity],[infinity])] [a.sup.-1] (x, t) b(x, t) [greater than or equal to] 1 and a(x, t) ~ b(x, t), if both a(x, t) [??] b(x, t) and a(x, t) [??] b(x, t). Hereafter, unless otherwise stated, the limit procedure is to let ([x.sub.1] [conjunction] [x.sub.2], t) [right arrow] ([infinity], [infinity]) but with [kappa][x.sub.1] [less than or equal to] [x.sub.2] [less than or equal to] [[kappa].sup.-1] [x.sub.1] for some positive constant [kappa] [member of] (0, 1), where [x.sub.1] [conjunction] [x.sub.2] = min{[x.sub.1], [x.sub.2]}.

With the above set-up, we are now ready to state our main result which gives asymptotic estimates of the finite-time ruin probabilities [[psi].sub.a] ([??], t) and [[psi].sub.b](x,t).

Theorem 1.4. Consider the two-dimensional risk model whose aggregate loss process is defined by (1.2). Suppose that Assumptions 1.1-1.3 hold. If [F.sub.1], [F.sub.2] [member of] [S.sub.*], then

(1.3) [mathematical expression not reproducible]

and

(1.4) [mathematical expression not reproducible].

By Theorem 1.4, we immediately obtain the following result, which generalizes Theorem 2 of Chen et al. (2011) and Theorem 3.1 of Lu and Zhang (2016).

Corollary 1.5. Let the conditions of Theorem 1.4 be valid. Then for any function f(x): [0, [infinity]) [??] [0, [infinity]) such that f(x) [right arrow] [infinity] as x [right arrow] [infinity],

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

The rest of this paper is organized as follows. In Section 2 we first develop some lemmas, then prove the main result given in Section 1. In Section 3, we extend the two-dimensional risk model to one with Brownian motion diffusions.

2. Proof of Theorem 1.4

2.1. Some Lemmas. In order to prove Theorem 1.4, we need some lemmas. The first lemma investigates some properties of long-tailed distributions, which has its own independent interest.

Lemma 2.1. Let F [member of] L. Then for any [epsilon] > 0, there exists some [x.sub.0] > 0 such that for all x > [x.sub.0] and all 0 [less than or equal to] u < x - [x.sub.0],

[e.sup.[epsilon](u-1)] [bar.F](x) [less than or equal to] [bar.F](x - u) [less than or equal to] [e.sup.[epsilon](u-1)] [bar.F](x).

Proof. We only prove the inequality on the right-hand side, and the one on the left-hand side can be proved similarly.

For any [epsilon] > 0, by F [member of] L, we may choose some [x.sub.0] > 0 such that for all x > [x.sub.0],

[bar.F](x - 1) [less than or equal to] [e.sup.[epsilon]] [bar.F](x).

Then for all x > [x.sub.0] and all positive integer n such that x - n +1 > [x.sub.0], we have

(2.1) [mathematical expression not reproducible].

For any non-negative number u such that 0 [less than or equal to] u < x - [x.sub.0], let [n.sub.u] be the largest integer which is no larger than u, then u - 1 < [n.sub.u] [less than or equal to] u < [n.sub.u] + 1. Therefore, for all x > [x.sub.0] and all 0 [less than or equal to] u < x - [x.sub.0], we have x - ([n.sub.u] + 1) + 1 = x - [n.sub.u] > [x.sub.0], thus by (2.1),

[mathematical expression not reproducible].

This completes the proof.

Lemma 1 and Theorem 1 of Korshunov (2002) stated the following uniform asymptotic result for the maximum of a random walk with a negative mean, which plays an important role in this paper. Here we make the convention that a summation over an empty set of indices is zero.

Lemma 2.2. Let {[X.sub.n], n [greater than or equal to] 1} be a sequence of i.i.d. r.v.s with a common distribution F and a common mean -[mu] < 0. Let {[S.sub.n] = [[summation].sup.n.sub.k=1] [X.sub.k], n [greater than or equal to] 0} be the random walk generated by {[X.sub.n], n [greater than or equal to] 1}.

(i) If F [member of] L, then

[mathematical expression not reproducible].

(ii) If F [member of] [S.sub.*], then

[mathematical expression not reproducible].

The following Lemma is due to Lemma 3.3 of Leipus and Siaulys (2007). The readers are referred to Lemma 4.4 of Wang et al. (2012) for a more general form in the dependent case.

Lemma 2.3. Let {[X.sub.n], n [greater than or equal to] 1} be a sequence of i.i.d. r.v.s with a common mean -[mu] < 0 and {[S.sub.n], n [greater than or equal to] 0} be the random walk generated by {[X.sub.n], n [greater than or equal to] 1}. If [X.sub.1] is bounded above, then there exist two constants r, M > 0 such that for all x > 0,

[mathematical expression not reproducible].

The following Lemma is due to Theorem 1(i) of Kocetova et al. (2009). See Lemma 4.6 of Wang et al. (2012) for its generalization to the dependent case.

Lemma 2.4. Consider the renewal counting process {N(t), t [greater than or equal to] 0} introduced in (1.1). For any a > [lambda], there exists some b > 1 such that

[mathematical expression not reproducible].

The last lemma is due to (3.2) of Chen et al. (2011).

Lemma 2.5. Let f (x) be a non-increasing function defined on [a, c], a < c. If a < b < c, then

[[integral].sup.c.sub.a] f(x) dx [less than or equal to] [(b - a).sup.-1] (c - a) [[integral].sup.b.sub.a] f(x) dx.

2.2. Proof of (1.3). In this subsection, we prove (1.3) in Theorem 1.4. It's sufficient to prove the following result:

(2.2) [mathematical expression not reproducible].

2.2.1. Proof of the upper limit part in (2.2). For any [epsilon] > 0, let

(2.3) [mathematical expression not reproducible].

We first deal with [[psi].sub.a2] ([??], t). By Lemma 2.4, there exists some [beta] > 0 such that

(2.4) [mathematical expression not reproducible].

Furthermore, by [F.sub.i] [member of] [S.sub.*] and Kesten's inequality, there exists some K := K([beta]) > 0 such that for all x [greater than or equal to] 0 and all n [greater than or equal to] 1,

(2.5) [mathematical expression not reproducible].

We know by (2.4) that there exists some [t'.sub.1] > 0 such that for all t [greater than or equal to] [t'.sub.1],

(2.6) [[mu].sub.i][lambda]t > 1, i = 1, 2

and

(2.7) [mathematical expression not reproducible].

By (2.5) and (2.7), it holds for all t [greater than or equal to] [t'.sub.1] and [x.sub.1], [x.sub.2] [greater than or equal to] 0 that

(2.8) [mathematical expression not reproducible].

Still by F [member of] [S.sub.*], there exists some [x'.sub.0] sufficiently large such that for all x > [x'.sub.0],

(2.9) [F.sub.i] (x + 1) [greater than or equal to] 1/2 [[bar.F].sub.i](x), i = 1, 2.

By (2.6), (2.8) and (2.9), it holds for [x.sub.i] [conjunction] [x.sub.2] > [x'.sub.0] and t > [t'.sub.1] that

(2.10) [mathematical expression not reproducible].

Next, we deal with [[psi].sub.a1]([??], t). Let c = [c.sub.1] [disjunction] [c.sub.2] = max{[c.sub.1], [c.sub.2]}. For any 0 < [delta] < min{1, [lambda][c.sup.-1]([[mu].sub.1] [conjunction] [[mu].sub.2])}, we write [[xi].sub.ij]([delta]) = [X.sub.ij] - [[lambda].sup.-1] [c.sub.i] (1 - [delta]), [[mu].sub.i[delta]] = -E[[xi].sub.ij]([delta]) = [[mu].sub.i] - [[lambda].sub.-1] [c.sub.i][delta], [S.sub.i](t, [delta]) = [sup.sub.0[less than or equal to]k[less than or equal to][lambda]t(1+[epsilon])] [[summation].sup.k.sub.j=1] [[xi].sub.ij] ([delta]), [[eta].sub.j]([delta]) = [[lambda].sup.-1] (1 - [delta]) - [[theta].sub.j], j [greater than or equal to] 1, i = 1, 2 and [eta]([delta]) = c [sup.sub.k[greater than or equal to]0] [[summation].sup.k.sub.j=1] [[eta].sub.j]([delta]). For any fixed [delta] > 0, since {[[eta].sub.j]([delta]), j [greater than or equal to] 1} is bounded above, it follows from Lemma 2.3 that there exists some 0 < [[gamma].sub.1] := [[gamma].sub.1]([delta]) < [epsilon] such that

[mathematical expression not reproducible].

Hence there exists some [l.sub.1] > 0 such that

(2.11) [mathematical expression not reproducible],

where 1(A) is the indicator function of the event A.

For some fixed 0 < [sigma] < [2.sup.-1], we have

(2.12) [mathematical expression not reproducible].

Firstly, we deal with [[psi].sub.a11] ([??], t). By F [member of] [S.sub.*], there exists some [x'.sub.1] [greater than or equal to] [x'.sub.0] such that for all x > [x'.sub.1],

(2.13) (1 - [epsilon]) [[bar.F].sub.i](x) [less than or equal to] [[bar.F].sub.i](x [+ or -] [l.sub.1]) [less than or equal to] (1 + [epsilon]) [[bar.F].sub.i](x), i = 1, 2.

Moreover, by Lemma 2.2(ii), we may assume that [x'.sub.1] is sufficiently large such that for all x [greater than or equal to] [x'.sub.1] and all t > 0,

(2.14) [mathematical expression not reproducible].

We know by (2.14), (2.13) and Lemma 2.5 that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.2] := [x'.sub.1] + [l.sub.1] and t > 0,

(2.15) [mathematical expression not reproducible].

Plugging (2.15) into [[psi].sub.all]([??], t), we obtain that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.2] and t > 0,

(2.16) [mathematical expression not reproducible].

Secondly, we deal with [[psi].sub.a12]([??], t). In view of Lemma 2.1, we know that for sufficiently large [x'.sub.2], the following relations hold for all x [greater than or equal to] [x'.sub.2] and any 0 [less than or equal to] u < x - [x'.sub.2]:

(2.17) [mathematical expression not reproducible].

It's clear that

(2.18) [mathematical expression not reproducible].

Take [x'.sub.3] sufficiently large such that (1 - [sigma]) [x'.sub.3] [greater than or equal to] [x'.sub.2], then we have [x.sub.i] - u [greater than or equal to] (1 - [sigma]) [x'.sub.3] [greater than or equal to] [x'.sub.2], i = 1, 2 for all [l.sub.1] [less than or equal to] u [less than or equal to] [sigma] ([x.sub.1] [conjunction] [x.sub.2]) if [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3]. Therefore, it follows from (2.14) that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3] and t > 0, it holds uniformly for all [l.sub.1] [less than or equal to] u [less than or equal to] [sigma]([x.sub.1] [conjunction] [x.sub.2]) that

(2.19) [mathematical expression not reproducible].

Combining (2.18)-(2.19) and Fubini's theorem, when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3] and t > 0, we have

(2.20) [mathematical expression not reproducible],

where D(t, [epsilon], [delta]) := {([[omega].sub.1], [[omega].sub.2]) : [x.sub.1] [less than or equal to] [[omega].sub.1] [less than or equal to] [x.sub.1] + [[mu].sub.1[delta]] [lambda]t(1 + [epsilon]), [x.sub.2] [less than or equal to] [[omega].sub.2] [less than or equal to] [x.sub.2]+ [[mu].sub.2[delta]] [lambda]t(1 + [epsilon])}. Noting that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3], we have [[omega].sub.i] - u [greater than or equal to] (1 - [sigma]) [x.sub.i] [greater than or equal to] [x'.sub.2] for all [x.sub.i] [less than or equal to] [[omega].sub.i] [less than or equal to] [x.sub.i] + [[mu].sub.i[delta]] [lambda]t(1 + [epsilon]), i = 1, 2 and [l.sub.1] [less than or equal to] u [less than or equal to] [sigma]([x.sub.1] [conjunction] [x.sub.2]). Therefore, by (2.17) and (2.11), when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3] and t > 0, we have

(2.21) [mathematical expression not reproducible].

Plugging (2.21) into (2.20) and using Lemma 2.5, we see that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3] and t > 0,

(2.22) [mathematical expression not reproducible].

At last, we deal with [[psi].sub.a13]([??], t). By Lemma 2.3, there exist two constants M := M([delta]) and r := r([delta]) such that for all [x.sub.1], [x.sub.2] > 0 and t > 0,

[mathematical expression not reproducible].

Recall that there is some 0 < [kappa] < 1 such that [kappa][x.sub.1] [less than or equal to] [x.sub.2] [less than or equal to] [[kappa].sup.-1] [x.sub.1], so we have [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [2.sup.-1] [kappa]([x.sub.1] + [x.sub.2]). Thus when [x.sub.1] [conjunction] [x.sub.2] > 0 and t > 0,

(2.23) [mathematical expression not reproducible].

Since [F.sub.i] [member of] [S.sub.*], we conclude by the proof of Lemma 1 of Embrechts et al. (1979) that any exponential function with negative parameters is a higher-order infinitesimal of [[bar.F].sub.i(x), i = 1, 2. Hence it is implied by (2.23) that there exists a constant [x'.sub.4] [greater than or equal to] [x'.sub.3] such that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.4] and t > 0,

[[psi].sub.a13] ([??], t) [less than or equal to] [epsilon][[bar.F].sub.1]([x.sub.1]) [[bar.F].sub.2]([x.sub.2]).

Using the same idea in the argument (2.8)-(2.10), we know that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.4] and t [greater than or equal to] [t'.sub.1],

(2.24) [mathematical expression not reproducible].

Plugging (2.16), (2.22) and (2.24) into (2.12), we see that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.4] and t [greater than or equal to] [t'.sub.1],

(2.25) [mathematical expression not reproducible].

Therefore, we conclude from (2.3), (2.10) and (2.25) that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.4] and t [greater than or equal to] [t'.sub.1],

[mathematical expression not reproducible].

Thus we prove the upper limit in (2.2) due to the arbitrariness of [epsilon] and [delta].

2.2.2. Proof of the lower limit part in (2.2). Recall that c = [c.sub.1] [disjunction] [c.sub.2]. For any [delta] > 0, let [mathematical expression not reproducible].

Noting that E[[??].sub.1]([delta]) > 0, so [??]([delta]) is proper. Thus for any 0 < [epsilon] < [2.sup.-1], there exists some [l.sub.2] > 0 large enough such that

(2.26) P([??]([delta]) > - [l.sub.2]) > 1 - [epsilon].

By Lemma 2.2(i) and [F.sub.1], [F.sub.2] [member of] [S.sub.*], there exists some [x'.sub.5] > 0 such that when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.5] and t > 0, it holds uniformly for all n [greater than or equal to] [lambda]t(1 - [delta]) that

[mathematical expression not reproducible],

where in the last step we used the long-tailed property of [F.sub.1], [F.sub.2] and Lemma 2.5. Thus by the independence of {[X.sub.1n], n [greater than or equal to] 1} and {[X.sub.2n], n [greater than or equal to] 1}, it holds uniformly for all [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.5], t > 0 and n [greater than or equal to] [lambda]t(1 - [delta]) that

(2.27) [mathematical expression not reproducible].

Since {N(t), t [greater than or equal to] 0} is a renewal counting process, we know that there exists some [t'.sub.2] > 0 sufficiently large such that for all t [greater than or equal to] [t'.sub.2],

(2.28) P(N(t) > (1 - [delta]) [lambda]t) > 1 - [epsilon].

Thus by (2.26) and (2.28), for all t [greater than or equal to] [t'.sub.2],

(2.29) [mathematical expression not reproducible].

Therefore, by (2.27), (2.29) and Lemma 2.5, when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.5] and t [greater than or equal to] [t'.sub.2],

[mathematical expression not reproducible].

Since [epsilon] and [delta] are arbitrarily fixed, we prove the lower limit in (2.2).

2.3. Proof of (1.4). Obviously, we have

(2.30) [mathematical expression not reproducible].

For any 0 < [epsilon] < [3.sup.-1], by (1.3) and the finiteness of E[X.sub.11], there exists some [x'.sub.6] > 0 and [t'.sub.3] > 0 large enough such that when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.6] and t [greater than or equal to] [t'.sub.3],

(2.31) [mathematical expression not reproducible].

By Corollary 2.1 of Wang et al. (2012), for sufficiently large [x'.sub.6] and [t'.sub.3], when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.6] and t [greater than or equal to] [t'.sub.3],

(2.32) [mathematical expression not reproducible].

Since {N(t), t [greater than or equal to] 0} is a renewal counting process, for sufficiently large [t'.sub.3], we know that

(2.33) (1 - [epsilon])[lambda]t [less than or equal to] EN(t) [less than or equal to] (1 + [epsilon])[lambda]t for all t [greater than or equal to] [t'.sub.3].

Combining (2.32), (2.33) and Lemma 2.5, we know that when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.6] and t > [t'.sub.3],

(2.34) [mathematical expression not reproducible].

By (2.30), (2.31) and (2.34), when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.6] and t > [t'.sub.3],

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Therefore, we prove (1.4) by the arbitrariness of [epsilon].

Remark 2.6. Chen et al. (2013) considered a risk model with extended negatively orthant dependent (ENOD, see Liu et al. (2009) for its definition) inter-arrival times [[[theta].sub.n], u [greater than or equal to] 1}, and obtained a similar result to Theorem 1.4 under the condition that the claim distributions belonged to the distribution class L [intersection] D. We see that if the inter-arrival times are ENOD, then Lemma 2.3 still holds by Lemma 4.4 of Wang et al. (2012), which implies that (2.11) and (2.23) are true; and by Lemma 4.6 of Wang et al. (2012), Lemma 2.4 holds, which implies (2.4); and by Theorem 4.2 of Chen et al. (2010), (2.28) holds true. Thus Theorem 1.4 still holds if Assumption 1.2 is replaced by the following Assumption 1.2 *.

Assumption 1.2*. The inter-arrival claim times {[[theta].sub.n], u [greater than or equal to] 1} are ENOD and identically distributed r.v.s with a common finite mean [[lambda].sup.-1].

3. The case with Brownian motion diffusions

In this section, we extend the two-dimensional risk model introduced in Section 1 to one with Brownian motion diffusions. By definition, a standard Brownian motion {B(t),t [greater than or equal to] 0} is a random process with independent increments and almost-surely continuous paths, and for each given t, B(t) is a centered Gaussian r.v. with variance t. We consider such a risk model that its aggregate loss process has the following form:

(3.1) [mathematical expression not reproducible],

where the two diffusion processes {[B.sub.i] (t), t [greater than or equal to] 0}, i = 1, 2 are mutually independent standard Brownian motions, [[sigma].sub.i], i = 1, 2 are positive constants and the other quantities are the same as in Section 1. The corresponding ruin probability can be defined as

[mathematical expression not reproducible]

or

[mathematical expression not reproducible].

Our main result is as follows.

Theorem 3.1. Consider the risk model introduced above. Suppose that Assumptions 1.1-1.3 hold. Suppose that {[([B.sub.1](t), [B.sub.2](t)).sup.T], t [greater than or equal to] 0} is independent of {[([X.sub.1n], [X.sub.2n]).sup.T], n [greater than or equal to] 1} and {[[theta].sub.n], n [greater than or equal to] 1}. If [F.sub.1], [F.sub.2] [member of] [S.sub.*], then

(3.2) [mathematical expression not reproducible]

and

(3.3) [mathematical expression not reproducible].

By Theorem 3.1, we immediately obtain the following result.

Corollary 3.2. Consider the two-dimensional risk model whose aggregate loss process is defined by (3.1). Suppose that Assumptions 1.1-1.3 hold. If [F.sub.1], [F.sub.2] [member of] [S.sub.*], then for any function f(x): [0, [infinity]) [??] [0, [infinity]) such that f(x) [right arrow] [infinity] as x [right arrow] [infinity],

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

In order to prove Theorem 3.1, we need the following lemma, which is due to Theorem 3.1 of Chapter X of Rolski et al. (1999).

Lemma 3.3. Let {B(t),t [greater than or equal to] 0} be a standard Brownian motion. Then for any [sigma], [delta] > 0 and x > 0,

[mathematical expression not reproducible].

Recall that if {B(t), t [greater than or equal to] 0} is a standard Brownian motion, then {-B(t), t [greater than or equal to] 0} is also a standard Brownian motion. Hence according to Lemma 3.3, we have the following result

(3.4) [mathematical expression not reproducible].

and

(3.5) [mathematical expression not reproducible]

Proof of Theorem 3.1. The proof of (3.3) is based on (3.2) and similar to that of (1.4), so we only prove (3.2). It suffices to prove

(3.6) [mathematical expression not reproducible]

and

(3.7) [mathematical expression not reproducible].

For any [delta] > 0 and i = 1,2, we set [a.sub.i[delta]] = [[lambda].sup.-1] ([c.sub.i] - [delta]) - E[X.sub.i1] and [b.sub.i[delta]] = [[lambda].sup.- 1]([c.sub.i] + [delta]) - E[X.sub.i1]. Since [lim.sub.[delta][right arrow]0] [a.sub.i[delta]] = [[mu].sub.i] > 0, there exists some [[delta].sub.0] > 0 such that [a.sub.i[delta]] > 0 for all 0 < [delta] < [[delta].sub.0]. For simplicity, we write [L.sup.+.sub.i] ([delta], t) = [L.sub.i](i) + [delta]t, [L.sup.-.sub.i] ([delta], t) = [L.sub.i](i) - [delta]t, [B.sup.+.sub.i] ([delta], t) = -[[sigma].sub.i][B.sub.i](t) + [delta]t, [B.sup.-.sub.i] ([delta], t) = - [[sigma].sub.i] [B.sub.i](t) - [delta]t, t [greater than or equal to] 0, i =1, 2.

By (3.4) and (3.5), for any [epsilon] > 0 and 0 < [delta] < [[delta].sub.0], there exists some [l.sub.3] > 0 and 0 < [[gamma].sub.2] < 2[delta]([[sigma].sup.-2.sub.1] [conjunction] [[sigma].sup.-2.sub.2]) such that

(3.8) [mathematical expression not reproducible]

and

(3.9) [mathematical expression not reproducible].

We first prove (3.6). By Theorem 1.4, there exists some [x'.sub.7] > [l.sub.3] and [t'.sub.4] > 0 such that for all [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.7] and t [greater than or equal to] [t'.sub.4],

(3.10) [mathematical expression not reproducible]

and

(3.11) [mathematical expression not reproducible].

We may assume that [x'.sub.7] is sufficiently large such that for all x [greater than or equal to] [x'.sub.7],

(3.12) (1 - [epsilon])[[bar.F].sub.i](x) [less than or equal to] [[bar.F].sub.i] (x [+ or -] [l.sub.3]) [less than or equal to] (1 + [epsilon])[[bar.F].sub.i](x), i = 1, 2.

Hereafter, denote [E.sub.it] = [sup.sub.0[less than or equal to]s[less than or equal to]t] [[??].sub.i](s) > [x.sub.i], i = 1, 2. For some fixed 0 < [sigma] < [2.sup.-1]. Choose some [x'.sub.8] > [[sigma].sup.-1] [x'.sub.7], when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.8], we decompose [[psi].sub.a] ([??], t) as follows.

(3.13) [mathematical expression not reproducible].

Recall that [x'.sub.7] > [l.sub.3], so if [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.8], then [x.sub.i] - [l.sub.3] > [x'.sub.7], i = 1, 2. Thus by (3.10) and (3.12), when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.8] and t [greater than or equal to] [t'.sub.4],

(3.14) [mathematical expression not reproducible].

Recall that [kappa][x.sub.1] [less than or equal to] [x.sub.2] [less than or equal to] [[kappa].sup.-1][x.sub.1] for some positive constant 0 < [kappa] < 1, so when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.8] and [l.sub.3] [less than or equal to] [y.sub.i] [less than or equal to] [sigma]([x.sub.1] [conjunction] [x.sub.2]), i = 1, 2, we have ([x.sub.1] - [y.sub.1]) [conjunction] ([x.sub.2] - [y.sub.2]) > [x'.sub.7] and (1 - [sigma])[kappa] [less than or equal to] [([x.sub.1] - [y.sub.1]).sup.-1] ([x.sub.2] - [y.sub.2]) [less than or equal to] [(1 - [sigma]).sup.-1] [[kappa].sup.-1]. Thus by Theorem 1.4 and Lemma 2.1, there exists some [x'.sub.9] > [x'.sub.8] and [t'.sub.5] > [t'.sub.4] such that when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.9] and t [greater than or equal to] [t'.sub.5], it holds uniformly for all [l.sub.3] [less than or equal to] [y.sub.i] [less than or equal to] [sigma]([x.sub.1] [conjunction] [x.sub.2]), i = 1, 2 that

(3.15) [mathematical expression not reproducible].

Hereafter, denote by Git(yi) the distribution of [sup.sub.0[less than or equal to]s[less than or equal to]t] [B.sup.-.sub.i] ([delta], s), i = 1, 2 and [G.sub.t] ([y.sub.1], [y.sub.2]) = [G.sub.1t]([y.sub.1])[G.sub.2t]([y.sub.2]). Then combining (3.15) with (3.9), we get

(3.16) [mathematical expression not reproducible].

Furthermore, by replacing [y.sub.2] with [l.sub.3] in (3.15) and using Lemma 2.1 and (3.12), we know that when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.9] and t > [t'.sub.5], it holds uniformly for all [l.sub.3] [less than or equal to] [y.sub.1] [less than or equal to] [sigma]([x.sub.1] [conjunction] [x.sub.2]) that

[mathematical expression not reproducible].

Thus when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.9] and t > [t'.sub.5], we have

(3.17) [mathematical expression not reproducible],

where (3.9) are used in the last step. Similarly, when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.9] and t > [t'.sub.5], we have

(3.18) [mathematical expression not reproducible].

Recall that [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [2.sup.-1] [kappa]([x.sub.1] + [x.sub.2]) since [kappa][x.sub.1] [less than or equal to] [x.sub.2] [less than or equal to] [[kappa].sup.-1] [x.sub.1], so by (3.4), for all [??] such that [x.sub.i] > 0, i = 1, 2, we have

(3.19) [mathematical expression not reproducible].

Suppose that [t'.sub.5] and [x'.sub.9] are sufficiently large, then by (3.19) and similar argument to (2.23)-(2.24), we know that when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.9] and t [greater than or equal to] [t'.sub.5],

(3.20) [mathematical expression not reproducible].

Combining (3.13), (3.14), (3.16)-(3.18) and (3.20), we pove (3.6) by the arbitrariness of [epsilon] and [delta].

Next, we prove (3.7). By (3.11) and (3.12), when [x.sub.1] [conjunction] [x.sub.2] > [x'.sub.7] and t [greater than or equal to] [t'.sub.4],

(3.21) [mathematical expression not reproducible].

Combining (3.8) and (3.21), we have

[mathematical expression not reproducible],

which implies (3.7) by the arbitrariness of [epsilon] and [delta].

4. ACKNOWLEDGEMENTS

We are grateful to Professor G. S. Ladde of Mathematics and Statistics Departments in University of South Florida for his valuable suggestions.

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DONGYA CHENG (1, 2) AND CHANGJUN YU (3)

(1) School of Mathematical Sciences, Soochow University, Suzhou 215006, China

(2) The Statistics and Operations Research Department, University of North Carolina at Chapel Hill, NC 27514, USA dycheng@suda.edu.cn

(3) School of Sciences, Nantong University, Nantong 226019, China ycj1981@163.com

* This research is supported by National Natural Science Foundation of China (No. 11401415), Tian Yuan foundation (No. 11426139), National Social Science Fund of China (15BTJ027), Postdoctoral Research Program of Jiangsu Province of China (No. 1402111C), Jiangsu Overseas Research and Training Program for Prominent University Young and Middle-aged Teachers and Presidents. Corresponding author: ycj1981@163.com