Stability of the Exit Time for Lévy Processes
aa r X i v : . [ m a t h . P R ] J un Applied Probability Trust (24 November 2018)
STABILITY OF THE EXIT TIME FOR L´EVY PROCESSES
PHILIP S. GRIFFIN, ∗ Syracuse University
ROSS A. MALLER, ∗∗ Australian National University
Abstract
This paper is concerned with the behaviour of a L´evy process when it crossesover a positive level, u , starting from 0, both as u becomes large and as u becomes small. Our main focus is on the time, τ u , it takes the process totransit above the level, and in particular, on the stability of this passage time;thus, essentially, whether or not τ u behaves linearly as u ↓ u → ∞ . Wealso consider conditional stability of τ u when the process drifts to −∞ , a.s.This provides information relevant to quantities associated with the ruin of aninsurance risk process, which we analyse under a Cram´er condition. Keywords:
L´evy process; passage time above a level; stability; insurance riskprocess; Cram´er condition; overshoot.2000 Mathematics Subject Classification: Primary 60G51; 60K05Secondary 91B30
1. Introduction
For a random walk S starting from 0 with a positive step length distribution and having finite mean, thenumber of steps required to first pass a positive level u , τ Su , say, is, for large u , asymptotic to a multiple of u , the constant of proportionality being the reciprocal of the mean step length. More precisely, τ Su /u → /c ∗ Postal address: Department of Mathematics, Syracuse University, Syracuse, NY, 13244-1150,USA ∗∗ Postal address: Centre for Financial Mathematics, and School of Finance, Actuarial Studies, & Applied Statistics, AustralianNational University, Canberra, ACT, AustraliaResearch partially supported by ARC Grant DP1092502 a.s. as u → ∞ , and, further, Eτ Su /u → /c as u → ∞ , where c ∈ (0 , ∞ ) is the expected step length.These express a kind of long term linearity of the passage time, and provide useful intuition in applications.Together with other “fluctuation” quantities related to passage over a level, such as the overshoot of thelevel, and various undershoots, etc., it constitutes one of many well known properties of the renewal theoryof random walks. More generally, properties such as stability of the passage time, etc., have been extendedto random walks on the line. (References and further discussion are given later.)It is natural to consider carrying the discrete time results over to a L´evy process ( X t ) t ≥ , and this has beendone in the literature for some of the fluctuation quantities, especially, see [17] for stability of the overshoot.Applications of this and related kinds of result abound; we have in mind, in particular, applications to theinsurance risk process: see, e.g., recent results in [5], [15], [19], [28], and [31]. These authors have tendedto concentrate on properties of the overshoot and undershoots, with less attention paid to the ruin time, τ u . But it could be argued that τ u is the most important or at least the most interesting variable, from apractical point of view.Our aim in this paper is to set out in detail a comprehensive listing of conditions for the stability of τ u , inthe L´evy setting. For “large time” stability, i.e., as u → ∞ , the discrete time (random walk) results can beconsulted to give useful guidance for some of the L´evy results; others are rather straightforward to transfer,but others again are challenging. We consider both stability in probability and almost sure (a.s.) stabilityof τ u /u , as u → ∞ , when lim sup t →∞ X t = ∞ a.s., and lim t →∞ X t = −∞ a.s.Even more interesting is the “small time” stability, i.e., as u →
0, of the passage time. Here there are ofcourse no corresponding random walks that can be used for guidance, but, remarkably, small time resultsfor L´evy processes often parallel large time results in certain ways. With this insight and some furtheranalysis we are able to give also a comprehensive analysis of the small time stability of τ u . Some curiousand unexpected results occur (see, e.g., Remark 2.3). Such results may be thought of as adding to ourunderstanding of the local properties of L´evy processes.The setting is as follows. Suppose that X = { X t : t ≥ } , X = 0, is a L´evy process defined on (Ω , F , P ),with triplet ( γ, σ , Π X ), Π X being the L´evy measure of X , γ ∈ R , and σ ≥
0. Thus the characteristicfunction of X is given by the L´evy-Khintchine representation, E ( e iθX t ) = e t Ψ X ( θ ) , whereΨ X ( θ ) = i θγ − σ θ / Z R ( e i θx − − i θx {| x |≤ } )Π X (d x ) , for θ ∈ R . (1.1) tability of the Exit Time Denote the maximum process by X t = sup ≤ s ≤ t X s , and let G t = sup { ≤ s ≤ t : X s = X s } be the time of the last maximum prior to time t . Our focus will be on the first passage time above level u ,defined by τ u = inf { t ≥ X t > u } , u > . (We adopt the convention that the inf of the empty set is + ∞ .) Also important will be the time of the lastmaximum before passage, G τ u − , and the position after transit above level u , X τ u . Throughout, we assumethat Π X is not identically zero and that X is not the negative of a subordinator (in which case τ u = ∞ for all u > L − t , H t ) t ≥ denote the bivariate ascending inverse local time–ladderheight subordinator process of X . The process ( L − , H ) is defective when, and only when, lim t →∞ X t = −∞ a.s. In that case, it is obtained from a nondefective process ( L − , H ) by exponential killing with rate q > L − , H ) is nondefective the killing is unnecessary and we set ( L − , H ) = ( L − , H ) and take q = 0. We denote the bivariate L´evy measure of ( L − t , H t ) t ≥ by Π L − ,H ( · , · ), and let Π L − and Π H be themarginal L´evy measures of L − and H . The Laplace exponent κ ( a, b ) of ( L − , H ) will play an importantrole in our analysis. It is defined by e − κ ( a,b ) = e − q Ee − a L − − b H (1.2)for values of a, b ∈ R for which the expectation is finite. We can write κ ( a, b ) = q + d L − a + d H b + Z t ≥ Z h ≥ (cid:0) − e − at − bh (cid:1) Π L − ,H (d t, d h ) , (1.3)where d L − ≥ H ≥ Theorem 1.1. (Laplace Transform Identity.)
Fix µ > , ρ ≥ , λ ≥ , ν ≥ , θ ≥ . If µ + λ = ρ , Z u ≥ e − µu E (cid:16) e − ρ ( X τu − u ) − λ ( u − X τu − ) − νG τu − − θ ( τ u − G τu − ) ; τ u < ∞ (cid:17) d u = κ ( θ, µ + λ ) − κ ( θ, ρ )( µ + λ − ρ ) κ ( ν, µ ) . (1.4) P.S.Griffin and R.A. Maller
In the present paper we apply these concepts to study the stability of the passage time, τ u , by which wemean that τ u /u has a finite and positive nonstochastic limit, where the convergence may be as u → u → ∞ , and the convergence may be in probability, almost sure (a.s.), or in mean. We will also consider, toa lesser extent, the position, X τ u , of X as it crosses the boundary. Some other results of interest, especially,that τ u /u are uniformly integrable as u → ∞ if X has a finite positive mean (see Lemma 5.2) are derived asby-products.The results relating to stability of τ u are given in Section 2. In contrast, in Section 3 we consider large timeconditional stability of τ u when P ( τ u < ∞ ) → u → ∞ . This is the usual setup in the L´evy insurancerisk model, for which see, e.g., [4], [3], [15] and [28] for background and references. Section 4 contains someconcluding remarks and references. All proofs are in Sections 5, 6, and the Appendix.
2. Stability
This section contains results relating to the stability of τ u as u → L where L = ∞ or L = 0. For stabilityto make sense when L = ∞ we need, at a minimum, to assume that P ( τ u < ∞ ) → u → ∞ . This isequivalent to lim sup t →∞ X t = + ∞ a.s., in which case τ u < ∞ a.s. for all u > τ u → ∞ a.s. as u → ∞ .The natural analogue of this condition when L = 0 is that P ( τ u < ∞ ) → τ u → u ↓
0. This isequivalent to 0 being regular for (0 , ∞ ); see [8] for an analytic equivalence. Thus the overriding assumptionsthroughout this section are: lim sup t →∞ X t = + ∞ a.s. when L = ∞ , and is regular for (0 , ∞ ) when L = 0 . Let Π X and Π ± X denote the tails of Π X , thusΠ + X ( x ) = Π X { ( x, ∞ ) } , Π − X ( x ) = Π X { ( −∞ , − x ) } , and Π X ( x ) = Π + X ( x ) + Π − X ( x ) , (2.1)for x >
0, and define a kind of truncated mean A ( x ) := γ + Π + X (1) − Π − X (1) + Z x (cid:16) Π + X ( y ) − Π − X ( y ) (cid:17) d y = γ + x (cid:16) Π + X ( x ) − Π − X ( x ) (cid:17) + Z < | y |≤ x y Π X (d y ) , x > . (2.2)The first theorem concerns the stability in probability of τ u . Consider first stability for large times, as u → ∞ , i.e., the property τ u /u P −→ /c as u → ∞ , for some c ∈ (0 , ∞ ). This is equivalent to the relativestability in probability of the process X itself, i.e., to X t /t P −→ c as t → ∞ . We prove it via an equivalenceof the stability of τ u with that of X , namely, X t /t P −→ c as t → ∞ , a trivial relationship. We then show thatthe latter holds iff X is relatively stable, which is not entirely obvious, but follows from similar (large time) tability of the Exit Time random walk working of [27], where the stability of the passage time of a random walk above a constantlevel is considered for general norming sequences. The stability of τ u is connected to the bivariate Laplaceexponent in (2.4), which is a new relationship, derived via Theorem 1.1, and the list of equivalences for thiscase is completed by that of (2.6) and (2.7), which is in Theorem 3.1 of [16].This list, for the case u → ∞ , c ∈ (0 , ∞ ), then sets the pattern we work from for the case u ↓ c ∈ (0 , ∞ ),and later results. Theorem 2.1 also considers the cases c = 0 and c = ∞ for completeness, though thesestrictly speaking do not give rise to stability conditions. Theorem 2.1. (Stability in Probability of the Exit Time.) (a) Fix a constant c ∈ (0 , ∞ ) and let L = 0 or ∞ ( /L = ∞ or ). Then the following are equivalent: τ u u P −→ c , as u → L ; (2.3)lim x → /L κ ( x, κ ( x, ξx ) = 11 + ξc , for each ξ >
0; (2.4) X t t P −→ c, as t → L ; (2.5) X t t P −→ c, as t → L. (2.6) In the case L = ∞ , (2.3) – (2.6) are equivalent to x Π X ( x ) → A ( x ) → c, as x → ∞ . (2.7) In the case L = 0 , (2.3) – (2.6) are equivalent to σ = 0 , x Π X ( x ) → A ( x ) → c, as x ↓ . (2.8) (b) Suppose c = 0 . If L = ∞ then (2.3) – (2.7) remain equivalent. If L = 0 then (2.3) – (2.5) remain equivalent,as do (2.6) and (2.8) . However while (2.6) implies (2.3) – (2.5) , the converse does not hold.(c) Suppose c = ∞ . Then (2.3) – (2.5) remain equivalent for L = 0 or ∞ in the following sense: τ u u P −→ , as u → L, iff lim x → /L κ ( x, κ ( x, ξx ) = 0 , for each ξ > , iff X t t P −→ ∞ , as t → L. (2.9) Again, while (2.6) implies (2.3) – (2.5) , it is not equivalent in either case, L = 0 or ∞ . Remark 2.1.
As mentioned above, the equivalence of (2.6) and (2.7) when L = ∞ is in [16], while theequivalence of (2.6) and (2.8) when L = 0 is in Theorem 2.1 of [16]. Both of these results hold for all c ∈ ( −∞ , ∞ ). We include them in the statement of Theorem 2.1 for completeness and for the convenienceof the reader. P.S.Griffin and R.A. Maller
The next theorem concerns the almost sure stability of τ u . We follow the pattern set by Theorem 2.1. Theconnection with the bivariate Laplace exponent is transmuted in this case to requiring finite first momentsof the ladder processes H and L − . Almost sure stability for large times requires a finite positive mean for X (for large times), and bounded variation with positive drift of X (for small times). Recall that when X is of bounded variation, we may write the L´evy-Khintchine exponent in the formΨ( θ ) = i θ d X + Z R ( e i θx − X (d x ) , (2.10)where d X = γ − R x {| x |≤ } Π X (d x ) is called the drift of X . Theorem 2.2. (Almost Sure Stability of the Exit Time.) (a) Fix c ∈ [0 , ∞ ) . (i) We have τ u u → c , a.s. as u → ∞ iff E | X | < ∞ and EX = c ≥ .(ii) We have τ u u → c , a.s., as u → iff X is of bounded variation with drift d X = c ≥ .(b) Fix c ∈ (0 , ∞ ) . Then (i) holds iff EH < ∞ and EL − < ∞ , in which case c = EH /EL − , while (ii)holds iff σ = 0 , d L − > and d H > , in which case c = d H / d L − . Remark 2.2.
Note that, under (2.7) and (2.8) respectively,lim x →∞ A ( x ) = γ + Z | y | > y Π X (d y ) and lim x ↓ A ( x ) = γ − Z < | y |≤ y Π X (d y ) . (2.11)Here existence of the limits is equivalent to conditional convergence of the integrals. Under the conditionsof parts (a)(i) and (a)(ii) of Theorem 2.2, these integrals converge absolutely and the limits are then givenby EX and d X respectively, thus confirming that the expressions for c in Theorems 2.1 and 2.2 agree. Thedifference between (2.7) and (2.8) and (i) and (ii) of Theorem 2.2 is essentially whether the integrals in (2.11)converge conditionally or absolutely.In the next theorem we examine the convergence of Eτ u /u as u → ∞ and as u ↓
0. Recall that Eτ u < ∞ for some, hence all, u ≥
0, iff X drifts to + ∞ a.s., iff EL − < ∞ (see, e.g., Theorem 1 of [18]). Theorem 2.3. (Stability of the Expected Exit Time.) (a) Fix c ∈ (0 , ∞ ) . Then(i) Eτ u < ∞ for each u > and lim u →∞ Eτ u u = 1 c iff < EX ≤ E | X | < ∞ . In this situation, EH < ∞ , EL − < ∞ , and c = EX = EH /EL − .(ii) Eτ u < ∞ for each u > and lim u ↓ Eτ u u = 1 c iff EL − < ∞ and d H > , and then c = d H /EL − .(b) (The case c = 0 ) (i) In Part (i) of the theorem, the case c = 0 cannot arise; when Eτ u < ∞ for each u > , lim u →∞ Eτ u /u exists and is in [0 , ∞ ) . tability of the Exit Time (ii) We have Eτ u < ∞ for each u > , and lim u ↓ Eτ u /u = ∞ iff EL − < ∞ and d H = 0 .(c) (The case c = ∞ ) (i) We have Eτ u < ∞ for each u > and lim u →∞ Eτ u /u = 0 iff EL − < ∞ and EH = ∞ .(ii) In Part (ii) of the theorem, the case c = ∞ cannot arise; when Eτ u < ∞ for each u > , we alwayshave lim inf u ↓ Eτ u /u > . Remark 2.3. (i) It is curious that the formula for c in Part (a)(ii) of Theorem 2.3 doesn’t agree with theversions in Theorem 2.1 or Theorem 2.2. We give an example to illustrate how the difference can arise. Let X t = at − N t where N t is a rate one Poisson process and a >
1. Thus lim t →∞ X t = ∞ a.s. Since τ u = ua − for sufficientlysmall u , it trivially follows that lim u ↓ τ u u = 1 a a . s . We claim that Eτ u u → Eτ a . (2.12)This is because, if ξ is the time of the first jump of N , then Eτ u = E ( τ u ; N ua − = 0) + E ( τ u ; N ua − ≥ ua − e − ua − + Z ua − E ( τ u | ξ = t ) P ( ξ ∈ d t )= ua − e − ua − + Z ua − ( t + Eτ − at + u ) e − t d t. Since τ x P → τ as x ↓
0, (2.12) now follows after dividing by u and taking the limit.We now check that this agrees with Part (ii) of Theorem 2.3. For the normalisation of L , the local timeat the maximum, we take L t = Z t I ( X s = X s ) d s. Then the ladder height process is linear drift, H t = at . Hence d H = a . By construction L − t = t + N t X R i where again N t is a rate one Poisson process and R i are iid random variables independent of N , withdistribution the same as that of τ . Hence d L − = 1 and EL − = 1 + Eτ , P.S.Griffin and R.A. Maller so the c in Part (ii) of Theorem 2.3 is d H /EL − = a/ (1 + Eτ ), giving agreement with (2.12). On the otherhand, the c in Part (ii) of Theorem 2.2 is d X = d H / d L − = a .We now turn to stability of the time of the last maximum before ruin. As may be expected, this is amore difficult object to study than τ u . We consider the three modes of convergence investigated in Theorems2.1–2.3. Theorem 2.4. (Stability of the Last Maximum before Ruin.)
Let L = 0 or ∞ ( /L = ∞ or ).(a) Fix c ∈ (0 , ∞ ) . We have G τ u − u P −→ c , as u → L, (2.13) if and only if lim x → /L κ (0 , x ) κ ( ξx, x ) = cc + ξ , for each ξ > . (2.14) (b) Fix c ∈ [0 , ∞ ) . (i) G τ u − u → c , a.s. as u → ∞ iff E | X | < ∞ and EX = c ≥ .(ii) G τ u − u → c , a.s., as u → iff X is of bounded variation with drift d X = c ≥ .(c) If < EX < ∞ then lim u →∞ EG τ u − u = 1 EX . Remark 2.4.
It’s not clear how the conditions of Theorem 2.1 relate to the stability in probability of G τ u − .We can show that (2.3)–(2.7) imply (2.13) and (2.14) but it’s not clear whether or not the converse holds.For almost sure convergence the results for G τ u − parallel those for τ u . For convergence in mean the situationremains largely unresolved.The final result, Theorem 2.5, belongs in the present section since it holds in a case when lim t →∞ X t = + ∞ a.s., but we apply it in the next section, in the case when lim t →∞ X t = −∞ a.s., to obtain results in theL´evy insurance risk model. Theorem 2.5. (Convergence of Expected Exit Times with Overshoot.)
Assume < EX ≤ E | X | < ∞ , and that X is not compound Poisson, or is compound Poisson with anonlattice jump distribution. Then for all ρ > u →∞ E (cid:18) G τ u − u e − ρ ( X τu − u ) (cid:19) = lim u →∞ E (cid:16) τ u u e − ρ ( X τu − u ) (cid:17) = 1 EX Ee − ρY , (2.15) where Y is the limiting distribution of the overshoot X τ u − u . Y has density Π H ( h )d h/EH on (0 , ∞ ) , andmass d H /EH at 0. tability of the Exit Time
3. Stability In the Insurance Risk Model
The aim of this section is to illustrate that stability questions are also of interest when X t → −∞ a.s. as t → ∞ . We phrase the discussion in terms of an insurance risk model. In this case X represents the excess inclaims over premium of an insurance company. The classical model in this context is the Cram´er-Lundbergmodel in which X is the sum of a compound Poisson process with positive jumps, representing claims, anda negative drift, representing premium inflow. The results in the present section will be given for a generalL´evy insurance risk model where no such restrictions are placed on X .The over-riding assumption throughout this section is the Cram´er condition , namely, that Ee ν X = 1 , for some ν > . (3.1)It’s well known that, under (3.1), EX is well defined, with EX +1 < ∞ , EX − ∈ (0 , ∞ ], and EX ∈ [ −∞ , t →∞ X t = −∞ a.s. Further, E ( X e νX ) is finite and positive for all ν in a left neighbourhood of ν , and µ ∗ := E ( X e ν X ) > µ ∗ = + ∞ ) . (3.2)Since lim t →∞ X t = −∞ a.s., we are in the situation that P ( τ u < ∞ ) < u >
0, and lim u →∞ P ( τ u < ∞ ) = 0. In an insurance risk context, we are interested in forecasting the ruin time τ u in a worst case scenario,i.e., conditional on τ u < ∞ (“ruin occurs”). Asymptotic properties of τ u and associated variables, conditionalon τ u < ∞ , often provide surprisingly good approximations of corresponding finite level distributions; cf, e.g.,[20]. In the present context we look at the stability of τ u , G τ u − and X τ u , showing they are asymptoticallylinear under mild conditions.We need some more infrastructure. Let ( X ∗ t ) t ≥ denote the Esscher transform of X defined by P (( X ∗ s , ≤ s ≤ t ) ∈ B, X ∗ t ∈ d x ) = e ν x P (( X s , ≤ s ≤ t ) ∈ B, X t ∈ d x ) , (3.3)for any Borel subset B of R [0 ,t ] . Equivalently X ∗ may be introduced by means of exponential tilting; thatis, define a new probability P ∗ , given on F t by dP ∗ dP = e ν X t . (3.4)Then X under P ∗ has the same distribution as X ∗ under P . It easily follows that Ef ( X ∗ t ) = E ∗ f ( X t ) = E ( f ( X t ) e ν X t ) , for any Borel function f for which the expectations are finite. X ∗ is itself a L´evy process with exponentΨ( θ − i ν ) and E ∗ X = µ ∗ . Since µ ∗ > X ∗ t drifts to + ∞ a.s., and hence ( H ∗ t ) t ≥ , the increasingladder height process associated with ( X ∗ t ) t ≥ , is proper.Our setup is that of Bertoin and Doney [9]. The main result in [9], which we give in the form proved inTheorem 7.6 of [29], see also Section XIII.5 of [3], is: Suppose (3.1) holds and the support of Π X is non-latticein the case that X is compound Poisson. Then lim u →∞ e ν u P ( τ u < ∞ ) = C ∈ [0 , ∞ ) , (3.5) where C := E ∗ e − ν Y > if and only if µ ∗ < ∞ . Here Y is the limiting distribution of the overshoot X τ u − u under P ∗ . To state the stability result for the general L´evy insurance risk model under (3.1), introduce the probabilitymeasure P ( u ) ( · ) = P ( · | τ u < ∞ ), and denote convergence in probability conditional on τ u < ∞ by P (u) −→ . Theorem 3.1.
Assume (3.1) holds and µ ∗ < ∞ (so that < µ ∗ < ∞ ). Then, as u → ∞ , X τ u u P (u) −→ , G τ u − u P (u) −→ µ ∗ and τ u u P (u) −→ µ ∗ . (3.6) Assume in addition that the support of Π X is non-lattice in the case that X is compound Poisson. Then lim u →∞ E ( u ) X τ u u = 1 , lim u →∞ E ( u ) G τ u − u = 1 µ ∗ and lim u →∞ E ( u ) τ u u = 1 µ ∗ . (3.7)Parts of our Theorem 3.1 are well known for the Cram´er-Lundberg model, and their extension to thegeneral L´evy insurance risk model is straightforward. Others appear to be new.
4. Concluding Remarks
There is of course a very large literature on (large time) renewal theorems for random walks, and, morerecently, some similar results have been proved for L´evy processes. Regarding the ruin time, most results sofar concern the infinite horizon ruin probability, P ( τ u < ∞ ), or, equivalently, the distribution of the overallmaximum of the random walk or L´evy process, and we do not attempt to summarise them here (other thanthe references mentioned in Sections 1–3). A web search turns up many such papers and books.The finite horizon ruin probability, P ( τ u < T ), is less studied, but important results are obtained in,e.g., [2], [5], [26], [6], [25], [12] (see also their references), and especially, in the insurance/actuarial literature(usually from a more applied point of view). These results of course give information on the long run tability of the Exit Time distribution of the ruin time, conditional on ruin occurring. A more recent result along these lines is in[21], assuming, like [12] and [25], convolution equivalent conditions on the tails of the process or its L´evymeasure. These authors are interested in the asymptotic distribution of τ u , rather than in its stability perse ; as mentioned earlier, results on stability such as we give are more akin to classical (large time) renewaltheory than to these, and small time versions, which make sense for L´evy processes but not for randomwalks, have previously been neglected, in the main.We turn now to the proofs.
5. Proofs for Section 2
We assume throughout this section that lim sup t →∞ X t = + ∞ a.s. when L = ∞ , and 0 is regular for(0 , ∞ ) when L = 0. In the former case, τ u < ∞ a.s. for all u >
0, while P ( τ u < ∞ ) → u → Proof of Theorem 2.1 . (a) and (b). Let c ∈ [0 , ∞ ) until further notice, with the obvious interpretationswhen c = 0. Assume (2.3) holds with u → L . From (1.4) we have, for u > θ > µ Z u ≥ e − µu E (cid:0) e − θτ u ; τ u < ∞ (cid:1) d u = 1 − κ ( θ, κ ( θ, µ ) . (5.1)Take y > µ by µ/y , u by uy , and θ by θ/y in this to get µ Z u ≥ e − µu E (cid:16) e − θτ uy /y ; τ uy < ∞ (cid:17) d u = 1 − κ ( θ/y, κ ( θ/y, µ/y ) . (5.2)By hypothesis, τ uy /y P −→ u/c as y → L for each u >
0, so letting y → L in (5.2) gives, by dominatedconvergence, lim y → L κ ( θ/y, κ ( θ/y, µ/y ) = 1 − µ Z u ≥ e − µu e − θu/c d u = θθ + µc . Replacing y by x = θ/y → /L and µ/θ by ξ gives (2.4) with x → /L .Conversely, assume (2.4) with x → /L . Then from (5.2) we see that, for θ > y → L µ Z u ≥ e − µu E (cid:16) e − θτ uy /y ; τ uy < ∞ (cid:17) d u = µcθ + µc . For each y > θ >
0, the function f y ( u, θ ) := E (cid:0) e − θτ uy /y ; τ uy < ∞ (cid:1) is monotone decreasing in u andbounded by 1. Given any sequence y k → L we can by Helly’s theorem find a subsequence e y k → L , possibly depending on θ but not on u , such that f e y k ( u, θ ) → e f ( u, θ ) for some function e f ( u, θ ) ∈ [0 , µ Z u ≥ e − µu e f ( u, θ )d u = µcθ + µc = µ Z u ≥ e − µu e − θu/c d u, and from the uniqueness of Laplace transforms we deduce that e f ( u, θ ) = e − θu/c , not dependent on the choiceof subsequence. Hence (taking u = 1 now)lim y → L E (cid:16) e − θτ y /y ; τ y < ∞ (cid:17) = e − θ/c , θ > , proving (2.3) with u → L .Since { τ u > t } ⊆ { X t ≤ u } ⊆ { τ u ≥ t } , t > , u > , (5.3)we easily see that (2.3) is equivalent to (2.5) in either case, L = ∞ or L = 0, for c ≥ L = ∞ . By Theorem 3.1 of [16], (2.6) with t → ∞ is equivalent to (2.7), and the first relation in (2.7) implieslim x →∞ V ( x ) x = 0 , (5.4)where V ( x ) := σ + Z | y |≤ x y Π X (d y ) . To deduce (2.5) from (2.7) and (5.4) in the case L = ∞ , decompose X into small and large jump componentsas in Lemma 6.1 of [16] to get X s = sν ( t ) + σB s + X (1) s + X (2) s , ≤ s ≤ t, where ν ( x ) := γ + Z < | y |≤ x y Π X (d y ) , x > ,σB s + X (1) s is a mean 0 martingale with jumps bounded in modulus by t and all moments finite, and X (2) s = X
0. Note that, for t > X t = X t ∨ (cid:18) sup t c − ε ) (cid:19)(cid:19) (1 + o (1))= (cid:18) P (cid:18) X t t ≤ c − ε (cid:19) − o (1) (cid:19) (1 + o (1)) . This shows that P ( X t > ( c − ε ) t ) →
1, and since also P (cid:18) X t t ≤ c + ε (cid:19) ≥ P (cid:18) X t t ≤ c + ε (cid:19) → , we have X t /t P −→ c . Hence (2.5) implies (2.6) for L = 0 or L = ∞ .(c) The equivalences in (2.9) follow by the same methods as used in Part (a), and clearly (2.6) in case c = ∞ implies (2.5) in case c = ∞ .All that remains is to give counterexamples showing (2.5) does not imply (2.6) when c = L = 0 or when c = ∞ and L = 0 or ∞ . Lemma 5.1.
There is a L´evy process for which 0 is regular for (0 , ∞ ) and with X t t P −→ , as t ↓ , but with X t t P −→ − , as t ↓ . (5.6) There is also a L´evy process with X t t P −→ ∞ , as t → L, but with X t t P −→ −∞ , as t → L, (5.7) for L = 0 or L = ∞ . Proof of Lemma 5.1 . This is given in the Appendix. ⊔⊓ With Lemma 5.1 we complete the proof of Theorem 2.1. ⊔⊓ Proof of Theorem 2.2 . By a simple pathwise argument using (5.3), it easily follows that for L = ∞ or L = 0, and c ∈ [0 , ∞ ], we have lim u → L τ u u = 1 c a . s . if and only iflim t → L X t t = c a . s . (5.8)(a)(i) If E | X | < ∞ and c = EX ≥ L = ∞ holds by the strong law. Conversely,by Theorem 15 of [14], (5.8) implies that at least one of EX +1 or EX − is finite or else X t → −∞ . Sincelim sup t →∞ X t = ∞ , the latter possibility is ruled out and so is the possibility that EX − = ∞ . Since c ∈ [0 , ∞ ), the strong law and (5.8) then force E | X | < ∞ and c = EX ≥ X is of bounded variation with d X ≥ X t /t → d X a.s. as t ↓ c = d X . Conversely, by the same result, (5.8) implies X is of bounded variation andnecessarily d X ≥ , ∞ ). It then follows that c = d X ≥ < EX ≤ E | X | < ∞ , we have EL − < ∞ (e.g., Theorem 1 of [18]). Thus letting t → ∞ in X L − t L − t = H t L − t = (cid:18) H t t (cid:19) (cid:18) tL − t (cid:19) , (5.9) tability of the Exit Time and using the strong law, we obtain H t /t → EX EL − as t → ∞ . This implies EH < ∞ and EX = EH /EL − . Conversely, EH < ∞ implies 0 ≤ EX ≤ E | X | < ∞ by Theorem 8 of [17], and EL − < ∞ implies X drifts to + ∞ a.s. by Theorem 1 of [18], so in fact 0 < EX ≤ E | X | < ∞ .(b)(ii) When X is of bounded variation, then σ = 0, and by taking limits as t ↓ L − d X = d H . (5.10)When d X > X t /t → d X > τ u /u → / d X < ∞ ( by (a)(ii)). Thus X τ u u = X τ u τ u τ u u → . s . as u ↓ . This implies d H > L − > c = d X = d H / d L − .Conversely assume d H >
0, d L − > σ = 0. We show lim u ↓ τ u u = d L − d H a.s. Let T u := inf { t > H t > u } , u >
0. Then τ u = L − T u . Hence by (5.9) X τ u u uτ u = (cid:18) H T u T u (cid:19) T u L − T u ! → d H d − L Since lim u → X τ u /u = 1 a.s. when d H > ⊔⊓ Proof of Theorem 2.3 . We begin by recalling that from Theorem 1 of [18], Eτ u < ∞ for some, hence all, u ≥
0, iff X drifts to + ∞ a.s., iff EL − < ∞ .Use identity (8) on p.174 of Bertoin (1996) to write Eτ u = lim λ ↓ κ ( λ, λ V H ( u ) = EL − V H ( u ) , u > , (5.11)where V H ( u ) = Z ∞ P ( H t ≤ u )d t is the renewal function associated with H .(a) (i) For u → ∞ : by the elementary renewal theorem (Kyprianou [29], Cor 5.3 p.114; note that there isno non-lattice restriction on the support of Π X , and the case EH = ∞ is covered, e.g., by Gut [23] Theorem4.1 p.51) we have lim u →∞ V H ( u ) u = 1 EH ∈ [0 , ∞ ) , (5.12)so we see that lim u →∞ Eτ u /u = 1 /c for some c ∈ (0 , ∞ ) iff EH < ∞ and EL − < ∞ , and then c = EH /EL − . Since EH < ∞ is equivalent to 0 < EX ≤ E | X | < ∞ when X t → ∞ by Theorem 8 of [17],we have only left to observe that by Wald’s equation for L´evy processes, [24], EH /EL − = EX . (ii) For u ↓
0: assume that EL − < ∞ and d H >
0. Since H t ≥ d H t , t ≥
0, it follows easily that V H ( u ) := Z ∞ P ( H t ≤ u )d t = Z u/ d H P ( H t ≤ u )d t ≤ e qu/ d H Z u/ d H e − qt P ( H t ≤ u )d t = e qu/ d H V H ( u ) , while trivially V H ( u ) ≤ V H ( u ). Thus by Theorem III.5 of Bertoin [7], which applies to proper subordinators,we have lim u ↓ V H ( u ) u = 1d H , (5.13)and so lim u ↓ Eτ u /u = EL − / d H by (5.11). Conversely, lim u →∞ Eτ u /u = 1 /c implies by (5.11) that (5.13)holds with d H replaced by cEL − >
0. By Lemma 4, p.52 of [14], we have V H ( u ) u ≍ H + R u Π H ( y )d y + uq , for all u > . (5.14)Hence d H >
0, since d H = 0 would imply lim u →∞ V H ( u ) /u = ∞ , a contradiction. ⊔⊓ (b) In case c = 0, we see from (5.11) and (5.12) that lim u →∞ Eτ u /u exists and is in [0 , ∞ ) when Eτ u < ∞ for all u >
0. When lim u → Eτ u /u = ∞ , (5.11) and (5.14) show that d H = 0, and conversely.(c) For the case c = ∞ , supposing Eτ u < ∞ for each u >
0, (5.11) and (5.12) show that lim u →∞ Eτ u /u = 0iff EL − < ∞ and EH = ∞ , while for u ↓
0, the case c = ∞ cannot arise; lim inf u → Eτ u /u > ⊔⊓ The following lemma, which is a L´evy process version of a result of [30] for random walk, is needed in theproofs of Theorems 2.4 and 2.5.
Lemma 5.2. (Uniform Integrability of τ u .) Suppose X is a L´evy process with < EX ≤ E | X | < ∞ and τ u = inf { t > X t > u } , u > . Then τ u /u are uniformly integrable as u → ∞ , i.e., lim x →∞ lim sup u →∞ E (cid:16) τ u u { τuu >x } (cid:17) = 0 . (5.15) Proof of Lemma 5.2 . The random walk result of [30] can be transferred using a stochastic bound due toDoney [13]. First consider the case when Π ≡
0. Then X t = tγ + σB t , where γ = EX > σ ≥
0, and ( B t ) t ≥ is a standard Browian motion. In this case τ u has an inverse Gaussian distribution and Eτ u = uσ /γ , whichimmediately implies uniform integrability of τ u /u . tability of the Exit Time So, assume Π is not identically 0. Then Π( x ) > x >
0, and by rescaling if necessary wecan assume c := Π(1) >
0. As in [18] , let σ = 0 and let σ n , n = 1 , , . . . , be the successive times atwhich X takes a jump of absolute value greater than 1. Then e i := σ i − σ i − are i.i.d exponential rvs with E ( e ) = 1 /c . Define S n := X σ n , n = 1 , , . . . , and τ Su = min { n ≥ S n > u } , u >
0. Then S n is a randomwalk with step distribution Y i := X σ i − X σ i − D = X e . By Wald’s equation, EY = EX /c >
0. Thus by[30], τ Su /u are uniformly integrable.Now we can use similar calculations as on p.287 of [18] to bound the expression on the left of (5.15) interms of a similar expression involving τ Su . For any Z ≥ a > E ( Z ; Z > a ) = Z z>a P ( Z > z )d z + aP ( Z > a ) . (5.16)Taking u > x > c > /c , we obtain E (cid:16) τ u uc { τuuc >x } (cid:17) = u − Z y>xu P ( τ u > yc ) d y + xP ( τ u > xuc ) . (5.17)By Theorem 2.1 we have τ u /u P −→ /EX as u → ∞ . So the second term on the righthand side of (5.17)tends to 0 as u → ∞ once xc > /EX . As on p.287 of [18] we have P ( σ j ≤ τ u < σ j +1 ) = P (cid:0) e m ≤ u, τ Su − e m = j (cid:1) , j ≥ , where e m is a finite rv independent of ( S n ) n =1 , ,... . The first term on the righthand side of (5.17) is boundedby u − X n ≥⌊ xu ⌋ P ( τ u > nc ) , (5.18)and now we argue as follows: u − X n ≥⌊ xu ⌋ P ( τ u > nc ) ≤ u − X n ≥⌊ xu ⌋ P ( τ u ≥ σ n ) + u − X n ≥⌊ xu ⌋ P ( σ n > nc )= u − X j ≥⌊ xu ⌋ ( j − ⌊ xu ⌋ + 1) P ( σ j ≤ τ u < σ j +1 )+ u − X n ≥ P n X i =1 e i > nc ! . (5.19)Since e i are i.i.d. with a finite exponential moment and Ee i = 1 /c < c , the sum in the second term onthe righthand side of (5.19) is convergent, and hence this term is o (1) as u → ∞ . The first term on the righthand side of (5.19) is u − X j ≥⌊ xu ⌋ ( j − ⌊ xu ⌋ + 1) P (cid:0) e m ≤ u, τ Su − e m = j (cid:1) = u − X j ≥⌊ xu ⌋ ( j − ⌊ xu ⌋ + 1) Z u P (cid:0) τ Su − y = j (cid:1) P ( e m ∈ d y )= u − X n ≥⌊ xu ⌋ Z u P (cid:0) τ Su − y ≥ n (cid:1) P ( e m ∈ d y ) ≤ u − X n ≥⌊ xu ⌋ P (cid:0) τ Su ≥ n (cid:1) ≤ u − Z y> ⌊ xu ⌋− P (cid:0) τ Su > y (cid:1) d y ≤ E (cid:18) τ Su u { τSuu > x } (cid:19) , if xu ≥
4, where the last inequality comes from (5.16). Since τ Su /u are uniformly integrable by Lai’s resultwe get (5.15). ⊔⊓ Remark 5.1.
Lemma 5.2 could be used to give an alternative proof of Part (a)(i) of Theorem 2.3 fromTheorem 2.2. This approach however would not work for Part (a)(ii) of Theorem 2.3, since τ u /u are notuniformly integrable as u ↓
0. This is because the almost sure limit in Theorem 2.2 does not agree with thelimit in mean in Theorem 2.3 as u ↓ Proof of Theorem 2.4 . (a) We first observe that (2.14) is equivalent tolim x → /L κ (0 , x ) − κ (0 , κ ( ξx, x ) = cc + ξ , for each ξ > . (5.20)When L = ∞ this because κ (0 ,
0) = q = 0 by our assumption that lim sup t →∞ X t = ∞ . When L = 0 wehave, by (1.3), that κ ( ξx, x ) → ∞ as x → /L for each ξ > H = d L − = 0 and Π L − ,H is a finitemeasure. But this is impossible since 0 is regular for (0 , ∞ ), so (5.20) is equivalent when L = 0 also.From (1.4) we have, for u > ν > µ Z u ≥ e − µu E (cid:0) e − νG τu − ; τ u < ∞ (cid:1) d u = κ (0 , µ ) − κ (0 , κ ( ν, µ ) . Take y > µ by µ/y , u by uy , and ν by ν/y in this to get µ Z u ≥ e − µu E (cid:16) e − νG τuy − /y ; τ uy < ∞ (cid:17) d u = κ (0 , µ/y ) − κ (0 , κ ( ν/y, µ/y ) . (5.21) tability of the Exit Time (2.13) implies G τ uy − /y P −→ u/c , as y → L , for each u >
0. So the lefthand side of (5.21) tends to c/ ( c + ν/µ )as y → L , and then (2.14) follows from (5.20) and the righthand side of (5.21). The proof that (2.14) implies(2.13) is analogous to that in Theorem 2.1.(b) By the strong law when L = ∞ and by Theorem 39 of [14] when L = 0, it suffices to prove that for c ∈ [0 , ∞ ) lim t → L X t t = c a . s ., (5.22)if and only if lim u → L G τ u − u = 1 c a . s . (5.23)A simple pathwise argument shows that (5.22) implies (5.23) when c >
0, but the case c = 0 is a littletrickier. Since the following argument works whenever c ∈ [0 , ∞ ), we prove it under that assumption. Soassume that (5.22) holds with c ∈ [0 , ∞ ). Then X t /t → c as t → L . If (5.23) fails, thenlim inf u → L G τ u − u < c , since under (5.22), G τ u − u ≤ τ u u → c as u → L , by Theorem 2.2. Now consider the random level Z u = X τ u − ,and observe that τ Z u = τ u , X G τu − = X τ u − = Z u . Writing X G τu − G τ u − = Z u u uG τ u − , it then follows that lim inf u → L Z u u < . Thus lim sup u → L X τ Zu Z u = lim sup u → L X τ u Z u ≥ lim sup u → L uZ u > . In particular it is not the case that lim v → L X τ v v = 1 a.s. This implies E | X | = ∞ when L = ∞ , by Theorem 8of [17], and X is not of bounded variation when L = 0, by Theorem 4 of [17]. In either case (5.22) fails tohold which is a contradiction.Conversely assume (5.23) holds for some c ∈ [0 , ∞ ). Then for any a > c , G τ u − > ua − eventually. Hence X ua − ≤ u eventually. This implies lim sup t → L X t t ≤ c. (5.24) If L = ∞ , then arguing as in the proof of Theorem 2.2 (a)(i), (5.24) implies that 0 ≤ EX ≤ E | X | < ∞ andso (5.22) holds with c = EX ≥
0. Since (5.22) implies (5.23), the constant c for which (5.23) was assumedto hold must also have been c = EX , completing the proof of (i). If L = 0 then (5.24) forces X to havebounded variation with d X ≥ , ∞ ). In that case (5.22) holds with c = d X , and soagain since (5.22) implies (5.23), the constant c for which (5.23) was assumed to hold must also have been c = d X , completing the proof of (ii).(c) Finally, suppose 0 < EX ≤ E | X | < ∞ . Then (5.23) holds with c = EX and, further, G τ u − /u areuniformly integrable as u → ∞ by Lemma 5.2. Thus we get lim u →∞ E ( G τ u − /u ) = 1 /EX . ⊔⊓ Proof of Theorem 2.5 . Since 0 < EX ≤ E | X | < ∞ , we have by Theorems 2.2 and 2.4lim u →∞ G τ u − u = lim u →∞ τ u u = 1 EX a . s . The assumptions on X imply that H does not have a lattice jump distribution, and hence it follows from[10] that X τ u − u D −→ Y, as u → ∞ , where Y is the rv defined in the statement of Theorem 2.5. Since τ u /u , and consequently G τ u − /u also, areuniformly integrable as u → ∞ , by Lemma 5.2, the result follows. ⊔⊓
6. Proofs for Section 3
We assume throughout this section the setup of Section 3. Let F τ u be the σ -algebra generated by X upto time τ u . By Corollary 3.11 of [29], for any Z u which is nonnegative and measurable with respect to F τ u ,we have E ( Z u ; τ u < ∞ ) = E ∗ (cid:0) Z u e − ν X τu (cid:1) . (6.1)This immediately yields the following lemma, which can be found in Theorem IV.7.1 of [4] for compoundPoisson processes with negative drift. Our proof is analogous to that in [4]. Lemma 6.1.
Suppose µ ∗ < ∞ and Y u are F τ u -measurable rvs such that Y u P ∗ −→ as u → ∞ . Then Y u P (u) −→ . Proof of Lemma 6.1 : For ε >
0, by (6.1), P ( u ) ( | Y u | > ε ) = P ( | Y u | > ε, τ u < ∞ ) P ( τ u < ∞ ) = E ∗ (cid:0) e − ν ( X τu − u ) ; | Y u | > ε (cid:1) e ν u P ( τ u < ∞ ) . tability of the Exit Time Since µ ∗ < ∞ we have by (3.5) that the denominator here is bounded away from 0, hence the result. ⊔⊓ Proof of Theorem 3.1
Since X is a L´evy process under P ∗ with E ∗ X = µ ∗ ∈ (0 , ∞ ), it follows easilyfrom the strong law that X τ u u → , G τ u − u → µ ∗ and τ u u → µ ∗ , P ∗ − a.s., as u → ∞ . (6.2)(3.6) is then immediate from Corollary 6.1.For (3.7), use Theorem 2.5 , (3.5) and (6.1) to deduce that, as u → ∞ , E ( u ) τ u u = E ( τ u ; τ u < ∞ ) uP ( τ u < ∞ ) = E ∗ (cid:0) τ u e − ν ( X τu − u ) (cid:1) ue ν u P ( τ u < ∞ ) → Cµ ∗ E ∗ e − ν Y = 1 µ ∗ . (6.3)The limit involving G τ u − is similar. For the final limit in (3.7), first observe that E ( u ) X τ u u = E ( X τ u ; τ u < ∞ ) uP ( τ u < ∞ ) ∼ E ∗ (cid:0) X τ u e − ν ( X τu − u ) (cid:1) Cu .
Now let O u := X τ u − u , u >
0. Then ( X τ u /u ) e − ν O u is uniformly integrable, because for x > c := sup y ≥ ( ye − ν y ) = ( eν ) − we have E ∗ (cid:18) X τ u u e − ν O u { Xτuu >x } (cid:19) = E ∗ (cid:18) O u u e − ν O u { O u > ( x − u } (cid:19) + E ∗ (cid:0) e − ν O u { O u > ( x − u } (cid:1) ≤ c u + e − ν ( x − u . Letting u → ∞ then x → ∞ shows the uniform integrability. Since X τ u − u D ∗ −→ Y and X τ u /u P ∗ −→ E ( u ) X τ u u ∼ C − E ∗ (cid:18) X τ u u e − ν O u (cid:19) → C − E ∗ e − ν Y = 1 , completing the proof. ⊔⊓ APPENDIXProof of Lemma 5.1 . We first construct a L´evy process satisfying (5.6). For the characteristics of X wetake γ = − σ = 0 and the L´evy measure given byΠ + X ( x ) = 1 x | ln x | , Π − X ( x ) = Π + X ( x ) + ln 2 x (ln x ) , < x < / + X ( x ) = Π − X ( x ) = 0 , x ≥ / . Then X is not of bounded variation since Z ( | x | ∧ x ) = ∞ , and consequently 0 is regular for (0 , ∞ ). Further one can easily check that (2.8) holds with c = −
1, and so X t t P −→ − t ↓ L = 0, showsthat under (2.8), P ( sup ≤ s ≤ t | X s + s | > εt ) → t ↓ . From this we conclude that X t t P −→ t ↓ , completing the example.We now construct a L´evy process satisfying (5.7). This is based on Example 3.5 in [27], which constructsa random walk S n = P ni =1 Y i with i.i.d. summands Y i which satisfying S n n P −→ − ∞ and S n n = max ≤ j ≤ n S j n P −→∞ . (6.4)This is done by finding a random walk which is negatively relatively stable (NRS) as n → ∞ , i.e., with S n D ( n ) P −→ − , ( n → ∞ ) , (6.5)for a norming sequence D ( n ) > D ( n ) /n → ∞ satisfyinglim n →∞ n X j = ℓ ( n ) P ( S > xD ( j )) = ∞ , (6.6)where ℓ ( n ) is inverse to D ( n ). The sequences D ( n ) and ℓ ( n ) are strictly increasing to ∞ as n → ∞ andsatisfy D ( n ) = − nA ( D ( n )) (where A ( · ) is defined in (2.2)) and ℓ ( n ) = − n/A ( n ). The function − A ( x ) ispositive for x large enough, slowly varying as x → ∞ , and tends to ∞ as x → ∞ .(i) Consider first the case L = ∞ . Let ( N t ) t ≥ be a Poisson process of rate 1 independent of the Y i andset X t := S N t , t ≥
0, where S n is as in (6.4). Then the compound Poisson process X t satisfies (5.7) with L = ∞ . This is fairly straightforward to check and we omit the details.(ii) Now consider the case L = 0. For this we have to modify Example 3.5 in [27] to work as t ↓
0. Detailsare as follows. tability of the Exit Time We construct a L´evy process X t which is NRS as t ↓
0, i.e., is such that X t b ( t ) P −→ − t ↓ , (6.7)for a nonstochastic function b ( t ) >
0, with b ( t ) ↓ b ( t ) /t → ∞ as t ↓
0. To do this, it will be usefulto summarize here some properties concerning (negative) relative stability at 0 of X t ; for reference, see [16](and replace X by − X ). We assume that Π X ( x ) > x >
0. We then have that X t ∈ N RS if andonly if σ = 0 and the function A ( x ) defined in (2.2) is strictly negative for all x small enough, x ≤ x , say,and satisfies lim x ↓ A ( x ) x Π X ( x ) = −∞ . (6.8)When A ( x ) < x ≤ x , there is an x ≤ x so that the function D ( x ) := sup (cid:26) y ≥ x : − A ( y ) y ≥ x (cid:27) , < x ≤ x , (6.9)is strictly positive and finite and satisfies D ( x ) = − xA ( D ( x ))for all x ≤ x . It is easily seen to be strictly increasing on x ≤ x (by the continuity of y
7→ − A ( y ) /y ) with D (0) = 0. Thus for small enough y we can define the inverse function ℓ ( y ) = sup { x : D ( x ) ≤ y } = inf { x : D ( x ) > y } . When (6.8) holds, − A ( x ) is slowly varying as x ↓
0, and as a consequence D ( x ) and ℓ ( x ) are both regularlyvarying with index 1 as x ↓ − A ( y ) /y is strictly decreasing for y small enough, so ℓ ( y ) is continuous andstrictly increasing, with ℓ (0) = 0, and ℓ ( y ) = y − A ( y ) , for y small enough. Finally, we can take b ( t ) = D ( t ) in (6.7) to get negative relative stability of X in theform: X t D ( t ) P −→ − t ↓ . (6.10)To construct the process required in the lemma, we will specify a L´evy measure Π X for X such that D ( t ) t → ∞ , as t ↓ , (6.11) consequently ℓ ( t ) /t →
0, and Z tℓ ( t ) Π + X ( xD ( s ))d s → ∞ , as t ↓ , (6.12)for all x >
0. (6.10) then implies X t /t P −→ − ∞ as t ↓
0. We claim that in addition (6.10)–(6.12) imply X t t P −→ ∞ as t ↓ . (6.13)To prove (6.13) from (6.10)–(6.12), fix t > x > P (cid:0) X t > xt (cid:1) = lim n →∞ P (cid:18) max ≤ j ≤ nt X ( j/n ) > xD ( ℓ ( t )) (cid:19) ≥ lim inf n →∞ X nℓ ( t ) ≤ j ≤ nt P (cid:18) X (( j − /n ) > − xD ( j/n ) , max j +1 ≤ k ≤ nt ∆( k/n ) D ( k/n ) ≤ x < ∆( j/n ) D ( j/n ) (cid:19) , (6.14)where ∆( k/n ) := X ( k/n ) − X (( k − /n ) , k = 1 , , . . . , are i.i.d. with distribution the same as that of X (1 /n ). Given ε ∈ (0 , t > P (cid:0) X (( j − /n ) > − xD ( j/n ) (cid:1) > − ε when j/n ≤ t ≤ t . Thus, keeping t ≤ t , P (cid:0) X t > xt (cid:1) ≥ (1 − ε ) lim inf n →∞ X nℓ ( t ) ≤ j ≤ nt P (cid:18) max j +1 ≤ k ≤ nt ∆( k/n ) D ( k/n ) ≤ x < ∆( j/n ) D ( j/n ) (cid:19) . Here the sum equals P (∆( j/n ) > xD ( j/n ) , for some j ∈ [ nℓ ( t ) , nt ]) = 1 − Y nℓ ( t ) ≤ j ≤ nt P (cid:0) X (1 /n ) ≤ xD ( j/n ) (cid:1) ≥ − exp − X nℓ ( t ) ≤ j ≤ nt P (cid:0) X (1 /n ) > xD ( j/n ) (cid:1) . Noting that n Z t +1 /nℓ ( t ) P (cid:0) X (1 /n ) > xD ( s ) (cid:1) d s ≤ n X nℓ ( t ) ≤ j ≤ nt Z ( j +1) /nj/n P (cid:0) X (1 /n ) > xD ( s ) (cid:1) d s ≤ X nℓ ( t ) ≤ j ≤ nt P (cid:0) X (1 /n ) > xD ( j/n ) (cid:1) d s, and employing the fact that lim n →∞ nP ( X (1 /n ) > a ) = Π + X ( a ) for all a > P (cid:0) X t > xt (cid:1) ≥ (1 − ε ) lim inf n →∞ − exp − n Z t +1 /nℓ ( t ) P (cid:0) X (1 /n ) > xD ( s ) (cid:1) d s !! = (1 − ε ) − exp − Z tℓ ( t ) Π + X (2 xD ( s )d s !! . tability of the Exit Time The last expression tends to 1 as t ↓ ε ↓
0, provided (6.12) holds. Thus (6.13) will follow from(6.10)–(6.12), as claimed.It remains to give an example where (6.10)–(6.12) hold. Define L ( x ) = e ( − log x ) β , < x < e − , and keep < β <
1. Choose a L´evy measure Π X which satisfiesΠ + X ( x ) = − L ′ ( x ) = 2 β ( − log x ) β − L ( x ) x and Π − X ( x ) = − L ′ ( x ) = Π + X ( x ) / , for x small enough, x ≤ x , say. (Note that Π + X ( x ) and Π − X ( x ) are infinite at 0 and decrease to 0 as x → ∞ .)A straightforward calculation using (2.2), gives, for x > A ( x ) = γ + Π + X (1) − Π − X (1) + 1 − L ( x ) . Thus A ( x ) → −∞ as x ↓
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