Asymptotic behaviour of first passage time distributions for subordinators

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results are the analogue of the results obtained by the authors for non-monotone L\'evy processes. For subordinators an approach different to that used for non-monotone L\'evy processes is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.


Introduction and main results
Let X be a subordinator, a stochastic process with non-decreasing càdlàg paths with independent and stationary increments, with Laplace exponent ψ, where b denotes the drift and Π the Lévy measure of X. We are interested in determining the local asymptotic behaviour of the distribution of Tx = inf{t > 0 : Xt > x}. More precisely, we would like to establish estimates for the density function hx(t), (if it exists: it does if b = 0), or more generally of P (Tx ∈ (t, t + ∆]), uniformly for ∆ in bounded sets and uniformly for x in certain regions, both as t → ∞ or as t → 0. This is a continuation of recent research in [3], where the same problem, in the t → ∞ case, has been solved for Lévy processes, excluding subordinators, that are in the domain of attraction of a stable law without centering. The reasons for excluding subordinators from that research were that the techniques used there rely heavily on excursion theory for the reflected process, which in this case does not make sense, and that in the subordinators case the rate of decrease of the tail distribution of the first passage time is typically exponential, while for other Lévy processes it is polynomial.
As can be seen in the paper [3], and in the present case, the distribution of the first passage time has different behaviour according to whether the process first crosses the barrier by a jump or continuously, that is by creeping. So, our results will describe the contributions of these events to the first passage time distribution separately. Of course if a subordinator has zero drift, it cannot creep, and moreover the distribution of Tx is absolutely continuous, so our results become somewhat simpler in that case.
In the present work we allow a more general behaviour than that of being in the domain of attraction of a stable law, namely for most of our results we only require X to be in the Feller class, said otherwise to be stochastically compact, either at infinity or at zero depending on whether x/t tends to b from above, or to E(X1) from below, or is bounded away from b and E(X1). A further difference from our work in [3] is that the results here obtained apply equally to subordinators which are stochastically compact with or without centering, while in [3] the assumption that the Lévy process is in the domain of attraction of a stable law without centering is in force.
In order to provide precise definitions of these notions we start by introducing some notation.
We will say that X is in a Feller class or is stochastically compact at infinity, respectively at 0, if It is known that this condition is equivalent to [SC'] ∃α ∈ (0, 2] and c ≥ 1 such that lim sup λz 0 yΠ(y)dy z 0 yΠ(y)dy ≤ cλ 2−α for λ > 1, as z → ∞, respectively as z → 0+; see [6] for a proof of this equivalence and background on the study of the Feller class for general Lévy processes. In this case we will say that the condition SC∞, respectively SC0, holds. For subordinators, there is a pioneering work by Jain and Pruitt [5], which is one of the main sources of this research, and where estimates for P (Tx > t) = P (Xt ≤ x) are obtained. Their main result will be recalled later, but first we quote some facts from the work by Maller and Mason in [7] and [6].
In the case where X is stochastically compact at infinity (respectively at zero), Maller and Mason proved that there exist functions c : [0, ∞) → (0, ∞) and b : [0, ∞) → [0, ∞) such that for any sequence (t k , k ≥ 0) tending towards infinity (respectively, towards 0) there is a subsequence where Y ′ is a real valued non-degenerate random variable, whose law may depend on the subsequence taken. A standard representation of the functions c and b are If in addition to the condition SC∞ (respectively SC0) the condition lim sup y→∞(y→0) holds, then the above defined functions satisfy lim sup so that the normalizing function b is not needed and hence can be assumed to be 0. In this case it is said that the process X is stochastically compact at zero (respectively at infinity) without centering. In all other cases, lim sup Throughout the paper we will work in one of the following frameworks on Π, t and x: always b < xt := x/t < µ and (SC0-I) the Lévy measure Π satisfies the condition SC0, t → ∞, xt → b; ((SC0-II) the Lévy measure Π satisfies the condition SC0, t → 0, xt → b, and (7) holds.
We start by providing some local estimates of the distribution of X. The following two results are an improvement of the main result by Jain and Pruitt for subordinators in the sense that we recover the precise estimate for P (Xt ≤ x) as t → ∞, obtained in [5], but we also prove that the estimate is uniform in x, and furthermore we provide a precise estimate for P (Xt ∈ (x − u, x]) which holds uniformly in x and u. The technique we use is also different to that of Jain and Pruitt [5], though both techniques involve normal approximations.
Throughout this note φ : R → R + , will denote the standard normal density.

(ii) In the settings (SC0-(I-II)) we have the estimate
uniformly in z > 0 and x.
Theorem 2 In the settings (SC0-(I-II)), (SC∞) and (G) the following estimates hold uniformly in u < x and uniformly in x.
From this, we deduce corresponding results for the passage time. Here denotes the density function of the first passage time on the event XT x > x, see [3] Lemma 1, and Theorem 3 Let ∆0 > 0 fixed. In the settings (SC0-(I-II)), (SC∞) and (G), the following estimates hold uniformly in 0 < ∆ < ∆0 and in x. Furthermore, under the settings (SC0-(I-II)) the more precise estimate hold uniformly in x.
When specialised to the case that Π is regularly varying, at infinity or zero, this gives the following.
Corollary 4 Let ∆0 > 0 fixed. Then the following estimates, hold uniformly in 0 < ∆ < ∆0 and in x such that either of the following conditions hold α ∈ (0, 1) and the function c is determined by the relation tΠ(c(t)) = 1, t > 0.
Remark 5 If (i) holds and b > 0, we see that h J x (t, ∆)/h C x (t, ∆) → 0, but note in this scenario Xt is not in the domain of attraction of an α-stable subordinator without centering as t → 0. If (ii) holds, and b > 0, Xt is in the domain of attraction of an α-stable subordinator without centering as t → ∞, and this ratio → ∞. In this situation, our forthcoming final result shows that it is possible for polynomial, rather than exponential decay to occur, but again this ratio → ∞.

Preliminaries
Most of our calculations involve an exponential change of measure, which we introduce now. For ψ ′ (∞) = b < x t := xt < µ = ψ ′ (0+) we denote by (Ys, s ≥ 0), a subordinator whose Laplace exponent is given by ψρ t , (19) In particular we have the following relation: Observe that in the above definition of Y we are deliberately excluding the dependence in xt of Y. We do this for notational convenience and also because we will mainly use the equality of measures in (20). The proof of our main results rely on the following technical results. The first of them is a consequence of Lemma 1, P109, of Petrov.
Remark 8 Our use of this result exploits the fact that, for any Lévy process, any t > 0, and any n ≥ 1, Xt is the sum of n independent and identically distributed summands.
The second relates the various quantities we will consider.

Lemma 9
We have the following relations , for u > 0. In particular, if X is stochastically compact at infinity, respectively at 0, then Proof. Just proceed as in Lemma 5.1 in [5]. The proof of Theorem 2 relies on the following proposition Proposition 10 In the settings (SC0-(I-II)), (SC∞) and (G), the estimate holds uniformly in 0 < h ≤ h0, y > h, and x.
The proof of Theorem 1 and Proposition 10 uses among other things the following Lemma.
Lemma 11 For t > 0, b < xt < µ, we have for any s > 0 Proof. The first three identities are proved by bare hands calculations, while the claimed upper bound is obtained as follows Lemma 12 In the settings (SC0-(I-II)), (SC∞) and (G), we have that tH(ρt) → ∞ uniformly in x.
Proof. The proof of the case (G) is a straightforward consequence of the fact that in this setting t → ∞ and as can be seen below, in the proof of Proposition 10 under the present assumptions. To deal with the cases (SC0-(I-II)), (SC∞) we use the Theorem 5.1 of Jain and Pruitt [5] which establishes that the condition tH(ρt) → ∞ is equivalent to P (Xt ≤ x) → 0, as t → ∞ or 0. For the case (SC0 − I) when (7) fails, the equality and an application of the weak law of large numbers for subordinators gives the result. To deal with the case (SC0 − II), we use the equality which together with the sequential convergence in (5) and the assumption that The case when (7) holds as well as the cases (SC∞) are proved with a similar argument. To show the uniformity observe that the function is increasing because the function z → 1 − e −z − ze −z is so. This implies that the function λ → H(ρ(λ)) is decreasing. The uniformity in the cases (G) and (SC0-(I-II)) follows easily from this fact. Indeed, it is enough to observe that tH(ρt) tends towards ∞ as soon as we take a x0 such that x0 > x and tH ρ x 0 t → ∞. To establish the uniformity in the case (SC∞ − I) when (7) holds, we observe that the hypotheses imply that there is a function D such that because by the assumption of stochastic compactness at ∞ we have that In the case (SC∞ − I) when (7) does not hold we proceed as above but using that there is a function j such that x ≤ bt + j(t) and We have all the elements to prove Theorem 1.
To conclude we observe that the following inequalities hold for θ large enough; here we used the assumption (SC0) and the equality (4). We infer that for θ > 0 large enough As a consequence, for n ≥ 0 R |θ| n |E(e iθXt )|dθ = R |θ| n exp{−tℜ(ψ(iθ))}dθ < ∞, and the conclusion follows from standard results.

Proof of (ii) in Theorem 1 and of Proposition 10 in the cases (SC 0 -(I-II))
Given that the result in (ii) in Theorem 1 is more precise than the one in Proposition 10 it will be enough to prove the former. We observe first that the assumption that b < xt < µ and xt → b implies that ρt → ∞, irrespective of whether t → 0 or t → ∞. We next establish that these conditions on xt, the fact that tH(ρt) → ∞, and the stochastic compactness at 0, imply that xρt → ∞, again irrespective of whether t → 0 or t → ∞. Indeed, the identities allow us to ensure that it is enough to justify that 0 < lim infz→∞ zψ ′ (z) ψ(z) . If the drift of X is positive this is straightforward. If the drift is zero this holds whenever lim sup z→0 zΠ(z) z 0 yΠ(dy) < ∞, which in turn holds by stochastic compactness at zero, The former claim is an easy consequence of the following inequalities which are obtained by barehand calculations. It is important to remark that the above facts and the Lemma 12 imply that xρt → ∞ uniformly in x. Furthermore, our previous remarks allow us to provide a unified proof of the cases t → 0 or t → ∞. We will apply Lemma 7 with n = [xρt] and for k ∈ {1, . . . , n}. We use the estimate (23) with s = t/n, thus µs = x/n, which together with our choice of n lead to the approximation for ρt large enough. It is then immediate from the definition of L that for ρt large enough √ tσ(ρt)L = nν which because of the assumption of stochastic compactness at 0 and Lemma 9 is √ tσ(ρt)L k1 tH(ρt) .
So the lemma tells us that (10) holds provided that To prove that this is indeed the case, observe the above estimate for L gives for ρt large enough. Applying the inequality (30) we obtain that for θ ≥ ℓ ≥ k2ρt ℜ(ψρ t (iθ)) = ∞ 0 (1 − cos(θy))e −ρty Π(dy) It follows from the above and the estimate (29) that for any 0 < α0 < α ≤ 2, with α as in (SC ′ 0 ), and for ρt large enough where in the last inequality we used Lemma 7. Observe that the uniformity follows from Lemma 12 and the fact that ρt tends to infinity uniformly as well because it is non-increasing.

Proof of Proposition 10 in the case (SC ∞ )
We will apply the Lemma 7 to W = Yt we see that |E(e iθW )| is in L1, so that W has a density, nt(·).
Observe that the arguments in the above proof apply equally well to establish that xρt → ∞, uniformly in x, even though ρt → 0, because in the present setting we assume stochastic compacity at infinity. Arguing exactly as in the proof of the case (SC0) we deduce that for 0 < h < h0, 0 < a < a0, and ρt small enough where in the last estimate we used Lemma 9 and that ρt → 0. So the lemma tells us that √ tσ(ρt)nt(z) = φ((z − x)/ √ tσ(ρt)) + o(1) Observe that as in the proof of the case (SC0) we have that there is a constant k3 such that ℓ ≥ k3ρt for ρt small enough. The hypothesis of stochastic compactness at infinity (SC∞), and Proposition 2.2.1 in [1] imply that for any α0 ∈ (2 − α, 2) there are constants k4 and k5 such that and thus We fix α0 ∈ (2 − α, 2), take ρ > sup {t>1} ρt, and choose v0 > 1 such that . Now, making a change of variables we bound γ as follows To describe the behaviour of I1 we start by lower bounding the exponent of the integrand as follows. For θ ∈ ((v0) −1 , (Lα 0 ρt) −1 ), or equivalently (1 − cos(θρty))e −ρty Π(dy) where in the last inequality we used (35). The later together with the inequality for t large enough, uniformly in x. Applying this in I1 and the results from Lemma 9 we obtain I1 ≤ √ tσ(ρt)ρt We next deal with the term I2. Proceeding as above we easily get that for θ > 1/ρtk5 Applying this estimate to I2 we get The argument is concluded by using the fact that tH(ρt) → ∞ to deduce from (35) that for all large enough t We infer therefrom that as t → ∞.
We have completed the proof that nt, the pdf of Yt + U h + ∆a, satisfies (34), uniformly for 0 < h < h0 and ε < a < a0. However, we also have so it follows by choosing a small, that uniformly for ε ′ < h < h0, for any and the proposition is proved. As in the previous proof the uniformity follows from Lemma 12 and the monotonicity of ρt.

Proof of Proposition 10 in the case (G)
The proof of this result follows the same line of argument as that of the previous section, so we will just point out the changes needed for that proof to apply in this setting. Observe that the function ψ ′ is strictly decreasing and continuous, and hence under our assumptions This implies in turn that Next we define n and W as in the previous proof and recall that where the above estimate is uniform in x, 0 < h < h0 and 0 < a < a0.
Using the above facts about ρ and arguing as in the previous proofs it is easily seen that uniformly in x, 0 < h < h0, 0 < a < a0. Next we prove that uniformly in x. The properties listed at the beginning of the proof and the definition of l imply that it can be bounded by below by a strictly positive constant, say l * . Also, as we have assumed X non-lattice, and since this is a property that is preserved under change of measure we have that lim inf θ→∞ ℜ(ψρ t (iθ)) = lim inf θ→∞ ∞ 0 (1 − cos(θy))e −ρty Π(dy) We denote ψ ρ (θ) = ∞ 0 (1 − cos(θy))e −ρy Π(dy), and m(s) = inf θ≥s ψ ρ (θ). The above observations and the continuity of ψ ρ (θ) imply that m(s) > 0, for all s > 0. It follows that for t > t0 The proof of Theorem 2 relies on Proposition 10 and it is the same for the three cases (SC0), (SC∞) and (G). We recall the identity, for t ≥ 0, The estimate in Proposition 10 implies that uniformly in x, z > h and 0 < h < h0. By making a change of variables uniformly in x and h ≤ h0 and v < x−h. In particular, for 0 < h < 2h0 ∧x, we get by taking v = h, using the uniform continuity of the normal density and making elementary manipulations that uniformly in x and 0 < h < 2h0 ∧ x. More generally, uniformly in x (42) uniformly in u < x. This estimate follows from (40) by splitting the interval (x − u, x] into disjoints intervals of length ≤ 1 := h0, and using again the uniform continuity of the normal density. We omit the details. Now, Fubini's theorem implies that the following identities hold for u < x, Applying the estimate in (42) into the first and second term of the latter expression, respectively, we obtain and Adding the two terms above we get the claimed estimate. We now get the estimate for P (Xt ≤ x). For ǫ > 0 there is a δ > 0 such that for 0 ≤ y < δσ(ρt) √ t, the inequality 1 − ǫ < e −{y 2 /2tσ 2 (ρt)} ≤ 1, holds. It follows from (41) that Now, the identity (38) can be used to obtain the inequality = e −tH(ρt) x−δσ(ρt) √ t 0 e tH(ρ) e −ρt(y−x) e ρt(y−x) P (Xt ∈ dy) From where it follows that √ tρtσ(ρt)e tH(ρt) P (Xt ∈ (0, x−δσ(ρt) since the fact that tH(ρt) → ∞ and Lemma 9 imply that ρtσ(ρt) √ t → ∞, uniformly. The estimates in (46) and (48) lead to (12).

Proof of estimate (13) in Theorem 3
We only prove here the estimate in (13) in the case where X is stochastically compact at 0. In the other two cases the proof is similar. The proof of the estimate (14) is given later.
We start by observing that The upper bound is thus obtained from the following estimate of the final term above when we recall that in (31) we proved that xρt → ∞, uniformly either as t → ∞ or t → 0.
To establish a lower bound, we note that the stochastic compactness of X allows to ensure that (52) Next we use that for ε > 0, there exists a δ > 0 such that Clearly the integral term here is bounded above by Π(δ √ tσ(ρt))/ρt. On the other hand, if x < δ √ tσ(ρt), we note that so that it suffices to show that if x * := x ∧ (δ √ tσ(ρt)) we have But the previous calculation, with x replaced by x * , gives an upper bound of C(x * ρt) κ 1 −1 for the LHS, and we know that x * ρt → ∞ uniformly, and so the result follows. (14) and (15) and Corollary 4

Proof of estimates
For the behaviour of the first passage time distribution on the event of creeping, we recall that the creeping probability is strictly positive iff the drift b of X is strictly positive and P (XT x = x) = bu(x), where u : R → R + denotes the density of the potential measure of X. From a result by Winkel [8], and generalized by Griffin and Maller [4], we know that in the case where b > 0 we have the identity In the case where the one dimensional distribution of X has a density, say ft, that is P (Xt ∈ dy) dy = ft(y), y ≥ 0, the former expression takes the form The estimate (15) follows immediately from this and Theorem 1.
We can also use this result and the former estimates to prove the estimate (14) in a straightforward way. Proof of the estimate (14) in the cases (SC∞) and (G). We know that Following the arguments in the proof of Theorem 2 it is easy to prove that uniformly in 0 ≤ h < x, and we recall that ρ(λ) is determined by the relation ψ ′ (ρ(λ)) = λ, λ > 0. We deduce therefrom the equality (1)).
Since s ∈ (t, t + ∆) and the error term is uniform in x/s, the claim follows.
We will next establish Corollary 4. Since the arguments used to prove (i) and (ii) are rather similar we will only provide those needed for the latter. Proof of (ii) in Corollary 4. In order to sharpen the estimate in (14) we will start by estimating the difference for t ≤ s ≤ t + ∆. Observe that ψ ′ is decreasing and regularly varying at 0 with index α − 1, which implies that its inverse, ρ, is non-increasing and regularly varying at ∞ with index 1/(α−1). Also, an easy calculation shows that the function λ → ψ(λ) ψ ′ (λ) , λ > 0 is a non-decreasing function and regularly varying at 0 with index 1. So the function G(λ) := ψ(ρ(λ)) ψ ′ (ρ(λ)) , λ > 0 is regularly varying at infinity with index 1/(α − 1) and non-increasing. uniformly as above. But this is a straightforward consequence of the uniformity properties of regularly varying functions because the function λ → σ(ρ(λ)) is regularly varying at infinity with index α−2 α−1 > 0 and hence uniformly for c in (0, A), for any A > 0. Putting the pieces together we conclude that uniformly in x/t → ∞ and 0 < ∆ < ∆0.

Proof of Proposition 6
Proof of estimate (17), the jump term. Write the LHS as I1 + I2, where where we use the fact that the previous fraction converges uniformly to (yt−z) −α on the range of integration. Since this function and its derivative are bounded on the range, we can integrate by parts and use the central limit theorem to see that lim ε→0 lim t→∞ |I1 − yt−ε 0 P (S1 ∈ dz)(yt − z) −α | = 0.
An integration by parts and the local limit theorem shows that P (S1 ∈ dz)Π S (yt − z), the conclusion follows because the normalisation tΠ(c(t)) = 1 implies that Π S (x) = x −α ; this can be seen by the fact that as t, x → ∞, P (Xt > c(t)x) ∽ tΠ(c(t)x) ∽ x −α .
Remark 13 Suppose now that X is in the domain of attraction of a stable subordinator of index α, at 0, with norming function γ, and put y ′ t = x/γ(t). Then the same result holds as t → 0, uniformly for y ′ t ∈ (D −1 , D).
Proof of estimate (18), the creeping term. As in the previous proof we have So, using the local limit theorem we get that for large t where o(1) is uniform in x. Using that c(s)/c(t) ∼ 1 uniformly in t ≤ s ≤ t + ∆0 and the uniform continuity of g it is deduced therefrom that Now, observe that the function r →