Cram\'er's estimate for stable processes with power drift

We investigate the upper tail probabilities of the all-time maximum of a stable L\'evy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable L\'evy process. We also study the lower tail probabilities of the integrated stable L\'evy process in the presence of a power positive drift.

Throughout, we assume that L takes positive values i.e. ρ = 0, and we exclude the degenerate case α = ρ = 1 where L is a unit drift. With these restrictions, L has no positive jumps if and only if α > 1 and ρ = 1/α.
x → ∞, (1.1) for α > 1, and it is well-known that the asymptotics is in fact an equality -see [20] or Corollary VII.2 in [4]. For more general power drifts and a class of Gaussian processes fulfilling a certain scaling property, we refer to [12] which, applied to the important case of Brownian motion with a parabolic drift, yields Let us mention that this estimate has been refined in Theorem 2.1 of [11], where a complete asymptotic expansion at infinity is obtained -see also Lemma .
In the specific case α ∈ (1/2, 2] and γ = 2, these estimates are somehow reminiscent of those previously obtained in [5] in the framework of Burgers turbulence with stable noise initial data. See Remark 2.6 below for more detail. Our arguments, quite different from those of [5], rely on the compensation formula for the case with positive jumps and on some ad hoc and rather involved estimates combined with Laplace's method in the spectrally negative case. In the second part of the paper we consider the Riemann-Liouville (or fractionally integrated) stable process with parameter β > 0, defined as t , t ≥ 0} is stable in the broad sense of [18], and by Proposition 3. 4 which can be viewed as an extension of M α,ρ,γ . Observe also from Theorem 10.5.1 in [18] and self-similarity that for every β > 0. It is also easy to check that M (β) α,ρ,γ d −→ M α,ρ,γ as β → 0 when γα > 1. As a rule, the non-Markovian character of a given process makes its passage times across a level more difficult to investigate and our second main result has a less precise character.
Here and throughout, we use the standard notation f (x) g(x) to express the fact that there exist two positive finite constants κ 1 , κ 2 such that κ 1 f (x) ≤ g(x) ≤ κ 2 f (x) as x → ∞ or as x → 0, the nature of the limit being clear from the context. Theorem B. Assume γα > 1.
(a) If L has positive jumps, one has The method to get these estimates differs here for the lower bound and the upper bound. The former uses a simple scaling argument, inspired by that of [12], and amounts to a comparison with the upper tails of L 1 . The latter relies on telescoping sums for the case with positive jumps, and on a simple yet powerful association lemma in the spectrally negative case -see Lemma 3.1.
In the last part of the paper, we study the lower tail problem for the integrated stable process with a power positive drift. In a Gaussian framework, lower tail probabilities have many applications described in [15]. In a self-similar framework they are connected to the persistence probabilities, whose applications are also manifold -see the recent survey [3]. We show the following.
Theorem C. Assume γα > 1 and ρ ∈ (0, 1). For every µ ≥ 0, one has Above, we have excluded the case ρ = 1, where the estimate amounts by monotonicity to the one-dimensional estimate P[L 1 +µ ≤ ε], which is exponentially small for µ = 0 -see e.g. (14.35) in [19] -and zero for µ > 0. Theorem C is an extension of Theorem A in [16] which deals with the case µ = 0. In this respect, we should mention that the condition γα > 1 on the drift power is optimal: in the Cauchy case α = γ = 1, the same Theorem A in [16] shows that the lower tail probability exponent depends on µ. Our argument here relies in an essential way on the strong Markov property of the bidimensional process EJP 24 (2019), paper 17.
t , L t ), t ≥ 0} and is hence specific to the case β = 1. The other cases are believed to be challenging. To give one example, for α = 2, µ = 0 and β ≥ 2 an integer, finding the exact values of the exponents κ n in the asymptotics as ε → 0 is still an open problem on Brownian motion -see Section 3.3 in [3]. In our proof the aforementioned association Lemma 3.1 plays also a significant role. Unfortunately, its one-sided character prevents us from dealing with the case of a negative power drift. We leave this question, whose connection to Burgers turbulence with stable Lévy process initial data in the case α > 1 and γ = 1 is precisely described in Section 4.1 of [3], to future research.

The case with positive jumps
We will use the standard notation for simplicity. Defining for every x > 0 the stopping time x for the overshoot at T x . For every f : R + → R + measurable and such that f (0) = 0, the compensation formula -see [4] p. 7 or Theorem 19.2 in [19] -implies Taking f (u) = 1 {u>0} and integrating in z, we obtain where the second equality follows by scaling, the convergence on the third line is obtained by bounded and monotone convergence (decomposing into {L s < 0} and {L s ≥ 0} inside the expectation), and the evaluation of the integral on the fourth line is EJP 24 (2019), paper 17. standard. To conclude the proof, it remains to show that L does not creep at T x , in other words that The latter is in accordance with the well-known fact that L does not creep at a fixed level x > 0 -see Theorem VI.19 and Lemma VIII.1 in [4]. However, this result does not apply here since we consider the first passage time above a moving boundary. To show (2.1), fix x > 0 and decompose where L is a copy of L which is independent of (T x , L Tx ), by the strong Markov property.
On the one hand, we see by scaling and e.g. Property 1.2.15 in [18] that On the other hand, we have and passing to the limit, we obtain lim inf Hence, we see that (2.1) is a consequence of Applying the compensation formula as above, we obtain Changing the variables z = s γ y and t = su, we see that c −1 + s γα−1 P 2 (s) equals which converges as s → ∞ to which is a standard and rigorous estimate -see Theorem 10.5.1 in [18] and Proposition VIII.4 in [4].
(b) Taking f (u) = 1 {u≥rx} for some r > 0 and applying as above the compensation formula leads to the estimate This implies the following limit theorem for the law of the renormalized overshoot: Recall, indeed, that the standard Pareto distribution with parameter δ > 0 has distribution function 1 − (r + 1) −δ on (0, ∞). This observation seems new even in the classical case of a linear drift γ = 1 with α > 1. Notice that still in the case of a linear drift, the limit behaviour of the overshoot is very different for Lévy processes having finite exponential moments. If we consider for example the tempered stable subordinator with negative unit drift and Lévy measure having density for some c ∈ (0, 1), then we are in the framework of [6] with ω ∈ (0, 1) and µ * < ∞ so that C > 0 in (5) therein. By Remark 2 of [6], this implies that K x converges at infinity to some proper random variable -see also Theorem 4.2 in [13] for more general results.
(c) In the case α > 1, ρ = 1 − 1/α and γ = 1, the Laplace transform of M α,1−1/α,1 can be computed with the help of Zolotarev's well-known general formula -see [20]: one finds This Laplace transform can be easily inverted and yields the identity in law both random variables being independent. This shows that the law of M α,1−1/α,1 is the so-called Mittag-Leffler distribution of parameter α − 1 which is studied e.g. in Exercise 34.4 of [19] -see also the references therein. In particular, there exists a closed expression for the survival function of M α,1−1/α,1 in terms of the classical Mittag-Leffler function, which leads to a complete and simple asymptotic expansion at infinity: one has where we have used Formula 18.1(7) in [8] and the standard notation for asymptotic expansions given e.g. in Appendix C of [1]. Observe from the complement formula for the EJP 24 (2019), paper 17.
Gamma function that the first term matches the one that can be derived from Theorem A (a), in this specific case. Notice also the following closed formula for the distribution function, as a convergent series: Let us finally refer to [10] for related results in the presence of a compound Poisson process.

The case with no positive jumps
Applying the strong Markov property at T x and using the absence of positive jumps, where we have set a + = max(a, 0) and, on the right-hand side, L is an independent copy of L. Integrating both sides on (0, is an increasing homeomorphism from (0, ∞) to (0, ∞), because αγ > 1 and α > 1. This implies the identity which extends to all ν > −1 by analyticity, since L + 1 has moments of every order. We will now study the asymptotic behaviour of both sides of (2.4), introducing the crucial parameter We begin with the left-hand side, which is easy.
at infinity, where p 1 stands for the density of the random variable L 1 . Making the change of variable y = x α α−1 and applying Watson's lemma -see e.g. Theorem C.3.1 in [1], this easily implies See also Theorem 2.5.3 in [21]. On the other hand, we can rewrite Plugging (2.5) into the right-hand side of (2.6), we obtain which yields the required asymptotic behaviour, by Laplace's method.
We will now analyze the right-hand side of (2.4), which is more involved. Introducing the function Taking ν = ν 0 and observing that ϕ −1 α, Therefore, since Φ α,γ,ν0 is continuous and positive on (0, ∞), we have Going back to (2.7) and using the facts that L + 1 has positive moments of every order and is independent of T x , we finally get from Lemma 2.3 the crude asymptotics . (2.8) In order to obtain an exact asymptotics and finish the proof, we will need the following technical lemma.
We can now finish the proof. Taking ν = ν 0 in (2.7), we first decompose the quantity

Applying Lemma 2.3 and the moment evaluation
which is e.g. a consequence of (2.6.20) in [21], we see that the proof will be complete as soon as But, decomposing according to {T x ≤ A} or {T x > A}, the left-hand side of (2.9) is bounded by and (2.9) follows by Corollary 2.5, the continuity of Φ α,γ,ν0 at zero, and dominated convergence.
Remark 2.6. As mentioned in the introduction, in the case γ = 2 our Theorem A echoes a large deviation estimate which had been previously obtained in [5]. More precisely, if we set then the main result of [5] states that if L has positive jumps, and that

The lower bound
This part is easy and relies essentially on the identity (1.3). Introducing we see by the scaling property {L a further scaling argument implies When L has positive jumps, applying Property 1.2.15 in [18] to the stable random variable L 1 and using (3.1) and (3.2) yield the required lower bound
for all t ≥ 0, which follows from γ + β ≥ 1. The next step is to write down the process decomposition L (β) where c β = 2 |β−1| and we have used (t + s) β ≤ c β (t β + s β ) for all t, s ≥ 0. The second term on the right-hand side is bounded by for some positive constant κ not depending on k, x. Settingκ = min{κ, 3 δ−1 c −1 β } > 0, and putting everything together, we finally obtain where the estimate follows at once from (2.3) and direct summation. This completes the proof for γ + β ≥ 1. The case γ + β < 1 follows along the same lines, except that (3.3) is not true anymore. We hence set Using the obvious inequality (t + r k ) γ+β ≥ (1 − λ)t γ+β + λr γ+β k leads first to Then, we can bound and the proof is finished similarly.

The upper bound in the case without positive jumps
The argument relies on the following well-known association lemma, which will also be used during the proof of Theorem C.
Proof. By càd-làg approximation, it is enough to consider the case when F, G depend only on a finite number of points. With the notation of Chapter 4.6 in [18], we are hence reduced to show that the random vector (L t1 , L t2 , . . . , L tn ) is associated for every n ≥ 2 and 0 < t 1 < . . . < t n . By independence of the increments we have (L t1 , L t2 , . . . , L tn ) = (X 1 , X 1 + X 2 , . . . , X 1 + . . . + X n ), where the X i 's are mutually independent real random variables, making the vector X = (X 1 , . . . , X n ) trivially associated. We can then apply e.g. Exercise 4.25 p. 220 in [18].
Let us now finish the proof. For simplicity, we will set T x for T (β) x . Let ε > 0 and fix δ small enough such that η = 1 − (1 − ε)(δ + 1) β+γ > 0. Using the absence of positive jumps, we obtain for some positive constants κ ε , c ε such that c ε → 1 as ε → 0 and, by Lemma 2.3, we first We shall now separate the proof according as β ≥ 1 or β < 1. Assume first β ≥ 1. Bounding the right-hand side of (3.5) leads to We next observe that the contribution of P[T x ≤ 1] in the right-hand side of (3.6) is negligible, using the obvious bound EJP 24 (2019), paper 17.
Above, the crude estimates are a consequence of (1.3), (2.5) and the last equality being well-known as the reflection principle for spectrally negative stable Lévy processes -see e.g. Exercises 29.7 and 29.18 in [19]. Finally, we notice that is an increasing functional of {L s , s ≥ 0}, because β ≥ 1. Since 1 {Tx<∞} is also an increasing functional of L, we deduce from Lemma 3.1 that for some κ > 0 not depending on x. Putting everything together, we get which, letting ε → 0, completes the proof in the case β ≥ 1.
Assume second β < 1. We set which is a positive increasing function on (0, ∞) such that σ t → 0 as t → 0. Replacing T β+γ we deduce using a change of variable that where h β (t) = 1 + t β − (1 + t) β is increasing in t. Going back to (3.5), and taking a < δ, the right-hand side is then greater than: is an increasing functional of L. We next observe that, cutting (3.7) in two as in (3.6), the second term will be negligible by taking δ small enough since EJP 24 (2019), paper 17.
Thus, it remains to deal with the term: From Theorem A and using the scaling of L, the second term behaves as which is negligible by taking a small enough. The proof is then concluded as in the case β ≥ 1 by applying Lemma 3.1 to the term P [F δ (L) ≥ −η/4, T x < ∞].
is again an homeomorphism from (0, ∞) to (0, ∞), and T x and L + 1 are independent. We can then bound using the crucial fact that the derivative of t → L (1) x , is a.s. non-negative. This leads to (1 + ψ −1 α,γ (x)) 1+ν0 − 1 is again bounded away from zero and ∞, by the fateful choice of ν 0 . We finally obtain for some κ + ∈ (0, ∞), and an appropriate modification of Lemma 1 yields at infinity, for some otherκ + ∈ (0, ∞). Unfortunately, the precise lower bound which can be derived from (2.5) is different: one gets for someκ − ∈ (0, ∞), and the exact polynomial speed before the exponential term remains unknown. We believe that this speed is given in the lower bound, and we refer to Remark 3.4 (c) below for a general conjecture.

A more precise estimate in the Brownian case
In this paragraph we specify the general results of [12] to the process L (β) in the case α = 2, and we get a refinement of Theorem B (b). Observe that in this framework we can also consider the wider range β > −1/2. The following proposition is a consequence of Theorem 1 in [12] but the exact asymptotics does not seem to have been written down anywhere, and we hence provide a detailed proof. It turns out that a transition phenomenon occurs at β = 1/2.

Remark 3.4.
(a) For β = 1/2, the transformation 2.1.4 (18) in [8] with m = 2 exhibits a logarithmic term: one has the non-trivial closed formula as t, s → s 0 , and we cannot apply the results of [12]. We believe that for some κ > 0 and δ = 0 to be determined, the logarithmic correction being heuristically due to the 1-self-similarity of EJP 24 (2019), paper 17.

Proof of Theorem C
Following the notation of [16], we will set θ = ρ α(1 − ρ) + 1 once and for all. The upper bound follows easily from for some κ ∈ (0, ∞), where the first inequality follows from µ ≥ 0 and the second one from Theorem A in [16] and scaling.
The lower bound is more involved and we will need the strong Markovian character of the two-dimensional process {(L (1) and observe first that, by scaling and translation, Notice also that P (x,y) [R ε ≤ R 0 ] = 1 for every x < 0 and y ∈ R, because µ ≥ 0. Applying the strong Markov property at R ε , we obtain {x=Rε} whose right-hand side is, by comparison, smaller than Indeed, the derivative of t → L x α α+1 R 0 ≥ t for every x, y, t ≥ 0. If we now assume µ ≤ 1 this implies, again by comparison, where R 0 is an independent copy of R 0 . Putting everything together, we obtain . Now by Theorem A in [16] we have for every x ∈ R and since α(γ + 1) > α + 1, we can also infer from Lemma 2 in [17] that This implies that there exists two finite constants κ 2 ≥ κ 1 > 0 independent of µ, ε such that P (−1,0) R ε ≥ ε − α α+1 ≥ κ 1 ε which completes the proof of the lower bound for µ ≤ µ 0 with µ 0 = (κ 1 /2κ 2 ) (γ+1)/θ > 0.
Observe that in the absence of self-similarity, the estimates (4.1) and (4.2) are different ones and cannot be deduced from one another, save for µ = 0.