Uniform Tail Asymptotics for the Stochastic Present Value of Aggregate Claims in the Renewal Risk Model

Transcription

1 Uniform Tail Asymptotics for the Stochastic Present Value of Aggregate Claims in the Renewal Risk Model Qihe Tang [a], Guojing Wang [b];, and Kam C. Yuen [c] [a] Department of Statistics and Actuarial Science, The University of Iowa 24 Schae er Hall, Iowa City, IA 52242, USA [b] Department of Mathematics, Suzhou University Suzhou 256, PR China [c] Department of Statistics and Actuarial Science, The University of Hong Kong Pokfulam Road, Hong Kong December 25, 29 Abstract Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same asymptotic formula holds for the nite-time and in nite-time ruin probabilities. Restricting our attention to the so-called constant investment strategy, we show how the insurer adjusts his investment portfolio to maximize the expected terminal wealth subject to a constraint on the ruin probability. Keywords: Asymptotics; Constant investment strategy; Lévy process; Portfolio optimization; Regular variation; Ruin probability; Uniformity. Mathematics Subject Classi cation: Primary 9B3; Secondary 6G5, 6K5, 9B28 Introduction Consider the renewal risk model in which successive claims, X ; X 2 ; : : :, form a sequence of independent, identically distributed (i.i.d.), and nonnegative random variables with generic Corresponding author: Guojing Wang.

2 random variable X and common distribution F on [; ), and their arrival times, 2, constitute a renewal counting process N t = # fn = ; 2; : : : : n tg ; t : For later use, we write = and =. To avoid triviality, throughout the paper, we assume that is a nonnegative random variable non-degenerate at. The amount of aggregate claims up to time t appears to be a compound sum of the form XN t S t = X k ; t ; (.) where the summation over an empty set of indices produces a value of. Suppose that the insurer is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process fe Rt ; t g; that is to say, fr t ; t g is a Lévy process which starts with, has independent and stationary increments, and is stochastically continuous. This assumption on price processes is widely used in mathematical nance. We refer the reader to the monograph of Cont and Tankov (24) and a recent survey paper of Paulsen (28). See also Paulsen (993, 22), Paulsen and Gjessing (997), Wang and Wu (2), Kalashnikov and Norberg (22), Cai (24), and Yuen et al. (24, 26), among others. For the general theory of Lévy processes, see Sato (999), Cont and Tankov (24), and Applebaum (24). As usual, we assume that all sources of randomness, fx ; X 2 ; : : :g, fn t ; t g, and fr t ; t g, are mutually independent. The stochastic present value of future aggregate claims up to time t can be expressed as D t = e R s ds s = X X k e R k ( k t); t : (.2) In this paper, we shall focus on the tail probability of D t and aim at a simple asymptotic formula which holds uniformly for all time horizons. We shall also pursue applications of our result to ruin theory. The rest of the paper consists of three sections. Section 2 presents our rst main result after recalling some preliminaries; Section 3 shows applications of this result to the calculation of the nite-time and in nite-time ruin probabilities and to a portfolio optimization problem with a constraint on the nite-time ruin probability; and Section 4 proves the results after presenting a series of lemmas. 2 Preliminaries and Main Result Throughout the paper, we assume that the Lévy process fr t ; t g in (.2) is right continuous with left limit with Lévy triplet (r; 2 ; ), where < r <, are two constants and is a measure on ( ; ), called the Lévy measure, satisfying (fg) = 2

3 and R (y2 ^ )(dy) <. Let ER >, so that R t drifts to almost surely as t. The Laplace exponent for fr t ; t g is de ned as If (z) is nite, then and (z) = log Ee zr ; z 2 ( ; ): (2.) (z) = 2 2 z 2 rz + Z e zy + zy ( ;) (y) (dy); Ee zrt = e t(z) < ; t ; see, for example, Proposition 3.4 of Cont and Tankov (24). We refer the reader to Cont and Tankov (24) and Klüppelberg and Kostadinova (28) for explicit expressions for the Laplace exponent () for some commonly used Lévy processes. Recall the renewal function of the renewal counting process fn t ; t g, de ned as X t = EN t = Pr ( k t) ; t : (2.2) In particular, if fn t ; t g is a Poisson process with intensity >, then t = t; and if follows a (2; ) distribution, then t = t=2 e 2t =4. More general examples of the renewal counting process fn t ; t g allowing explicit forms of the renewal function t can be found in Asmussen (23, pages 88 and 48). Denote by the set of all t for which < t. With t = infft : Pr ( t) > g, it is clear that [t; ]; if Pr ( = t) > ; = (t; ]; if Pr ( = t) = : For notational convenience, we write T = [; T ] \ for every xed T 2. We shall assume that the claim-size distribution F is regularly varying tailed, hence heavy tailed; that is, F (x) = F (x) > holds for all x and there is some constant, < <, such that the relation F (xy) lim x F (x) = y (2.3) holds for all y >. We use F 2 R to signify the regularity property in (2.3) and use R to denote the union of all R over the range < <. The class R contains a lot of popular distributions such as Pareto, Burr, Loggamma, and t distributions. Hereafter, all limit relationships are for x unless stated otherwise. For two positive functions a() and b(), we write a(x). b(x) if lim sup a(x)=b(x), a(x) & b(x) if lim inf a(x)=b(x), and a(x) b(x) if both. Furthermore, for two positive bivariate functions a(; ) and b(; ), we say that the asymptotic relation a(x; t) b(x; t) holds uniformly for t in a nonempty set if lim sup a(x; t) x b(x; t) = : t2 3

4 Clearly, the asymptotic relation a(x; t) b(x; t) holds uniformly for t 2 if and only if a(x; t) lim sup sup x t2 b(x; t) and lim inf inf a(x; t) x t2 b(x; t) ; which mean that the relations a(x; t). b(x; t) and a(x; t) & b(x; t), respectively, hold uniformly for t 2. We are now ready to state the main result of this paper: Theorem 2.. Consider the insurance risk model introduced in Section in which the claimsize distribution F belongs to the class R for some < < and the Laplace exponent of the Lévy process fr t ; t g satis es ( ) < for some >. Then, it holds uniformly for all t 2 that Pr (D t > x) F (x) (2.4) Taking t = in relation (2.4) yields a more transparent asymptotic formula: Ee () Pr (D > x) F (x) Ee : () Roughly speaking, the condition ( ) < in Theorem 2. means that the impact of the insurance claims dominates that of the nancial uncertainty. This is also con rmed by relation (2.4), which shows that the tail probability of the claim-size distribution determines the exact decay rate while the nancial uncertainty and the claim frequency only contribute to the coe cient of the asymptotic formula. From (2.) it is easy to verify that (z) is convex in z for which (z) is nite. Since () =, we see that the condition ( ) < implies that (z) < for all z 2 (; ]. In addition, by Jensen s inequality, the condition ( ) < implies that ER >. Hence, R t drifts to almost surely as t. As shown in Lemma 4.2 below, relation (2.4) with xed t 2 is an easy consequence of the one-dimensional version of Theorem 2. given in Resnick and Willekens (99). However, it is much harder to prove the claimed uniformity of relation (2.4), which is in essence the scienti c value of the present work. Note that the result of Resnick and Willekens (99) has recently been extended in many ways by Goovaerts et al. (25), Wang and Tang (26), Zhang et al. (29), and Chen and Yuen (29). Therefore, starting with these extended results, it should be possible and routine, but rather laborious, to further extend Theorem 2. to a somewhat broader class of heavy-tailed distributions (for example, the class of distributions with extended regularly varying tails), and to the case that claim sizes possess a certain dependence structure (for example, pairwise asymptotic independence). We shall not pursue such extensions in this paper. However, it would be interesting to establish results similar to Theorem 2. in the presence of certain dependence structures among the sources of randomness, fx ; X 2 ; : : :g, fn t ; t g, and fr t ; t g. 4

5 3 Applications to Ruin Theory 3. Finite- and In nite-time Ruin Probabilities Consider an insurance business commencing at time with initial wealth x. The cash ow of premiums less claims is modeled as a compound renewal process with the form C t = ct S t ; t ; (3.) where c is a xed rate of premium payment and fs t ; t g is a compound renewal process given in (.). Recall that the price process of the investment portfolio is the geometric Lévy process fe Rt ; t g. Thus, the wealth process of the insurer is described as U t = e x Rt + e R s dc s ; t : (3.2) As usual, de ne the ruin time of this risk model as T (x) = inf ft > : U t < j U = xg ; with the convention that inf? =. Then, the probability of ruin by a nite time t is and the probability of ultimate ruin is (x; t) = Pr (T (x) t) ; (x; ) = Pr (T (x) < ) : Yuen et al. (24, 26) studied the in nite-time ruin probability and related quantities of this renewal risk model. For the special case that fn t ; t g is a Poisson process, following their approach, it is not hard to establish an integro di erential equation and an integral equation for the in nite-time ruin probability. We shall not extend such a discussion here as we are mainly interested in the asymptotic behavior of the nite-time and in nite-time ruin probabilities. Theorem 3.. Consider the insurance risk model introduced above. Under the conditions of Theorem 2., it holds uniformly for all t 2 that (x; t) F (x) In particular, putting t = gives (3.3) Ee () (x; ) F (x) : (3.4) () Ee Paulsen (22) obtained relation (3.4) for the special case that fn t ; t g is a Poisson process and fr t ; t g is a Brownian motion with positive drift; see Proposition 4. of 5

6 Paulsen (22). The reader is also referred to Heyde and Wang (29) for a result for the nite-time ruin probability similar to (3.3) but with a xed-time horizon and fn t ; t g being a Poisson process. Due to the uniformity of relation (3.3), we can easily derive an explicit asymptotic expression for the Laplace transform of the ruin time T (x). Corollary 3.. Under the conditions of Theorem 2., it holds for every r that Ee (() Ee rt (x) (T (x)<) F (x) : (3.5) (() r) Ee Proof. When r =, relation (3.5) coincides with relation (3.4). Thus, we only need to consider r >. For this case the indicator (T (x)<) in (3.5) can be eliminated. By the uniformity of relation (3.3), we have Z Z Ee rt (x) = r (x; t)e rt dt rf (x) e ()s d s e rt dt: Using Fubini s theorem to interchange the order of integrals, Z Z Z e ()s d s e rt dt = e rt dt e ()s d s = r Therefore, (3.5) holds. s r) Z 3.2 Portfolio Optimization with a Constraint on Ruin e (() r)s d s : It is commonly acknowledged that risky investments may impair the insurer s solvency just as severely as large claims do; see Kalashnikov and Norberg (22) and Tang and Tsitsiashvili (23). Frolova et al. (22) also pointed out that disasters may arrive during the period when the market value of assets is low and the company will not be able to cover losses by selling these assets. In this section, we consider a so-called constant investment portfolio and determine the optimal investment strategy that maximizes the insurer s expected terminal wealth and maintains the insurer s solvency. Such optimization problems have been considered by several authors; see, for example, Schmidli (22), Paulsen (23), and Kostadinova (27). For simplicity, we assume that a nancial market consists of two assets, which can be traded continuously. One of them is a risk-free asset with price process satisfying dp () t = r P () t dt; t > ; with P () = and r >, while the other is a risky asset with price process satisfying dp () t = P () t dq t ; t > ; (3.6) with P () = and fq t ; t g being a Lévy process with Lévy triplet (r Q ; 2 Q ; Q), as described in Section 2. Suppose that the insurer continuously invests a constant fraction 6

7 2 [; ] of his wealth in the risky asset and keeps the remaining wealth in the risk-free asset. This constant investment strategy is commonly used in mathematical nance and actuarial science; see, for example, Björk (998), Emmer et al. (2), Emmer and Klüppelberg (24), Kostadinova (27), and Klüppelberg and Kostadinova (28). The price process of this investment portfolio satis es the stochastic di erential equation dp () t = P () t dq () t ; t > ; (3.7) with P () = and Q () t = ( )r t + Q t for t. From Theorem 4. of Cont and Tankov (24) (see also Proposition. of Sato (999)), the Lévy triplet of the Lévy process fq () t ; t g is given by 8 < : r = ( )r + r Q + R y ( ;)(y) (y) (dy); 2 = 2 2 Q ; (A) = Q (fx : x 2 Ag) for every Borel set A; where = fx : jxj < g. Hence, by Proposition 8.22 of Cont and Tankov (24) or Theorem 37 in Chapter 2 of Protter (25), we solve (3.7) to get where P () t = e R() t ; t ; R () t = ( )r t + Q t 2 2 Q 2 t + X with Q s = Q s Q s. The solution P () t exponential of the Lévy process fq () t (24), the Lévy triplet of fr () t 8 < : <st (ln( + Q s ) Q s ) ; given above is recognized as the stochastic ; t g. By Proposition 8.22 of Cont and Tankov ; t g is given by r = r R ln( + y) ( ;)(ln( + y)) y ( ;) (y) (dy); 2 = 2 ; (A) = (fy : ln( + y) 2 Ag) for every Borel set A: Using the Lévy triplet of fq t ; t g, one obtains the Laplace exponent of fr () t ; t g as (z) = Z Qz 2 ( )r + r Q Q y Q (dy) z + Z (( + y) z ) Q (dy); (3.8) provided that the second integral in (3.8) is nite. Note that Q (( ; ]) = since the price process fp () t ; t g in (3.6) is positive. It follows from (3.8) that where = r Q r + R y Q (dy). ( ) = r + ; (3.9) 7

8 Consider a xed-time horizon t >, say t = 5. Our goal is to determine a value 2 [; ] that maximizes the expected value of the terminal wealth = e R() t x + e R() s dc s ; (3.) U () t subject to the following constraint on the ruin probability: (x; t ) = Pr inf U s () < st U = x p; (3.) where fc t ; t g is given in (3.) and p is some small positive number, say p = 5%. Let the conditions of Theorem 3. hold, and let > so that the i.i.d. claims have a nite mean = EX. Simple calculation gives EU () t = xe ( )t + e ( )(t s) d (cs s ) : (3.2) Since there is usually no closed-form expression for (x; t ) available, we use its approximation given in (2.4); that is to say, we replace (3.) by (x; t ) F (x) e ()s d s p: (3.3) For given x >, >, t >, and < p <, de ne = : 2 [; ] and F (x) e ()s d s p : Suppose? 6= f : () < g. Every 2 is called an admissible investment strategy. With > given, we see from (3.8) that (), as a function of, is convex in 2 [; ] with () = r <. Hence, is a closed interval in [; ]. Let = [a x ; b x ] [; ]. To simplify the optimization problem, we assume that s is absolutely continuous with respect to the Lebesgue measure; that is to say, there exists a nonnegative and measurable function s such that s = R s udu for s 2 [; ). We further assume that the relation c s (3.4) holds almost everywhere for s 2 [; ). Relation (3.4) can be interpreted as the safety loading condition of the insurance portfolio. It is veri able for many interesting cases, for example, when follows an exponential distribution or a (2; ) distribution. Under (3.4), the expectation EU () t, as expressed in (3.2), is increasing in ( ). Hence, the optimization problem becomes maximizing ( ) subject to 2 : (3.5) From (3.9), we see that ( ) is increasing in 2 [; ] when > and decreasing in 2 [; ] when <. Hence, if >, the solution to (3.5) is = b x ; (3.6) 8

9 while if <, the solution to (3.5) is = a x : (3.7) When =, however, ( ) = r does not depend on. Thus, every admissible strategy 2 could be used as a solution to (3.5). Nevertheless, in order to reduce uncertainty from the risky asset, we may choose (3.7) as the solution. The optimization solutions (3.6) and (3.7) are intuitively clear. The condition > means that the expected return rate of the risky asset is higher than that of the risk-free asset. Hence, as (3.6) shows, the insurer will invest as much as he is allowed in the risky asset. The optimization solution (3.7) can be explained in a similar way. Moreover, we have the following observation. In modern portfolio theory, the quantity 2 Q, which is the volatility when fq t ; t g is a Brownian motion with drift, is often used to measure the risk of the risky asset. Recall relation (3.8) and the de nition of. Clearly, () is increasing in 2 Q. Hence, both a x and b x are decreasing in 2 Q, meaning that the riskier the risky asset is, the less the insurer will invest in it. 4 Proofs 4. Lemmas Let us rst recall some properties of distributions with regularly varying tails. By Theorem.5.2 of Bingham et al. (987), the convergence in relation (2.3) is uniform over ["; ) for every xed " > ; that is to say, lim sup x y2[";) F (xy) F (x) y = : (4.) By Theorem.5.6 of Bingham et al. (987), the well-known Potter s bounds for distributions in the class R are stated as follows. If F 2 R for some < <, then for arbitrarily xed b > and " >, there exists some x > such that, for all x; y x, b (y=x) +" ^ (y=x) " F (y) F (x) b (y=x) +" _ (y=x) " : (4.2) For arbitrarily xed < < < <, by xing the variable y to x and " < ( ) ^ ( ) in (4.2), we see that F (x) = o x ; x = o F (x) : (4.3) Let X and Y be two independent random variables with X following a distribution F 2 R for some < < and Y being a nonnegative random variable satisfying EY < for some >. Then, for every xed M, with x > given in (4.2), we have Pr (XY > x; Y > M) lim x F (x) = lim x Pr (XY > x; Y 2 (M; x=x ] [ (x=x ; )) F (x) = EY (Y >M) ; (4.4) 9

10 where we used the dominated convergence theorem guaranteed by (4.2) in dealing with the rst part corresponding to Y 2 (M; x=x ], and used Markov s inequality and the second relation of (4.3) in dealing with the second part. Relation (4.4) with M = is well known, and is usually referred to as Breiman s theorem; see Breiman (965). The following result is the one-dimensional version of Theorem 2. of Resnick and Willekens (99): Lemma 4.. Consider the randomly weighted sum S (W ) = X W k X k ; where fx ; X 2 ; : : :g is a sequence of i.i.d. nonnegative random variables with common distribution F 2 R for some < <, and fw ; W 2 ; : : :g is another sequence of nonnegative random variables independent of fx ; X 2 ; : : :g. We have Pr S (W ) > x F (x) if one of the following assumptions holds:. < < and for some < " < ^ ( ), X 2. < and for some < " <, X EWk ; E W +" k _ W " k < ; X E W +" k _ W " +" k < : Lemma 4.2. Under the conditions of Theorem 2., relation (2.4) holds for every xed t 2. Proof. We apply Lemma 4. to the series D t given in (.2). For < <, we choose some < " < ^ ( ) ^ ( ) so that X E e (+")R k _ e ( Z X E e (+")Rs + e ( = Ee(+") Ee + Ee( ") (+") Ee < : ")R k ( k t) ( ") ")Rs Pr ( k 2 ds)

11 Similarly, for <, we choose some < " < ^ ( ) to justify that X Therefore, by Lemma 4., Pr (D t > x) F (x) E e (+")R k _ e ( ")R k +" ( k t) < : X Ee R k ( k t) = F (x) This proves that relation (2.4) holds for every xed t 2. Lemma 4.3. Under the conditions of Theorem 2., for every xed T 2, it holds uniformly for all t 2 T that Pr Xe R (t) > x F (x) e s() Pr ( 2 ds) : (4.5) Proof. Conditioning on, we have Pr Xe R (t) > x = Pr Xe Rs > x Pr ( 2 ds) : For an arbitrarily xed large number M >, according to (jr s j M), (R s > M), and (R s < M), we split the right-hand side of the above into three parts as I (x; t) + I 2 (x; t) + I 3 (x; t). By relation (4.), it holds uniformly for all t 2 that I (x; t) = F (x) = F (x) Pr Xe Rs > x; jr s j M Pr ( 2 ds) Ee Rs (jrsjm) Pr ( 2 ds) e s() Pr ( 2 ds) F (x) Ee Rs (jrsj>m) Pr ( 2 ds) : Thus, it su ces to show that the terms I 2 (x; t), I 3 (x; t), and the last term above, denoted as I 4 (x; t), are negligible for large M >, uniformly for all t 2 T, in comparison to F (x) R t e s() Pr ( 2 ds). More precisely, we are going to prove that, for j = 2; 3; 4, For I 2 (x; t), we have lim sup M I 2 (x; t) = lim sup x I j (x; t) sup t2 T F (x) R t = : (4.6) e s() Pr ( 2 ds) Pr Xe Rs > x; R s > M Pr ( 2 ds) Pr Xe M > x Pr ( t) e M F (x) Pr ( t) :

12 Trivially, Pr ( t) e T () This proves relation (4.6) with j = 2. For I 3 (x; t), we have e s() Pr ( 2 ds) : (4.7) I 3 (x; t) = Pr Xe Rs > x; R s < M Pr ( 2 ds) Pr Xe inf st R s > x; inf s < st M Pr ( t) : (4.8) Note that, for all T and x > x >, Pr inf s > x Pr sup R s > st st x Pr ( R T > x x ) ; see the lemma of Willekens (987). This, together with Ee R T = e T () <, implies that Ee inf st R s < : Hence, applying (4.4) to (4.8), it holds uniformly for all t 2 T that I 3 (x; t). Ee inf st R s (infst R s< M) F (x) Pr ( t) : This, together with (4.7), proves relation (4.6) with j = 3. Finally, for I 4 (x; t), by Hölder s inequality, Ee Rs (jrsj>m) Pr ( 2 ds) Ee R s = Pr (jrs j > M) Pr sup st jr s j > M = Pr ( t) : = Pr ( 2 ds) Hence, by (4.7), relation (4.6) with j = 4 holds and we conclude the proof. Lemma 4.4. Under the conditions of Theorem 2., for every xed T 2 and n = ; 2; : : :, it holds uniformly for all t 2 T that Pr X k e R k ( k t) > x F (x) e s() Pr ( k 2 ds) : Proof. By Lemma 4.3, it su ces to prove that, uniformly for all t 2 T, Pr X k e R k ( k t) > x Pr X k e R k ( k t) > x : (4.9) 2

13 We rst prove the lower-bound version of relation (4.9). Clearly, Pr X k e R k ( k t) > x Pr n[ X k e R k ( k t) > x Pr X k e R k ( k t) > x X k<jn Pr X k e R k ( k t) > x; X j e R j ( j t) > x : Thus, it remains to verify that the second term above is negligible, uniformly for all t 2 T, in comparison to the rst term. Actually, for arbitrarily xed k < j n and M >, Pr X k e R k ( k t) > x; X j e R j ( j t) > x Pr X k e Rs > x; e Rs X j e R j +R k ( j t) > x; e Rs x k = s Pr ( k 2 ds) M + Pr e Rs > x Pr ( k 2 ds) M = I (x; t) + I 2 (x; t): Note that e Rs and e R j +R k in I (x; t) conditional on ( k = s) are independent. It holds for arbitrarily xed " > and all large M > that I (x; t) Pr X j sup e Rs > M Pr X k e Rs > x Pr ( k 2 ds) st " Pr X k e R k ( k t) > x : For I 2 (x; t), by Markov s inequality, the second relation of (4.3), relation (4.7), and Lemma 4.3, we have I 2 (x; t) x Ee R s Pr ( k 2 ds) M x Pr ( k t) M = o()f (x) e s() Pr ( k 2 ds) = o() Pr X k e R k ( k t) > x : Therefore, it holds uniformly for all t 2 T that Pr X k e R k ( k t) > x; X j e R j ( j t) > x = o() Pr X k e R k ( k t) > x : (4.) We next prove the upper-bound version of relation (4.9). For arbitrarily xed < <, we 3

14 derive Pr Pr + Pr X k e R k ( k t) > x n[ X k e R k ( k t) > x x X k e R k ( k t) > x; = I 3 (x; t) + I 4 (x; t): n[ X k e R k ( k t) > x ; n n\ X k e R k ( k t) x x By Lemma 4.3, it holds uniformly for all t 2 T that I 3 (x; t) Pr X k e R k ( k t) > x x ( ) Pr X k e R k ( k t) > x : Thus, it remains to verify that I 4 (x; t) is negligible, uniformly for all t 2 T, in comparison to I 3 (x; t). Indeed, we have I 4 (x; t) k e R k ( k t) > x n ; X X j e R j ( j t) > xa X k6=jn = o() j2f;:::;ngnfkg Pr X k e R k ( k t) > x n ; X je R j ( j t) > x n Pr X k e R k ( k t) > x ; where in the last step we used relation (4.) and Lemma 4.3. This ends the proof. Lemma 4.5. Under the conditions of Theorem 2., for every xed T 2, it holds for every " >, for all large n, and uniformly for all t 2 T that X Pr X k e R k ( k t) > x. "F (x) k=n+ Proof. Choose some > such that ( ) >. Then, X Pr X k e R k ( k t) > x + k=n+ X Pr e Rs X k e R k +Rn ( k t) > x; e Rs x k=n+ Pr e Rs > x Pr ( n 2 ds) = I (x; t) + I 2 (x; t): n = s Pr ( n 2 ds) 4

15 For I (x; t), note that, by Lemma 4.2, it holds uniformly for all s t 2 T that X Pr X k e R k +Rn ( k t) > x n = s = Pr (D t s > x) k=n+ Pr (D T > x) F (x) Z T e u() d u : Thus, by further conditioning on e Rs, it can be shown that, uniformly for all t 2 T, I (x; t). Pr Xe Rs > x; e Rs x Z T e u() d u Pr ( n 2 ds) T F (x) T Pr Xe Rs > x Pr ( n 2 ds) F (x) T Pr ( n t) ; e s() Pr ( n 2 ds) where in the third step we used Lemma 4.3. Following the proof of Lemma 5.3 of Tang (24), we have lim sup EN t (Nt>n) = : (4.) n t2 T t Therefore, it holds for all large n and uniformly for all t 2 T that I (x; t). "F (x) By Markov s inequality and the second relation of (4.3), it holds uniformly for all t 2 T that This ends the proof. I 2 (x; t) x ( x ( ) ) = o()f (x) Ee R s Pr ( n 2 ds) e s() d s Lemma 4.6. Let Z be an exponential functional of a Lévy process fr t ; t g de ned as Z = Then, we have the following two results: Z e Rt dt: (4.2). Z < almost surely if and only if R t almost surely as t ; 2. If > and () <, then EZ <. Proof. See Subsection 2. of Maulik and Zwart (26). 5

16 4.2 Proof of Theorem 2. By Lemma 4.2, relation (2.4) holds for every xed t 2. We formulate the proof of the uniformity into two steps. First, we establish the local uniformity of relation (2.4); that is, for arbitrarily xed T 2, relation (2.4) holds uniformly for all t 2 T. For arbitrarily xed < < and n = ; 2; : : :, we have Pr (D t > x) Pr X k e R k ( k t) > x = I (x; t) + I 2 (x; t): x + Pr X X k e R k ( k t) > x k=n+ By Lemma 4.4, it holds uniformly for all t 2 T that I (x; t). ( ) F (x) By Lemma 4.5, for arbitrarily xed " >, it holds for all large n and uniformly for all t 2 T that I 2 (x; t). " F (x) (4.3) It follows that, uniformly for all t 2 T, Pr (D t > x). ( ) + " F (x) By the arbitrariness of " and, we prove that, uniformly for all t 2 T, Pr (D t > x). F (x) The corresponding lower bound can be constructed in a similar way. Actually, by Lemma 4.4, it holds uniformly for all t 2 T that Pr (D t > x) Pr X k e R k ( k t) > x F (x) e s() d s X k=n+ F (x) Ee n() Pr ( k 2 ds) Since Ee n() tends to as n, it follows that, uniformly for all t 2 T, Pr (D t > x) & F (x) 6

17 Next, we extend the uniformity of relation (2.4) to. For arbitrarily xed < " <, choose some large T 2 such that Z e s() d s " Z T T Let t 2 [T; ] and apply (4.4) and Lemma 4.2. On the one hand, (4.4) Pr (D t > x) Pr (D > x) F (x) and on the other hand, Z e s() d s ( + ")F (x) e s() d s ; Pr (D t > x) Pr (D T > x) F (x) Z T + " F (x) + " F (x) e s() d s Z e s() d s By these two estimates and the arbitrariness of ", we see that relation (2.4) holds also uniformly for all t 2 [T; ]. 4.3 Proof of Theorem 3. Substituting (3.) into (3.2) yields that, for every t 2, Z s (x; t) = Pr x + c e Ru du D s < ; (4.5) inf st where s t is understood as s < when t =. Therefore, it follows immediately from Theorem 2. that, uniformly for all t 2, (x; t) Pr (D t > x). F (x) It remains to derive the corresponding uniform asymptotic lower bound for (x; t). Still start from (4.5) and let T 2 be arbitrarily xed. It holds for arbitrarily xed > and all T t that (x; t) Pr D t c Z Pr (D t > ( + )x) e Ru du > x By Theorem 2., it holds uniformly for all t 2 that Pr c Z Pr (D t > ( + )x) ( + ) F (x) 7 e Ru du > x : (4.6)

18 By Markov s inequality, Pr c Z e Ru du > x Z x E e Ru du ; c where the niteness of E R e Ru du is guaranteed by Lemma 4.6. Substituting these two estimates into (4.6) and using the arbitrariness of, we see that, uniformly for all T t, (x; t) & F (x) (4.7) We now focus on the uniformity of (4.7) over t 2 T. Clearly, Z Nt (x; t) Pr D Nt c e Ru du > x X Z k = Pr X k e R k c e Ru du ( k t) > x : k It follows that, for arbitrarily xed > and n = ; 2; : : :, (x; t) Pr X k e R k ( k t) > ( + 2)x Pr c Pr c X k=n+ Z k k e Ru du ( k t) > x Z k k e Ru du ( k t) > x = I (x; t) I 2 (x; t) I 3 (x; t): (4.8) By Lemma 4.4, it holds uniformly for all t 2 T that I (x; t) ( + 2) F (x) e s() Pr ( k 2 ds) ( + 2) Ee n() F (x) Introduce a random variable Z independent of fr t ; t g, fn t ; t g and equal in distribution to R e Rt dt. By Lemma 4.6, EZ <. Hence, by Markov s inequality and the second relation of (4.3), Pr(Z > x) = o(f (x)). It holds uniformly for all t 2 T that Z k I 2 (x; t) Pr e Ru du ( k t) > x k cn Z s Pr e Ru du > x Pr ( k 2 ds) cn Pr Z > x Pr ( k t) cn = o()f (x) 8

19 For arbitrarily xed < " <, further introduce a nonnegative random variable Z " independent of fr t ; t g, fn t ; t g and with tail given by Pr (Z " > x) = "F (x) _ Pr(Z > x); x : Hence, Pr (Z " > x) "F (x) and the distribution of Z " belongs to the class R too. Clearly, X Z k I 3 (x; t) Pr ce Rn ( nt) e (Ru Rn) du > x k=n+ k = Pr Ze Rn ( nt) > x c Pr Z " e Rn ( nt) > x : c By Lemma 4.3, it holds uniformly for all t 2 T that I 3 (x; t). Pr Z " > x e s() Pr ( n 2 ds). "c F (x) c Substituting these estimates into (4.8), we obtain that, uniformly for all t 2 T, (x; t) & ( + 2) Ee n() "c F (x) By the arbitrariness of, ", and n, relation (4.7) holds uniformly for all t 2 T. Acknowledgments. The authors would like to thank the anonymous referee for his/her careful reading and useful comments. Qihe Tang acknowledges support of the 28 Old Gold Summer Fellowship from the University of Iowa, Guojing Wang acknowledges support of the Natural Science Foundation of Jiangsu Province (Grant No: KB2855), and Kam C. Yuen acknowledges support of a research grant from the University of Hong Kong. References [] Applebaum, D. Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge, 24. [2] Asmussen, S. Ruin Probabilities. World Scienti c, London, 23. [3] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular Variation. Cambridge University Press, Cambridge, 987. [4] Björk, T. Arbitrage Theory in Continuous Time. Oxford University Press, New York, 998. [5] Breiman, L. On some limit theorems similar to the arc-sin law. Theory Prob. Appl. (965)

20 [6] Cai, J. Ruin probabilities and penalty functions with stochastic rates of interest. Stochastic Process. Appl. 2 (24), no., [7] Chen, Y.; Yuen, K. C. Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25 (29), no., [8] Cont, R.; Tankov, P. Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL, 24. [9] Emmer, S.; Klüppelberg, C. Optimal portfolios when stock prices follow an exponential Lévy process. Finance Stoch. 8 (24), no., [] Emmer, S.; Klüppelberg, C.; Korn, R. Optimal portfolios with bounded capital at risk. Math. Finance (2), no. 4, [] Frolova, A.; Kabanov, Y.; Pergamenshchikov, S. In the insurance business risky investments are dangerous. Finance Stoch. 6 (22), no. 2, [2] Goovaerts, M. J.; Kaas, R.; Laeven, R. J. A.; Tang, Q.; Vernic, R. The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuar. J. (25), no. 6, [3] Heyde, C. C.; Wang, D. Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims. Adv. in Appl. Probab. 4 (29), no., [4] Kalashnikov, V.; Norberg, R. Power tailed ruin probabilities in the presence of risky investments. Stochastic Process. Appl. 98 (22), no. 2, [5] Klüppelberg, C.; Kostadinova, R. Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom. 42 (28), no. 2, [6] Kostadinova, R. Optimal investment for insurers when the stock price follows an exponential Lévy process. Insurance Math. Econom. 4 (27), no. 2, [7] Maulik, K.; Zwart, B. Tail asymptotics for exponential functionals of Lévy processes Stochastic Process. Appl. 6 (26), no. 2, [8] Paulsen, J. Risk theory in a stochastic economic environment. Stochastic Process. Appl. 46 (993), no. 2, [9] Paulsen, J. On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Probab. 2 (22), no. 4, [2] Paulsen, J. Optimal dividend payouts for di usions with solvency constraints. Finance Stoch. 7 (23), no. 4, [2] Paulsen, J. Ruin models with investment income. Probab. Surv. 5 (28), [22] Paulsen, J.; Gjessing, H. K. Ruin theory with stochastic return on investments. Adv. in Appl. Probab. 29 (997), no. 4,

21 [23] Protter, P. E. Stochastic Integration and Di erential Equations. Second edition. Springer-Verlag, Berlin, 25. [24] Resnick, S. I.; Willekens, E. Moving averages with random coe cients and random coe cient autoregressive models. Comm. Statist. Stochastic Models 7 (99), no. 4, [25] Sato, K. Lévy Processes and In nitely Divisible Distributions. Cambridge University Press, Cambridge, 999. [26] Schmidli, H. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 2 (22), no. 3, [27] Tang, Q. Uniform estimates for the tail probability of maxima over nite horizons with subexponential tails. Probab. Engrg. Inform. Sci. 8 (24), no., [28] Tang, Q.; Tsitsiashvili, G. Precise estimates for the ruin probability in nite horizon in a discrete-time model with heavy-tailed insurance and nancial risks. Stochastic Process. Appl. 8 (23), no. 2, [29] Wang, D.; Tang, Q. Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22 (26), no. 2, [3] Wang, G.; Wu, R. Distributions for the risk process with a stochastic return on investments. Stochastic Process. Appl. 95 (2), no. 2, [3] Willekens, E. On the supremum of an in nitely divisible process. Stochastic Process. Appl. 26 (987), no., [32] Yuen, K. C.; Wang, G.; Ng, K. W. Ruin probabilities for a risk process with stochastic return on investments. Stochastic Process. Appl. (24), no. 2, [33] Yuen, K. C.; Wang, G.; Wu, R. On the renewal risk process with stochastic interest. Stochastic Process. Appl. 6 (26), no., [34] Zhang, Y.; Shen, X.; Weng, C. Approximation of the tail probability of randomly weighted sums and applications. Stochastic Process. Appl. 9 (29), no. 2,