Extended Factorizations of Exponential Functionals of L\'evy Processes

In [16], under mild conditions, a Wiener-Hopf type factorization is derived for the exponential functional of proper L\'evy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed L\'evy processes. As a by-product, we derive some interesting new distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar properties for the law of maxima and first passage time of some stable L\'evy processes. Thus, for example, we show that for a large class of stable processes the first passage time has a bounded and non-increasing density on the positive half-line. We also generate many instances of integral or power series representations for the law of the exponential functional of L\'evy processes with one or two-sided jumps. The proof of our main results requires different devices from the one developed in [16]. It relies in particular on a generalization of a transform recently introduced in [8] together with some extensions to killed L\'evy process of Wiener-Hopf techniques.


Introduction and main results
Let ξ = (ξ t ) t≥0 be a possibly killed Lévy process starting from 0. We denote by Ψ q its Lévy-Khintchine exponent which takes the form, for any z ∈ iR, where q ≥ 0 is the killing rate, σ ≥ 0, b ∈ R and Π is a sigma-finite positive measure satisfying the condition R (y 2 ∧ 1)Π(dy) < ∞. In this paper, we are interested in both characterizing the distribution and deriving some fine distributional properties of the so-called exponential functional of ξ, which is defined by where e q is the lifetime of ξ, i.e. it is an exponential random variable of parameter q (with the convention that e 0 = ∞) independent of ξ. When q = 0 we simply write Ψ = Ψ 0 and we assume that ξ drifts to −∞. The motivation for studying this positive random variable finds its roots in probability theory but has some strong connections with issues coming from other fields of mathematics such as functional and complex analysis. Besides their inherent interest, problems of this type have also ties with other areas of sciences, e.g. astrophysics, biology, insurance and mathematical finance. It is also worth mentioning that there exists a close connection between the law of the exponential functional of some specific Lévy processes and the one of the maxima of stable processes offering a way to study the fluctuation of these processes from a perspective different from the classical Wiener-Hopf techniques. We refer to [16] for a thorough description of the recent methodologies which have been developed to investigate the distribution of I Ψq .
In particular, we mention that, in that paper, it is shown under a mild assumption that, when q = 0 and −∞ < E [ξ 1 ] < 0, the variable I Ψ factorizes into the product of two independent exponential functionals of Lévy processes defined in terms of the ladder height processes of ξ. The purpose of this paper is to extend this Wiener-Hopf type factorization by first relaxing the finite moment condition on the underlying Lévy processes and then by deriving similar factorization identities for the exponential functional of killed Lévy processes. We emphasize that the approach carried out in [16] can not be used to deal with this generalization. Indeed, therein, the main identity is obtained by means of the functional equation (2.9), satisfied by the Mellin transform of I Ψ combined with the characterization of its distribution as the stationary measure of some generalized Ornstein-Uhlenbeck processes. Indeed the law of I Ψq , for any q > 0, cannot be identified as a stationary measure of some Markov process anymore whereas when q = 0 and the first moment of ξ 1 is not finite, the functional equation (2.9) does not hold even on the imaginary line and therefore we cannot directly guess the existence of a probabilistic factorization. In order to circumvent these difficulties our strategy relies on a transformation between Laplace exponents of Lévy processes which allows to establish a connection between the study of the exponential functional for unkilled and killed Lévy processes. This will be achieved by generalizing to our context a mapping recently introduced by Chazal et al. [8] and by providing some interesting results concerning the Wiener-Hopf factorization of killed Lévy processes. We also indicate that our extended factorizations of exponential functionals allow us to identify some fine distributional properties enjoyed by a large class of these random variables, such as the smoothness of their distribution, the monotonicity, complete monotonicity of their density, etc. We will be using these type of results to provide some new distributional properties enjoyed by the density of first passage times for some stable Lévy processes.
Our first theorem is the main result in our paper. Equation (1.6) extends [16, (1.6), Theorem 1.2] to the killed case as well to the case when E [ξ 1 ] = −∞ and it is the backbone of all our applications.
Theorem 1.1. Let q ≥ 0 and assume that ξ drifts to −∞, when q = 0. Then the law of the random variable I Ψq is absolutely continuous with density which we denote by m Ψq . Next, assume that one of the following conditions holds: (1) P+: Π + (dy) = Π(dy)I {y>0} ∈ P, Then, in both cases, there exists an unkilled spectrally positive Lévy process with a negative mean such that its Laplace exponent ψ q + takes the form Furthermore, for any q ≥ 0, we have the factorization where × stands for the product of independent random variables and φ q − (z) = −Φ − (q, z) is the Laplace exponent of a negative of a subordinator which is killed at the rate given by the expression (1) We mention that when q = 0, in comparison to [16,Theorem 1.1], here we also include the case when E [ξ 1 ] = −∞. We recall that under such a condition, the functional equation (2.9) below does not even hold on the imaginary line iR.
(2) We emphasize that the main factorization identity (1.6) allows to build up many examples of two-sided Lévy processes for which the density of I Ψq can be described as a convergent power series. This is due to the fact that the exponential functionals on the right-hand side of the identity are easier to study as we have, for instance, simple expressions for their positive or negative integer moments. More precisely, the positive entire moments of I φq − , for any q ≥ 0, are given in (2.22) below and we have from [4] that the law of 1/I ψ q + is determined by its positive entire moments as follows with the convention that the right-hand side is −(ψ q + ) ′ (0 − ) when m = 1. Some specific examples will be detailed in Section 3.

3
(3) Assuming that we start with bivariate Laplace exponents Φ + and Φ − such that their Lévy measures satisfy condition P q ± with Φ + (q, 0)Φ − (q, 0) > 0, then from Lemma 2.9 below we can construct a killed Lévy process with Laplace exponent Ψ q given by identity (1.4) and such that factorization (1.6) holds. (4) We point out that (1.6) holds even when φ q − (z) − φ q − (0) = 0, for all z ∈ iR, i.e. when ξ is a subordinator. In this case We postpone the proof of the theorem to Section 2. We proceed instead by providing some consequences of our factorization identity (1.6) concerning some interesting distributional properties of the exponential functional. Before stating the results, we recall that the density m of a positive random variable is completely monotone if m is infinitely continuously differentiable and (−1) n m (n) (x) ≥ 0, for all x ≥ 0 and n = 0, 1, . . . . Note in particular, that m is nonincreasing and thus the distribution of the random variable is unimodal with mode at 0, that is its distribution is concave on [0, ∞). Corollary 1.3. (i) Let us assume that either condition (1) or (2) of Theorem 1.1 and |Ψ q (s)| < +∞, for s ∈ [−1, 0], holds true. Then, for any q > 0, such that Ψ q (−1) ≤ 0, the density m Ψq is non-increasing, continuous and a.e. differentiable on R + with m Ψq (0) = q. (ii) Let ξ be a subordinator with Lévy measure Π + ∈ P. Then, for any q > 0, the density of I Ψq is completely monotone and bounded with m Ψq (0) = q. Moreover, recalling that, in this case, the drift b of ξ is non-negative, we have, for any x < 1/b (with the convention that where a n (Ψ q ) = q n k=1 −Ψ q (−k) with a 0 (Ψ q ) = q. If b > 0, we have for any x > 0 (iii) Let ξ be a spectrally positive Lévy process and we denote, for any q > 0, by γ q the only positive root of the equation Ψ q (−s) = 0. Then, we have where B(1, γ q ) is a Beta random variable and ψ q + (z) = zφ q + (z). Moreover, if Ψ q (−1) ≤ 0 then I Ψq has a non-increasing density. (iv) Finally, let ξ be a spectrally negative Lévy process. Denoting here, for any q > 0, by γ q the only positive root of the equation Ψ q (s) = 0, we have where Γ stands for the Gamma function. In particular, we get the precise asymptotic for the density at infinity, i.e.
lim x→∞ Then, for any β ≥ γ q + 1, the mapping x → x −β m Ψq (x −1 ) is completely monotone. In particular, the density of the random variable I −1 Ψq is completely monotone, whenever γ q ≤ 1. (1) We point out that a positive random variable with a completely monotone density is in particular infinitely divisible, see [22,Theorem 51.6].
(2) A positive random variable with a non-increasing density is strongly multiplicative unimodal (for short SMU), that is the product of this random variable with any independent positive random variable is unimodal and in this case the product has its mode at 0, see [9, Proposition 3.6]. (3) We mention that the two cases (ii) and (iii), i.e. when the Lévy process has no negative jumps, have not been studied in the literature so far. (4) One may recover from item (iv) the expression of the density found in [18] in this case.
Furthermore, we point out that in [18] it is proved that the density extends actually to a function of a complex variable which is analytical in the entire complex plane cut along the negative real axis and admits a power series representation for all x > 0. (5) Note that the series (1.8) is easily amenable to numerical computations since a k (Ψ q ) can be computed recurrently. We stress that (1.8) would be difficult to get from (2.9) because the functional equation holds on a strip in the complex plane which might explain why such simple series has not yet appeared in the literature.
The proof of this corollary and of the following one will be given in Section 3. We will also describe therein some examples illustrating these results. As a by-product of Corollary 1.3, we get the following new distributional properties for the law of maximum and first passage times of some stable Lévy processes.
be an α-stable Lévy process starting from 0 with α ∈ (0, 2]. Let us write S 1 = sup 0≤t≤1 Z t and T 1 = inf{t > 0; Z t ≥ 1} and recall that the scaling property of Z yields the identity T 1 d = S −α 1 . Then, writing ρ = P(Z 1 > 0), we have the following claims: (i) The density of T 1 is bounded and non-increasing for any α ∈ (0, 1) and ρ ∈ 0, 1 α − 1 . In particular, this property holds true for any α ∈ (0, 1 2 ] and for symmetric stable Lévy processes, i.e. ρ = 1 2 , with α ∈ (0, 2 3 ]. (ii) The density of S α 1 is completely monotone if α ∈ (1, 2] and ρ = 1 − 1 α , that is when Z is spectrally positive. Remark 1.6. When α ∈ (0, 1) and ρ = 1, that is Z is a stable subordinator, we have the obvious identity Z 1 d = S 1 . Thus, we get from the first item that the density of Z −α 1 is non-increasing on R + if α ∈ (0, 1 2 ]. From Remark 1.4 (2) this means that for these values of α, Z −α 1 is SMU with mode at 0. Note that this result is consistent with the main result of Simon in [23] where it is shown that Z 1 is SMU with a positive mode if and only if α ∈ (0, 1 2 ]. The positivity of the mode implies that any non-zero real power of Z 1 , and in particular T 1 2. Proof of Theorem 1.1 2.1. The case P+. We start by extending to two-sided Lévy processes a transformation which has been introduced in [8] in the framework of spectrally negative Lévy processes. This mapping turns out to be very useful in the context of both the Wiener-Hopf factorization of Lévy processes and their exponential functionals. For its statement we need the following notation T β + : There exists β + > 0 such that for all β ∈ [0, β + ), |Ψ(β)| < +∞ and e βy Π + (y)dy ∈ P.
Also if Ψ satisfies T β + , we write, for all q ≥ 0, Let us assume that T β + holds. Then, for any β ∈ (0, β + ), the linear mapping T β defined by is the Laplace exponent of an unkilled Lévy process ξ (β,q) = (ξ and Lévy measure Π β given by Furthermore, if we assume that ξ drifts to −∞ when q = 0, then for any q ≥ 0, β * q > 0, and, for any β ∈ (0, β * q ), we have Proof. First, by linearity of the mapping T β , one gets where we recognize, on the right-hand side, the Laplace exponent of a negative of a compound Poisson process with parameter q > 0, whose jumps are exponentially distributed with parameter β > 0. Next we observe that one can write and Ψ − is the Laplace exponent of a Lévy process without positive jumps. The description of T β Ψ − as the Laplace exponent of a Lévy process without positive jumps follows from [8, Proposition 2.2]. Thus, from the linearity of the transform it remains to study its effect on Ψ + . An integration by parts gives us that which provides the expression (2.2) since the mapping y → e βy Π(y) is non-increasing by assumption and plainly the condition T β + gives that ∞ 0 (1 ∧ y 2 )(−e βy Π + (y)) ′ dy < +∞. Next, when q > 0 then β * q > 0 since Ψ(0) = 0 and the mapping s → Ψ(s) is continuous on [0, β + ). When 6 q = 0, the condition T β + combined with the fact that ξ drifts to −∞ implies that Ψ ′ (0+) < 0, where Ψ ′ (0+) can be −∞. Clearly then we have that Ψ(β) < 0, for any β ∈ (0, ǫ) and some ǫ > 0, and hence β * 0 > 0. Moreover, we observe that, for any q ≥ 0, which is clearly finite and negative, for any β ∈ (0, β * q ). The proof of the proposition is completed.
Remark 2.2. We note that, for any q > 0 and any 0 < β < β + , the Lévy process ξ (β,q) can be decomposed as ξ is a Lévy process with Laplace exponent T β Ψ and (N q t ) t≥0 is an independent compound Poisson process with parameter q whose jumps are exponentially distributed with parameter β.
We shall need the following alternative representation of the bivariate ladder exponents as well as an interesting application of the transform T β in the context of the Wiener-Hopf factorization of Lévy processes.
is the Laplace exponent of (resp. the negative of ) a subordinator. More precisely, they take the form where we recall that µ q ± (dy) = ∞ 0 e −qy 1 µ ± (dy 1 , dy) and q ± = k ± +η ± q+ Consequently, the Wiener-Hopf factorization (1.4) takes the form Moreover, assume that T β + holds and that ξ drifts to −∞ when q = 0. Then, for any β ∈ (0, β * q ), we have Proof. Since, for any q > 0, we have that we deduce the first claim from the fact that q ± > 0, for any q > 0. Next, we have, under the T β + condition that s → φ q + (s) is well-defined on (−∞, β + ), see [16,Lemma 4.2]. Also, for any β ∈ (0, β * q ), φ q + (β) < 0 as clearly both Ψ q (β) < 0 and φ q− (β) < 0. Thus, for such β the mapping s → φ q + (s + β) is the Laplace exponent of a killed subordinator. Moreover, it is not difficult to check that, for any fixed q ≥ 0 and β ∈ (0, β * q ), z → T β φ q − (z) is the Laplace exponent of the negative of a proper subordinator. Moreover, since β ∈ (0, β * q ) we deduce, from the item (2) of Proposition 2.1, that the proper Lévy process with characteristic exponent T β Ψ q drifts to −∞ and hence its descending ladder height process is also the negative of a proper subordinator, see e.g. [10]. We conclude by observing the identities and by invoking the uniqueness for the Wiener-Hopf factors, see [22,Theorem 45 The T β transform turns out to be also very useful in proving the following claim which shows, in particular, that the family of exponential functional of Lévy processes is invariant under some length-biased transforms. In particular, the law of I Ψq admits such a representation in terms of a perpetual exponential functional. Although, a similar result was given in [8] for one-sided Lévy processes, its extension requires deeper arguments.
Theorem 2.4. Let us assume that ξ drifts to −∞ when q = 0. Then the following claims hold: (1) The law of the random variable I Ψq is absolutely continuous.
(2) Assume further that T β + holds. Then, for every β ∈ (0, β * q ), there exists a proper Lévy process with a finite negative mean and Laplace exponent T β Ψ q , such that Proof. First, we point out that the absolute continuity of I Ψq in the case q = 0 is well-known and can be found in [3,Theorem 3.9]. Thus, we assume that q > 0. The case when ξ is with infinitely many jumps can be recovered from [3, Theorem 3.9 (b)]. Indeed, for any v > 0, denote by g(s) = e s and dY is strictly increasing we have that condition (3.12) in [3] is satisfied. Moreover, for ǫ < v, we have that the density of the absolutely continuous part of the measure dY (v) t restricted to [0, ǫ] has a density which equals 1. According to [3, Theorem 3.9 (b)] this suffices to show that v 0 e ξs ds has a law which is absolutely continuous with respect to the Lebesgue measure. Then for any Borel set A ⊂ (0, ∞) we have that t 0 e ξs ds ∈ A dt and our statement follows in this case. Next, assume that ξ = ξ (1) + B where ξ (1) is a compound Poisson process and B a Brownian motion with given mean and variance, which can be both zero. We denote by (T n ) n≥1 (resp. (Y n ) n≥1 ) the sequence of inter-arrival times (resp. the sequence) of the jumps of ξ (1) . Define the measures Υ andΥ respectively on R N + + = {ω = (t 1 , t 2 , ...) : t i > 0, for i ≥ 1} and R N + = {ω = (y 1 , ...) : y i ∈ R, for i ≥ 1} to be induced by the sequences (T n ) n≥1 and (Y n ) n≥1 . For any ω andω, we set S 0 (ω) =S 0 (ω) = 0 and we write S j (ω) = j i=1 t i , S j (ω) = j i=1 y i , and P j (ω) = P(S j (ω) ≤ e q < S j+1 (ω)) = P(A j (ω)).
Denote by Γ j,ω (dx) = P (e q ∈ dx; A j (ω)|ω) = P j (ω)P (e q ∈ dx|A j (ω)) and note that Γ j,ω are absolutely continuous with respect to the Lebesgue measure. We also set e Bs ds. 8 Now, we pick A ⊂ R + such that the Lebesgue measure of A is zero and write Next, denote by D k = D S k (ω) the full set of continuous functions images of the Brownian motion up to time S k (ω) and note that and Θ is the measure on D k induced by B. Furthermore since the setsÃ j (ω) have zero Lebesgue measure it suffices to show that eq−S k (ω) 0 e B ′ s ds is absolutely continuous on A j (ω). Indeed it follows because the law of t 0 e B ′ s ds is absolutely continuous for any t > 0 and non trivial Brownian motion, see [26]. When B ′ s = as the same claims follows as the measures Γ j,ω are absolutely continuous and hence so is holds for any z ∈ C such that 0 < ℜ(z) < β * q , where we recall from Proposition 2.1, that for any q ≥ 0, β * q > 0 is valid. We point out that all quantities involved are finite since for every q ≥ 0 for which I Ψq is well-defined, we have using [21, Lemma 2] and a monotone argument, that Thus, for any β ∈ (0, β * q ), we have, for any −β < ℜ(z) < β * q − β, We note in particular from (2.10) that E I β Ψq < ∞. On the other hand, we have from Proposition 2.1, that for any q ≥ 0 and any β ∈ (0, β * q ), T β Ψ q is the Laplace exponent of a Lévy process with a finite negative mean and thus the random variable I T β Ψq is well-defined. We deduce from (2.9) and the definition of the transformation T β , that, for any −β < ℜ(z) < β * q − β, Next, since the distribution of I Ψq is absolutely continuous, we have that the function m β,q given by x β m Ψq (x), a.e. for x > 0, is well-defined and determines a probability density function. We denote by I β,q the random variable with density m β,q . Then, clearly from (2.10) and (2.11), the functional equation holds for −β < ℜ(z) < β * q − β and E I −1 β,q < ∞ and E I δ β,q < ∞, for some δ > 0. Thanks to the existence of these moments we can use [16,Lemma 4.4] to deduce that in the notation of [16], Lm β,q (x) = 0 a.e. Moreover, as E I −1 β,q < ∞ we can apply [16,Theorem 3.7] to get in fact that (2.8) holds. Indeed, otherwise, both the law of I T β Ψq and I β,q will be different stationary measures of the generalized Ornstein-Uhlenbeck process associated to the Lévy process with exponent T β Ψ q as defined in [16,Theorem 3.7] which is impossible. The proof of the last claim follows readily from the limit lim β→0 E I β Ψq = 1 combined with identity (2.8).
The next result is in the spirit of the result of [16,Theorem 1] in the case E + and thus can be seen as its extension.
Proof. We consider only the case when q > 0 since when q = 0 we are in the setting of the classical Vigon's equation amicale. Next, the latter applied to the unkilled Lévy process ξ (β,q) as defined in Proposition 2.1 with β ∈ (0, β * q ), yields, with the obvious notation, where, from (2.2), we have .
From the latter we immediately deduce by comparing the Laplace transforms that Next, we have, using identity (2.2) and the fact that condition T β + holds, the existence of a constant C > 0 such that, for all β small enough, Using this inequality and recalling that, for any q > 0, U − is a positive finite measure on R + , as a potential measure of a negative of a killed subordinator, that is a transient Markov process, we obtain, with where the constant C q > 0 is also uniform for all β small enough. This gives us Since for all β small enough ∞ y e βr Π + (r)dr ≤ C 1 , with C 1 > 0, we have, from (2.18), at the points of continuity of Π + (dy), that − defines a finite measure, we conclude from (2.20) that lim β→0 µ β q + (y) = µ q + (y) and hence (2.16) holds. Next, the fact that the mapping s → φ q + (s) is well defined on (0, β + ) follows readily from [16,Lemma 4.2] since Ψ q satisfies the condition T β + . Then, for any 0 < β < β + , (2.16) gives us that The claim that e βy µ q + (y) ∈ P now follows from the fact that for every fixed r > 0, the mapping y → e β(y+r) Π + (y + r) is non-increasing on R + . Hence φ q + also satisfies T β + . Assume now that e βy Π + (dy) ∈ P, then one may write Π + (dy) = π + (y)dy and the equation which is a differentiated version of (2.16) shows that e βy µ q + (dy) ∈ P. To rigorously justify the exchange of differentiation and integration in the differentiated version above note that under T β + the differentiated version is clearly valid if q > 0 since U (q) − defines a finite measure. Moreover, when q = 0 and β > 0, e −βr U − (dr) is a finite measure due to the sublinearity of the potential function U − ((0, r)), see [2, p 74]. Finally when both q = 0 and β = 0 the differentiated version follows from [16,Lemma 4.11].
In order to complete the proof of Theorem 1.1 in the case P+ we will resort to some approximation procedures for which we need the following results.  (b) Let (Ψ (n) ) n≥1 be a sequence of characteristic exponents of Lévy processes such that, for all z ∈ iR, where Ψ is the characteristic exponent of a Lévy process. Assume further that for all n ≥ 1, Ψ (n) (0) = Ψ(0) = 0. Then, for all q > 0, Remark 2.8. A case similar to (a) was treated in Lemma 4.8 in [16]. However there it is assumed that the subordinators are proper. Note that case (b) is far simpler than Lemma 4.8 in [16] as we are strictly in the killed case and the exponential functional up to a finite time horizon is continuous in the Skorohod topology.

The case P q
± . Since the case q = 0 was treated in [16], we assume in the sequel that q > 0. In what follows, we provide a necessary condition on the Lévy measures of the characteristic exponent of bivariate subordinators in order that they correspond to the Wiener-Hopf factors of a killed Lévy process. We mention that Vigon [24] provides such a criteria for proper Lévy processes and our condition relies heavily on his approach. Lemma 2.9. Let us consider φ q + and φ q − as defined in Proposition 2.3. Assume that µ q + ∈ P and µ q − ∈ P with q = q + q − > 0. 14 (1) There exists a characteristic exponent of a killed Lévy process Ψ q such that (2) If in addition for any 0 < β < β + , for some β + > 0, −∞ < φ q + (β) < 0 and e βy µ q + (dy) ∈ P, then Ψ q satisfies the condition T β + .
Proof. From Proposition 2.3, writing φ q ± (z) = φ ± (z) − q ± , we observe that Then, from Vigon's philantropy theory, we know that −(φ + (z) − q + )φ − (z) is the characteristic exponent of an unkilled Lévy process that drifts to −∞. It is also clear that q − φ + (z) is the characteristic exponent of an unkilled subordinator. From the inequality q + q − > 0 we complete the proof of the first item. Next, from the form of φ q − in Proposition 2.3 and carefully using the same techniques as in deriving (2.2) we deduce that T β φ q − is the Laplace exponent of a negative of an unkilled subordinator whose Lévy measure has the form µ β q − (dy) = e −βy (µ q − (dy)+ βµ q − (y)dy). Similarly, due to our assumption, i.e. −∞ < φ q + (β) < 0, the mapping s → φ q + (s + β) is the Laplace exponent of a killed subordinator with Lévy measure e βy µ q + (dy). As µ q − ∈ P, we easily check that µ β q − (dy) ∈ P and since, by assumption, e βy µ q + (dy) ∈ P, we have from the first item that there exists a characteristic exponent Ψ β , of an unkilled Lévy process drifting to −∞, which is defined by Moreover, as we deduce, by means of an uniqueness argument, that T β Ψ q = Ψ β q . Then, by the mere definition of condition T β + we check that Ψ q satisfies condition T β + .
Remark 3.1. The previous example illustrates nicely the fact that our main factorization allows to get exact asymptotics for both large and small values of the argument as soon as one is able to expand as a series the density of the exponential functionals involved in the identity.
Since Ψ α 1 − 1 α − q = 0 we have 0 < γ q = 1 − 1 α ≤ 1 2 , we deduce from Corollary 1.3 (iv) that the density of 1/T 1 is completely monotone which means that the density of S α 1 is completely monotone. Note that the law of S 1 has been computed explicitly as a power series by Bernyk et al. [1]. We end up the paper by pointing out that in the Brownian motion case, i.e. α = 2, the density of S 2 1 is well-known to be m S