Asymptotic Ruin Probabilities for a Bivariate Lévy-driven Risk Model with Heavy-tailed Claims and Risky Investments

Transcription

1 Asymptotic Ruin robabilities for a Bivariate Lévy-driven Risk Model with Heavy-tailed Claims and Risky Investments Xuemiao Hao and Qihe Tang Asper School of Business, University of Manitoba 181 Freedman Crescent, Winnipeg, Manitoba R3T 5V4, Canada xuemiao.hao@ad.umanitoba.ca Department of Statistics and Actuarial Science, University of Iowa 241 Schaeffer Hall, Iowa City, IA 52242, USA qihe-tang@uiowa.edu December 6, 212 Abstract Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y = x >, evolves according to dy t = Y t dr t d t for t >, where and R are two independent Lévy processes representing, respectively, a loss process in a world without economic factors and a process describing return on investments in real terms. Motivated by a conjecture of aulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x, which confirms aulsen s conjecture. Keywords: Asymptotics; Extended regular variation; Finite-time and infinitetime ruin probabilities; Lévy process; Stochastic difference equation; Tail probabilities Mathematics Subject Classification: rimary 91B3; Secondary 6G51, 91B28 1 Introduction Consider a bivariate Lévy-driven risk model in which the surplus process, Y, of an insurance company is modeled by Y t = x t + Corresponding author. hone: ; fax: Y s dr s, t >, 1.1 1

2 with Y = x > the initial surplus level, and and R two independent Lévy processes representing, respectively, a loss process in a world without economic factors and a process describing return on investments in real terms. See aulsen 1998a, 1998b, 22, 28 for detailed explanations. This model does not mean that the surplus must be completely invested in a risky asset. An understanding on the stochastic process R is as follows. Consider a financial market consisting of a risk-free bond with price S,t and d risky stocks with prices S k,t for k = 1,..., d. Denote by π the proportion of the surplus invested in the bond and by π k the proportion invested in stock k. Thus, π = π, π 1,..., π d, called a relative investment portfolio and assumed to be time invariant, satisfies π + π π d = 1. Then the differential in 1.1 is dr t = d k= π k ds k,t S k,t, t >. In particular, consider the Black Scholes market with { ds,t = rs,t dt, t >, ds k,t = S k,t µ k dt + σ k dw k,t, t >, where W t = W 1,t,... W d,t is a d dimensional Wiener process, r, < µ k < and σ k > for k = 1,..., d. Then dr t = π rdt + d π k µ k dt + σ k dw k,t, t >. Therefore, the assumption that R is a Lévy process is fulfilled in this particular case. The solution of 1.1 is given by Y t = e R t x e R s d s, 1.2 where R, also a Lévy process, is the logarithm of the stochastic exponential also called the Doléans-Dade exponential of R; see, e.g. rotter 25 for details. For simplicity, we write Z t = e R s d s, so that Y t = e R t x Z t. The stochastic process Z is usually called the discounted net loss process. We shall start with 1.2 instead of 1.1, as has been done by many researchers including Kalashnikov and Norberg 22 and Klüppelberg and Kostadinova 28. We are interested in the asymptotic behavior of the finite-time and infinite-time ruin probabilities of this bivariate Lévy-driven risk model. As usual, the finite-time ruin probability is defined as ψx, T = inf Y t < t T Y = x, T, 2

3 and the infinite-time ruin probability as ψx, = lim ψx, T = inf Y t < T t< Y = x. Introduce ZT = sup Z t = sup e R s d s, T, t T t T where the supremum is taken over t < if T =. Noticing that Y t < and Z t > x are equivalent, we have ψx, T = Z T > x, < T. 1.3 In this paper, we aim at a simple and unified asymptotic expression for ψx, T with < T as the initial surplus level x becomes large. 2 Main Results Hereafter, all limit relationships are for x unless otherwise stated. For two positive functions a and b, we write ax bx if lim ax/bx = 1, write ax bx or bx ax if lim sup ax/bx 1 and write ax bx if < lim inf ax/bx lim sup ax/bx <. For a real number x, we write x + = x and x = x as the positive and negative parts of x, respectively. For a general Lévy process L, its characteristic exponent, Ψ L u = log Ee 1 iul, has the following Lévy-Khintchine representation: Ψ L u = ilu σ2 u e iux + iux1 x <1 νdx 2.1 with l R, σ, and Lévy measure ν on R \ {} satisfying x2 1 νdx <. We further denote by ϕ L u = Ψ L iu = log Ee ul 1 the Laplace exponent of L. Clearly, log Ee ult = tϕ L u for every t. Now we turn to the loss process, which we assume to be a Lévy process. When its Lévy measure satisfies ν 1 = ν 1, >, introduce Π = ν 1 1, /ν 1, which is a proper probability measure on 1,. We assume that Π is of extended regular variation ERV. Formally, a distribution F belongs to the class ERV α, β for some α β < if F x = 1 F x > holds for all x and the relations v β lim inf F vx F x lim sup F vx F x v α 2.2 hold for all v 1. Note that relations 2.2 with α = β define the class R α of distributions of regular variation. Here comes our main result: 3

4 Theorem 2.1 Consider the bivariate Lévy-driven risk process Y given by 1.2, where and R are two independent Lévy processes. Assume Π ERV α, β for some < α β <. 1 If E e β+ε R 1 < 1 for some ε >, then it holds for every T, that T ψx, T λ Xe R t > x dt, 2.3 where λ = ν 1 and X is distributed by Π and independent of and R. 2 Furthermore, if E e α ε R 1 e β+ε R 1 < 1 for some ε, α, then relation 2.3 also holds for T =. The condition Π ERV α, β means that insurance claims are heavy tailed. Roughly speaking, the exponential moment condition in each case means that insurance claims control the uncertainty of negative returns of investments. When α = β, it is easy to see that the exponential moment conditions in both cases are equivalent. For this case, the statement of Theorem 2.1 is simplified to the following: Corollary 2.1 Let α = β in Theorem 2.1. Then it holds for every T, ] that ψx, T 1 eϕ RαT ϕ Rα ν x. 2.4 Relation 2.4 with T = was ever conjectured by aulsen 22, Theorem 3.2b and Remark 3.2b. Therefore, our work shows that aulsen s conjecture is indeed true. Admittedly, the class ERV is marginally larger than the class R, but it incurs a lot more technicalities to the study. A self-contained proof targeting Corollary 2.1 only can be much simpler than the proof of Theorem 2.1 given below. However, we have reasons to believe that the ruin probabilities for the more general subexponential case will asymptotically behave in the form of 2.3, rather than 2.4. Therefore, we carry out our research within the class ERV, hoping that it will offer insights in the subexponential case. The asymptotic behavior of the ruin probabilities in presence of investments has been extensively investigated in the literature. Early works focused on a special case in which is a compound oisson process with subexponential jumps and R is a deterministic linear function corresponding to a constant rate of compound interest; see Klüppelberg and Stadtmüller 1998, Asmussen 1998, Kalashnikov and Konstantinides 2 and Konstantinides et al. 22. Later on, for the subexponential case Tang 25 established a formula for the finite-time ruin probability, which is in line with the formula for the infinitetime ruin probability. The formulas obtained in these papers are essentially the same as 2.3 with R replaced by a deterministic linear function. 4

5 It is more interesting to consider that the insurance company makes both risk-free and risky investments. For a very general case, Kalashnikov and Norberg 22 proved that the infinite-time ruin probability necessarily decays to as a power function as the initial surplus level increases no matter what tail behavior the jumps of have. Frolova et al. 22 obtained an explicit asymptotic formula for the infinite-time ruin probability for the case in which is a compound oisson process with exponential hence light-tailed jumps and R is a Wiener process with drift. Most of recent research has been done for the case in which has regularly varying jumps. For example, aulsen 22 showed relation 2.4 with T = for a special case in which is a compound oisson process with regularly varying jumps and R or, equivalently, R is a Wiener process with drift. Klüppelberg and Kostadinova 28 extended this result to a general Lévy process R. Heyde and Wang 29 obtained similar results to our relations 2.3 and 2.4 with T < for the case in which is a compound oisson process with heavy-tailed jumps and R is a general Lévy process. Tang et al. 21 established a result similar to our relation 2.3 for both T < and T = for the case in which is a compound renewal process with regularly varying jumps and R is a general Lévy process. Albrecher et al. 212 also investigated the asymptotic behavior of the infinite-time ruin probability and related quantities for the case in which is a compound renewal process having light-tailed or heavy-tailed jumps and R is a Brownian motion with drift. Hult and Lindskog 211 considered a more general case in which is a Lévy process with regularly varying jumps and R is a semimartingale and they established an asymptotic formula for the finite-time ruin probability, which holds uniformly for all R with stochastic exponential e R t fulfilling a certain moment condition. See also Asmussen and Albrecher 21, Sections VIII.5-6 for a brief review of ruin theory in presence of investments. We are going to prepare some lemmas in Section 3 and then prove Theorem 2.1 and Corollary 2.1 in Section 4. 3 Lemmas In this section we prepare some lemmas for the main result. The first lemma below describes some well-known properties of distributions of extended regular variation; see Bingham et al. 1987, roposition and Tang and Tsitsiashvili 23, Lemma 3.5: Lemma 3.1 Suppose F ERV α, β for some < α β <. 1 For every ε, α and every b > 1, there is some x > such that the inequalities 1 y α ε y β+ε F xy b F x b y α ε y β+ε hold whenever x > x and xy > x. 5

6 2 It holds for every ε > that F x = o x α ε and x β+ε = o F x. A feature of the following lemma is that inequality 3.1 holds uniformly for all nonnegative random variables Y independent of X: Lemma 3.2 Let X be a real-valued random variable whose distribution belongs to the class ERV α, β for some < α β <. Then for every ε, α and every b > 1, there is some x = x ε, b > such that, for all x > x and all nonnegative random variables Y independent of X, XY > x be Y α ε Y β+ε. 3.1 X > x roof. For arbitrarily chosen b 1, b, by Lemma 3.11 there is some x > such that, for all x > x, XY > x XY > x, Y x/x + Y > x/x b X > xe Y α ε Y β+ε + x/x β+ε EY β+ε. Then inequality 3.1 follows since x β+ε = ox > x by Lemma By going along the same lines of the proof of Lemma of Samorodnitsky and Taqqu 1994, we obtain the following: Lemma 3.3 Let X, Y be jointly distributed random variables. If the distribution of X belongs to the class ERV α, β for some α β < and Y > x = o X > x, then X + Y > x X > x. The tail behavior of randomly weighted sums of heavy-tailed random variables has become a hot topic in applied probability since Resnick and Willekens A very recent work on the topic is Olvera-Cravioto 212. The next lemma summarizes several known results: Lemma 3.4 Let {X k, k N} be a sequence of independent and identically distributed i.i.d. random variables with common distribution F and let {ω k, k N} be another sequence of nonnegative random variables, non-degenerate at zero and independent of {X k, k N}. Assume F ERV α, β for some < α β < and one of the following two conditions holds: 1 in case β, 1, there is some ε, α such that E ω α ε k ω β+ε k < ; 6

7 Then 2 in case β [1,, there is some ε, α such that max 1 n< E ω α ε k 1 ω β+ε β+ε k <. n ω k X k > x ω k X + k > x Furthermore, the distributions of max n 1 n< ω kx k and ω kx + k class ERV α, β. ω k X k > x. 3.2 both belong to the roof. The second relation in 3.2 is proved by Theorem 3.1b of Zhang et al. 29. Let us check the extended regular variation of the distribution, denoted by F ω +, of ω kx + k. It is easy to prove that, for each k N, the relation ω kx k > x F x holds and the distribution of ω k X k belongs to the class ERV α, β; see Cline and Samorodnitsky 1994, Theorem 3.5 for these facts. By Lemma 3.2, for every b > 1 there is some x > such that the inequality ω k X k > x bf xe ω α ε k ω β+ε k holds for all k N and all x > x. Hence, for arbitrarily given δ >, all x > x and all large n, ω k X k > x bf x E ω α ε k ω β+ε k δ ω 1 X 1 > x. 3.3 k=n k=n By the second relation in 3.2, inequalities 3.3 and the fact that each ω k X k follows a distribution in ERV α, β, it holds for arbitrarily fixed v > 1 and some large n that F ω + vx F ω + x ω kx k > vx n ω kx k > x 1 + δ ω kx k > vx n ω kx k > x 1 + δv α. Symmetrically, F + ω vx F + ω x δ v β. The arbitrariness of δ implies that F ω + ERV α, β. The extended regular variation of the distribution of max n 1 n< ω kx k easily follows from the extended regular variation of F ω + and the first relation in 3.2. It remains to verify the first relation in 3.2. Since the inequality ω kx + k max n 1 n< ω kx k is obvious, we only need to prove that n ω k X k > x ω k X k > x. 3.4 max 1 n< 7

8 Actually, by Theorem 3.2 of Chen and Yuen 29, it holds for all n N that n n ω k X k > x ω k X k > x. 3.5 By 3.5 and 3.3, it holds for arbitrarily given δ > and some large n that n n max ω k X k > x ω k X k > x 1 ω k X k > x. 1 n< 1 + δ Then the arbitrariness of δ implies 3.4. Consider the compound oisson process N t C t = X k, t, 3.6 where {X k, k N} is a sequence of i.i.d. random variables with generic random variable X and common distribution F, while {N t, t } is a oisson process, independent of {X k, k N}, with intensity λ >. The following lemma plays a crucial role in proving Theorem 2.1: Lemma 3.5 Let C be a compound oisson process given by 3.6 and let L be a Lévy process independent of C. If F ERV α, β for some < α β < and ϕ L β + ε < for some ε, α, then it holds for every T, ] that sup t T e Ls dc s > x λ T Xe Lt > x dt. Furthermore, sup t T e Ls dc s follows a distribution in ERV α, β. roof. Let τ k, k N, be the arrival times of the oisson process {N t, t }. Then sup t T e Ls dc s = sup t T N t X k e Lτ k = max n< n X k e Lτ k 1τk T, where, by convention, the summation over an empty set of indices produces a value. This enables us to apply Lemma 3.4. By its convexity, ϕ L u < for all u, β + ε]. Then it is straightforward to verify the corresponding conditions in order to apply Lemma 3.4. Therefore, sup t T e Ls dc s > x X k e Lτ k 1τk T > x T = λ Xe Lt > x dt, where in the last step we used the fact that τ k dt = λdt. The following lemma, which is a natural generalization of Lemma 2 of Grey 1994, does not require any information on the dependence structure of A, B or on the left tail of A: 8

9 Lemma 3.6 Let A, B and Q be three random variables, with Q independent of A, B. If each of A and Q follows a distribution in the class ERV α, β for some < α β <, and B is nonnegative satisfying EB β+ε < for some ε, α, then the distribution of A + QB belongs to the class ERV α, β and A + QB > x A > x + QB > x. 3.7 roof. We only focus on the proof of relation 3.7 since the other conclusion follows immediately from relation 3.7 and the fact that the distribution of QB belongs to the class ERV α, β. Arbitrarily choose an increasing function l :,, satisfying lx < x/2, lx = ox and lx. Moreover, arbitrarily choose constants δ, 1/2 and p β/β + ε, 1. We split the left-hand side of 3.7 as A + QB > x = A > 1 + δx A > 1 + δx, A + QB x + 1 δx < A 1 + δx, A + QB > x + A < lx or lx < A 1 δx, A + QB > x + A lx, A + QB > x = I 1 x I 2 x + I 3 x + I 4 x + I 5 x. 3.8 We estimate the five terms on the right-hand side of 3.8, in turn. Clearly, 1 + δ β A > x I 1 x 1 + δ α A > x. Noticing that, by Lemma 3.12, x pβ+ε = o A > x, we have I 2 x = A > 1 + δx, A + QB x, B x p B > x p A > 1 + δx Q < δx 1 p + B > x p = o A > 1 + δx + O x pβ+ε = o A > x. By the definition of the class ERV α, β, I 3 x 1 δx < A 1 + δx 1 δ β 1 + δ β A > x. By conditioning on A and applying Lemma 3.2, it holds for every b > 1 and all large x that I 4 x A > lx, QB > δx b Q > δx E B α ε B β+ε 1 A >lx = o Q > x. 9

10 As in dealing with I 2 x above, we further split I 5 x into two parts according to B x p and B > x p. For the first part, we condition it on A, B and apply the uniformity over y lx of the asymptotic relation Q > x + y Q > x. We have I 5 x = A lx, A + QB > x, B x p B > x p = 1 + o1 A lx, QB > x, B x p + O1 B > x p = 1 + o1 A lx, QB > x + o1 Q > x = 1 + o1 QB > x A > lx, QB > x + o1 Q > x = 1 + o1 QB > x + o1 Q > x QB > x, where in the last but one step we used A > lx, QB > x = o1 Q > x, as in the treatment of I 4 x above, and in the last step we used QB > x Q > x. Simply plugging these estimates for I 1 x,..., I 5 x into 3.8 and noticing the arbitrariness of δ, we obtain 3.7, as desired. The following lemma partially extends Theorem 1 of Grey 1994: Lemma 3.7 Let A, B be a random pair satisfying E log A 1 <, B = 1 and E log B <. Let Q be a random variable independent of A, B. 1 Then there is exactly one distribution for Q satisfying the stochastic difference equation where D = denotes equality in distribution. Q D = A + QB, Furthermore, if the distribution of A belongs to the class ERV α, β for some < α β < and E B α ε B β+ε < 1 for some ε, α, then Q > x B j > x, 3.1 A i i 1 where {A k, B k, k N} is a sequence of i.i.d. copies of A, B and, by convention, the multiplication over an empty set of indices produces a value 1. roof. 1 The existence and uniqueness of the weak solution of 3.9 are justified by Theorem 1.6b, c and Theorem 1.5i of Vervaat Let {Q k, k N} be a sequence of random variables defined recursively by Q k = A k + Q k 1 B k, k N,

11 where Q is an arbitrary starting random variable independent of {A k, B k, k N}. Then by Theorem 1.5i of Vervaat 1979, the sequence {Q k, k N} weakly converges with a limit distribution which does not depend on Q and coincides with the distribution of Q in 3.9. See also Goldie 1991 for these statements. To prove relation 3.1, we apply the method developed by Grey 1994 to establish the following two relations: i 1 i 1 Q > x A i B j > x, Q > x A i B j > x Let us prove the first relation in 3.12 now. Introduce a nonnegative random variable Q independent of A, B and satisfying Q > x c A > x for some constant c > 1 E B α ε B β+ε 1. By Lemmas 3.6 and 3.2, A + Q B > x 1 + c E B α ε B β+ε A > x. Since 1 + c E B α ε B β+ε < c, there is some x > such that, for all x > x, A + Q B > x Q > x. Construct a starting random variable Q which is independent of A, B and follows the distribution of Q conditional on Q > x, which belongs to the class ERV α, β. Substituting Q to 3.11 and applying Lemma 3.6, the distribution of Q 1 belongs to the class ERV α, β and Q 1 > x = A 1 + Q B 1 > x A 1 > x + Q B 1 > x. We claim that Q 1 is stochastically not greater than Q, written as Q 1 st Q. Actually, for x > x, Q 1 > x = A + Q B > x Q > x A + Q B > x Q > x Q > x Q > x = Q > x, while for x x, Q 1 > x 1 = Q > x. Using 3.11 with k = 2 and Q 1 st Q, one easily infers that Q 2 st Q 1. Furthermore, by using Lemma 3.6 twice, the distribution of Q 2 belongs to the class ERV α, β and Q 2 > x A 2 > x + Q 1 B 2 > x = A 2 > x + A 1 B 2 + Q B 1 B 2 > x A 2 > x + A 1 B 2 > x + Q B 1 B 2 > x. Repeating this procedure, we can prove that the sequence {Q k, k N}, starting with Q, is stochastically non-increasing and that, for each k N, the distribution of Q k belongs 11

12 to the class ERV α, β with tail satisfying k k Q k > x B j > x + = k A i j=i+1 B j > x + A i i 1 Q Q k B j > x k B j > x. It follows that Q > x k B j > x + A i i 1 Q k B j > x. The last tail probability is negligible as k becomes large because, by Lemma 3.2, k B j > x c E B α ε B β+ε k A > x Q and E B α ε B β+ε < 1. This proves the first relation in We turn to the second relation in The fact Q > > is explained in the proof of Theorem 1 of Grey Construct another starting random variable Q independent of A, B with tail Q > x = Q > A > x1 x + Q > x1 x<. Clearly, the distribution of Q belongs to the class ERV α, β and Q st Q. It is easy to see that the sequence {Q k, k N}, starting with this Q, is stochastically bounded from above by Q. Similarly as before, applying Lemma 3.6 and relation 3.11 recursively, we obtain that, for each k N, Q > x Q k > x k i 1 A i B j > x + B j > x A i i 1 i=k+1 Q k B j > x B j > x. A i i 1 By Lemma 3.2, the last sum above is negligible as k becomes large. This proves the second relation in Finally, we list several useful martingale inequalities. The proof of the following lemma is an excise of Doob s inequality; see also the proof of Lemma 3.2 of aulsen 22: Lemma 3.8 Let L be a Lévy process with Laplace exponent ϕ L. If ϕ L u < for some u >, then E sup t T e ult < for every fixed T,. 12

13 For a stochastic process M, denote by [M, M] and M, M its quadratic variation and predictable quadratic variation, respectively. The following lemma recalls some well-known martingale inequalities of which the first one is the Burkholder-Gundy inequality. Their proofs can be seen, for example, in Liptser and Shiryayev Lemma 3.9 For a local martingale M, write M T = sup t T M t for T. 1 For every q 1,, there are positive constants c q and c q such that, uniformly for all local martingales M with M = and all T, c qe[m, M] q/2 T E M T q c q E[M, M] q/2 T. Moreover, if M t is continuous, then the inequalities above hold for all < q <. 2 If M is a local square integrable martingale with M =, then it holds for every q, 2 that E MT q 4 q E M, M q/2 T. 2 q 4 roofs of the Main Results For the process, the Lévy-Khintchine representation 2.1 for its characteristic exponent becomes Ψ u = ipu σ2 u e iux + iux ν dx + 1 e iux ν dx. x <1 Consequently, its Lévy-Itô decomposition is given by x 1 t = pt + σ W t + M t + C t, 4.1 where W is a standard Wiener process, M is a square integrable martingale with almost surely countably many jumps of magnitude less than 1, and C is a compound oisson process in the form of 3.6 in which the oisson intensity is λ = ν R \ 1, 1 and F is given by ν 1 R\ 1,1 /λ. In particular, W, M and C are three independent Lévy processes. See, e.g. Kyprianou 26 for more details. 4.1 roof of Theorem 2.11 Recall relation 1.3 with T <. The basic idea of our proof is that, when considering the tail behavior of ZT, the Wiener process and small jumps of the process are negligible. Clearly, by 4.1 it holds that 3 inf I j,t + sup I 4,t ZT t T t T 4 sup t T I j,t,

14 where I 1,t = p e R s ds, I 2,t = σ e R s dw s, I 3,t = e R s dm s and I 4,t = e R s dc s. By Lemma 3.5, the distribution of sup t T I 4,t belongs to the class ERV α, β and T sup I 4,t > x λ X e R t > x dt, 4.3 t T where X, independent of R, follows the distribution F as defined above. Note that the right-hand side of 4.3 with x > is identical to the right-hand side of 2.3. If E sup I j,t β+ε <, j = 1, 2, 3, 4.4 t T then by Lemma 3.12 and Lemma 3.3, all terms except sup t T I 4,t appearing in the upper and lower bounds for ZT in 4.2 are negligible and it follows from 4.3 that T ZT > x sup I 4,t > x λ Xe R t > x dt, t T yielding relation 2.3. Therefore, it suffices to prove 4.4. By Lemma 3.8, we have E sup e β+ε R t <. 4.5 t T Then relation 4.4 with j = 1 follows trivially from 4.5. Since I 2,t is a continuous martingale and [I 2,t, I 2,t ] = σ 2 t e 2 R s ds, we use Lemma 3.91 and relation 4.5 again to obtain that, for some constant c 1 >, T β+ε/2 E sup I 2,t β+ε c 1 E e 2 R t dt c 1 T β+ε/2 E sup e β+ε R t <. t T t T Similarly, by Lemma 3.9 and relation 4.5, it holds for some constant c 2 > that β+ε/2 E sup I 3,t β+ε c 2 x 2 ν dx E sup e β+ε R t <. t T x 1 t T 4.2 roof of Theorem 2.12 Recall 1.3 with T =, that is, ψx, = Z > x. The basic idea of our proof is to construct two discrete-time processes, fulfilling a certain recursive structure, whose limits serve as the stochastic upper and lower bounds, respectively, for the ultimate supremum of the discounted net loss process. This idea is from Grey To derive an asymptotic upper bound for Z, we observe that Z Z 1 + e R 1 sup 1 t< 1 e R s R 1 ds d = Z 1 + Z e R 1,

15 where on the right-hand side Z is independent of Z 1, R 1. Consider the stochastic difference equation Q = d Z1 + Q e R 1, 4.7 where on the right-hand side Q is independent of Z 1, R 1. By Theorem 2.11, the distribution of Z1 belongs to the class ERV α, β and Z 1 > x λ 1 Xe R t > x dt. 4.8 By comparing 4.6 with 4.7 and applying Lemma 3.7, we have i 1 Z > x Q > x A i B j > x, 4.9 where {A k, B k, k N} is a sequence of i.i.d. copies of the random pair Z 1, e R 1. It follows from 4.9 and 4.8 that, for arbitrarily fixed δ > and some large x, Z > x i 1 i 1 A i B j > x, B j x + x 1 + δλ = 1 + δλ i i 1 i 1 B j > x x Xe R t R i 1 β+ε i 1 x B j > x dt + E Xe R t > x dt + x/x β+ε. 1 Ee β+ε R 1 Since the last term above is negligible and δ can be arbitrarily small, we obtain ψx, λ x Xe R t > x dt. i 1 B β+ε j To derive an asymptotic lower bound, by Theorem 2.11 we have, for every T >, Z > x ZT > x λ Xe R t > x dt. 4.1 By Lemma 3.2, it holds for every b > 1 and all large x that Xe R t > x dt b E e α ε R t e β+ε R t dt T X > x T b e ϕ Rα εt + e ϕ Rβ+εt dt T e ϕ Rα εt = b ϕ Rα ε + eϕ Rβ+εT ϕ Rβ + ε 15 T

16 since ϕ Rα ε < and ϕ Rβ + ε <. This means that, as T becomes large, the second term on the right-hand side of 4.1 is negligible when compared with X > x, hence with Xe R t > x dt. It follows that ψx, λ 4.3 roof of Corollary 2.1 Xe R t > x dt. We start with 2.3, which holds for T, ]. By Lemma 3.2, it holds for every b > 1 and all large x that Xe R t > x be e α ε R t e α+ε R t b e ϕ X > x Rα εt + e ϕ Rβ+εt, the right-hand side of which is integrable with respect to dt over [,. Therefore, by the dominated convergence theorem, Xe R t > x lim x ψx, T T X > x = λ lim x X > x T dt = λ Ee α R t 1 eϕ RαT dt = λ ϕ, Rα where the second step is due to the well-known Breiman s 1965 theorem. Acknowledgments. The authors would like to thank the editor and an anonymous referee for their careful reading and useful comments. Hao acknowledges support of the Natural Sciences and Engineering Research Council of Canada grant no and Tang acknowledges the support by a Centers of Actuarial Excellence CAE research grant from the Society of Actuaries. References [1] Albrecher, H.; Constantinescu, C.; Thomann, E. Asymptotic results for renewal risk models with risky investment. Stoch. rocess. Appl , no. 11, [2] Asmussen, S. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. robab , no. 2, [3] Asmussen, S.; Albrecher, H. Ruin robabilities. Second edition. World Scientific ublishing Co. te. Ltd., Hackensack, NJ, 21. [4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular Variation. Cambridge University ress, Cambridge, [5] Breiman, L. On some limit theorems similar to the arc-sin law. Theory robab. Appl , no. 2,

17 [6] Chen, Y.; Yuen, K. C. Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25 29, no. 1, [7] Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic rocess. Appl , no. 1, [8] Frolova, A.; Kabanov, Y.; ergamenshchikov, S. In the insurance business risky investments are dangerous. Finance Stoch. 6 22, no. 2, [9] Goldie, C. M. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. robab , no. 1, [1] Grey, D. R. Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. robab , no. 1, [11] Heyde, C. C.; Wang, D. Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims. Adv. in Appl. robab , no. 1, [12] Hult, H.; Lindskog, F. Ruin probabilities under general investments and heavy-tailed claims. Finance Stoch , no. 2, [13] Kalashnikov, V.; Konstantinides, D. Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27 2, no. 1, [14] Kalashnikov, V.; Norberg, R. ower tailed ruin probabilities in the presence of risky investments. Stochastic rocess. Appl , no. 2, [15] Klüppelberg, C.; Kostadinova, R. Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom , no. 2, [16] Klüppelberg, C.; Stadtmüller, U. Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuar. J. 1998, no. 1, [17] Konstantinides, D.; Tang, Q.; Tsitsiashvili, G. Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom , no. 3, [18] Kyprianou, A. E. Introductory lectures on fluctuations of Lévy processes with applications. Springer-Verlag, Berlin, 26. [19] Liptser, R. Sh.; Shiryayev, A. N. Theory of Martingales. Kluwer Academic ublishers Group, Dordrecht, [2] Olvera-Cravioto, M. Asymptotics for weighted random sums. Adv. Appl. robab , no. 4, [21] aulsen, J. Ruin theory with compounding assets a survey. The interplay between insurance, finance and control. Insurance Math. Econom a, no. 1, [22] aulsen, J. Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stochastic rocess. Appl b, no. 1,

18 [23] aulsen, J. On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. robab , no. 4, [24] aulsen, J. Ruin models with investment income. robab. Surv. 5 28, [25] rotter,. E. Stochastic Integration and Differential Equations. Second edition. Springer-Verlag, Berlin, 25. [26] Resnick, S. I.; Willekens, E. Moving averages with random coefficients and random coefficient autoregressive models. Comm. Statist. Stochastic Models , no. 4, [27] Samorodnitsky, G.; Taqqu, M. S. Stable Non-Gaussian Random rocesses. Stochastic Models with Infinite Variance. Chapman & Hall, New York, [28] Tang, Q. The finite-time ruin probability of the compound oisson model with constant interest force. J. Appl. robab , no. 3, [29] Tang, Q.; Tsitsiashvili, G. recise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic rocess. Appl , no. 2, [3] Tang, Q.; Wang, G.; Yuen, K. C. Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. Insurance Math. Econom , no. 2, [31] Vervaat, W. On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. robab , no. 4, [32] Zhang, Y.; Shen, X.; Weng, C. Approximation of the tail probability of randomly weighted sums and applications. Stochastic rocess. Appl , no. 2,