PROBABILITY THEORY EXPONENTIAL FUNCTIONAL OF LÉVY PROCESSES: GENERALIZED WEIERSTRASS PRODUCTS AND WIENER-HOPF FACTORIZATION P. PATIE AND M.

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1 PROBABILITY THEORY EXPONENTIAL FUNCTIONAL OF LÉVY PROCESSES: GENERALIZED WEIERSTRASS PRODUCTS AND WIENER-HOPF FACTORIZATION P. PATIE AND M. SAVOV Abstract. In this note, we state a representation of the Mellin transform of the exponential functional of Lévy processes in terms of generalize Weierstrass proucts. As by-prouct, we obtain a multiplicative Wiener-Hopf factorization generalizing previous results obtaine by the authors in [14] as well as smoothness properties of its istribution. Résumé. Fonctionnelle exponentielle es processus e Lévy : prouits e Weierstrass généralisés et factorisation e Wiener-Hopf. Dans cette note, nous énonçons une représentation e la transformée e Mellin e la fonctionnelle exponentielle es processus e Lévy sous la forme e prouits e Weierstrass généralisés. Nous en éuisons une factorisation multiplicative e Wiener-Hopf généralisant un résultat obtenu récemment par les auteurs ans [14] ainsi que es propriétés e régularités pour sa loi. 1. Introuction Let ξ = (ξ t t be a possibly kille real-value Lévy process with a positive mean if it is conservative. That means that ξ is a process with stationary an inepenent increments with m = E [ξ 1 ] > if not kille at an inepenent exponential time. We refer to the excellent monographs [1] an [15] for backgroun. The law of ξ is characterize by its characteristic exponent, i.e. ln E [ ] e zξt = Ψ(zt where Ψ : ir C amits, in our context, the following Lévy-Khintchine representation (1.1 Ψ(z = σ2 2 z2 + bz + R ( e zr 1 zri { r <1} Π(r q, where q is the killing rate, σ, b R, m = b + rπ(r (, ] if q =, an, the Lévy measure r >1 Π satisfies the integrability conition R (1 r2 Π(r < +. We use the convention that ξ t = for any t e q, where e q stans for an inepenent (of ξ exponential variable of parameter q >. The aim of this note is to escribe the istribution of the positive ranom variable I Ψ = e ξt t which is calle the exponential functional of the Lévy process ξ. More specifically, we provie a representation on a strip of the Mellin transform of this variable in terms of generalize Weierstrass proucts. We euce from this result a multiplicative Wiener-Hopf factorization as well as some smoothness properties of its istribution. We refer to the book of Yor [16] an the paper of Bertoin an Yor [4] for references on the topic an for motivations for stuying the law of I Ψ. We woul also like to mention that from these publications, there have been numerous investigations of fine istributional properties of this ranom variable revealing new interesting connections with several fiels. For instance, we inicate that its analysis is intimately connecte with the stuy of special functions, see e.g. Kuznetsov an Paro [8] for Barnes multiple gamma functions an also Patie [12], Patie an Savov [14] for generalization of hypergeometric functions. Hirsch an Yor [7] establishe an interesting connection between the exponential functional of some suborinators an Key wors an phrases. Lévy processes, Functional equation, Weierstrass prouct, Wiener-Hopf factorization 2 Mathematical Subject Classification: 6G51, 33C15, 91B28. 1

2 the class of multiplicative infinitely ivisible ranom variables. Haas an Rivero [6] stuy Yaglom limits of positive self-similar Markov processes an extreme value theory by means of specific properties of I Ψ. Paro et al. [11] an Patie an Savov [14] erive a first Wiener-Hopf type factorization for its istribution by resorting to probabilistic evices. Finally, in [13], where the proofs of the results state in the next Section can be foun, the law of the variable I Ψ turns out to be a key object in the evelopment of the spectral theory of a class of non-selfajoint invariant Feller semigroups. 2. Main results We shall provie a representation of the Mellin transform of the positive ranom variable I Ψ which we enote, for some z C, as follows M IΨ (z = E [ I z 1 ] Ψ. Before stating the main result an its main consequences, we introuce a few further notation. First, by efining the following subspaces of the negative of continuous negative efinite functions N = {Ψ of the form (1.1}, we have I Ψ < + a.s. if an only if Ψ N. Inee, this is plain if q > an if q =, by the strong law of large numbers for Lévy processes we have the equivalence I Ψ < + a.s. m >. We have that Ψ N amits an analytical extension in the strip C (a,b = {z C; a < Re(z < b} with a < < b, if an only if E [ e zξ 1 ] <, for all z C(a,b. Uner this conition, the restriction of Ψ on the real interval (a, b is clearly convex an zero free on (, b. Next, with the usual convention inf{ } =, we set θ Ψ = inf u> {Ψ( u = } (, ], a Ψ = sup u> {Ψ is analytical on C ( u, } [, an Ψ = a Ψ θ Ψ [,. We shall also nee the space of Bernstein functions B efine as the set of functions φ : C (, C which amit the representation (2.1 φ(z = κ + δz + (1 e zr µ(r where κ, δ an µ is a Lévy measure such that (1 rµ(r <. It is easy to see that if φ B then φ N an, in such a case, we simply write I φ for I φ. We procee by recalling that the analytical form of the Wiener-Hopf factorization of Lévy processes leas, for any Ψ N, Ψ(z = φ + ( z (z, z ir, where φ ± B with φ + ( = if q = an φ ± ( > otherwise. We point out that in the case q >, the parameters of the Wiener-Hopf factors epen on q. In particular, in this case we write simply, in the representation (2.1, with the obvious notation, µ ± (r = e qr1 µ ±,q (r 1, r, where µ ±,q (r 1, r is the Lévy measure of the bivariate ascening an escening laer height an time processes. Finally, for a function φ : C C, we write formally the generalize Weierstrass prouct where W φ (z = e γ φz φ(z γ φ = lim n ( n k=1 φ(k φ(k + z e φ k=1 φ (k φ(k log φ(n (k φ(k z We observe that if φ(z = z, then W φ correspons to the Weierstrass prouct representation of the Gamma function Γ, vali on C/{, 1, 2,...}, an γ φ is the Euler-Mascheroni constant, see e.g. [9], justifying both the terminology an notation. We are now reay to state our main result which provies an explicit representation, in terms of generalize Weierstrass proucts, of the Mellin transform of I Ψ for general Lévy processes. 2.

3 Theorem 2.1. For any φ B, we have γ φ <. Moreover, for any Ψ N, we have (2.2 M IΨ (z = (Γ(z W (1 z, z C (,Ψ+1, W φ+ (z where the prouct W φ+ (resp. W φ is absolutely convergent on C (, (resp. C ( φ,. We point out that besies its probabilistic nature, this result seems to have some analytical interests. Inee, on the one han, it reveals that the class of Bernstein functions appears to be a natural set to generalize the Weierstrass prouct of the Gamma function. In this vein, we mention that writing φ(z = Γ(z+α Γ(z, one has φ B for any < α < 1 an one recovers the Weierstrass prouct representation of ouble Gamma functions, see e.g. [8]. On the other han, in the case < m <, we are able to show, by eveloping a precise stuy of the asymptotic ecay of the Mellin transform M IΨ on imaginary lines in its strip of efinition, that the right-han sie of (2.2 is the unique solution of the functional equation, erive by Maulik an Zwart [1], (2.3 M IΨ (z + 1 = z Ψ( z M I Ψ (z, M IΨ (1 = 1, which hols on the omain {z C; Ψ(Re( z }. We emphasize that our representation remains vali in the case m = although, to the best of our knowlege, there oes not exist any characterization of the Mellin transform in this case. By Mellin ientification, we obtain as a straightforwar consequence of the representation (2.2, the following factorization of the istribution of the exponential functional. Corollary 2.2. For any Ψ N, we have the following multiplicative Wiener-Hopf factorization (2.4 I Ψ = Iφ+ X φ where stans for the prouct of two inepenent ranom variables an [ X φ ] is a positive variable whose istribution is eterminate by its negative entire moments as follows, E X 1 = (, an for any n 2, [ ] E X n = ( n 1 k=1 (k. Finally, we have ψ(z = z (z {Ψ N ; q = an Π(rI {r>} } together with the ientity in istribution (2.5 X φ = Iψ if an only if the Lévy measure µ in the representation (2.1 of has a non-increasing ensity. It is worth mentioning that recently Hirsch an Yor [7] provie an infinite prouct representation, which iffers from our Weierstrass prouct, of the Mellin transform of the variable I φ+ when q =, which is state vali on the positive real line. They also have a similar type of representation for the positive moments of a variable, introuce by Bertoin an Yor [3], whose istribution is closely connecte to the one X 1 when ( = a situation exclue in our work. We also point out that the factorization (2.4 is a generalization of the result obtaine first by Paro et al. [11] an improve in Patie an Savov [14] uner the sufficient conitions that either the Lévy measure Π(rI {r<} or both Lévy measures in the representation (2.1 of φ + an, have a ecreasing ensity. In fact, in the aforementione papers the ientity (2.4 was obtaine with X φ = Iψ, an, the methoologies evelope therein stem heavily on the connection between the istribution of the exponential functional an the stationary measure of a family of Markov processes an aitional specific properties of the exponential functional. It is not clear to us how these techniques coul be use to erive both the general version of the factorization (2.4 an the necessary conition for the ientity (2.5 to hol. We also point out that, in [13], this Wiener-Hopf factorization allows us to erive general intertwining relationships between the Feller semigroups of positive self-similar Markov processes, extening the work of Carmona et al. [5]. For at least this purpose, it woul be interesting to give an interpretation of X φ, in the most general situation, in terms of the unerlying Lévy process ξ. We woul also like to take the opportunity to raise the question whether one can provie a pathwise interpretation of the multiplicative Wiener-Hopf factorization (2.4. 3

4 In [13], relying on the representation (2.2, as mentione above, we evelop a etaile stuy of the asymptotic behavior of the moulus of the Mellin transform M IΨ (z along the imaginary lines. In particular, we provie sufficient conitions for its asymptotic ecay to be nearly exponential, that is ecay faster than any inverse polynomials. These estimates combine with Mellin inversion theorem allows to euce smoothness properties of the istribution of I Ψ. We recall that, in the case q =, Bertoin et al. [2], see also [14] for the case q >, have shown that the istribution of the positive variable I Ψ for any Ψ N is absolutely continuous with a ensity which we enote by p ψ. To state our last results, we write, for any φ B an b, ( φ(b(y + i H φ (b = ln y, φ(by H φ = lim inf b H φ (b an H φ = lim sup b H φ (b. Corollary 2.3. The following assertions hol true. 1. We have for all b, H φ± (b π p Ψ C (R + if for any n N (2.6 lim sup b n e b( π 2 H φ + (b+h φ (b =. b 3. (2.6 hols if, for instance, at least one the following conitions is satisfie. (a δ >, where δ is the rift term of. Π (b For all λ big enough an some α >, lim sup (λx x Π (x Π(rI {r<}. < 1 λ 1+α, where Π is the tail of the measure (c ɛ > such that lim inf x µ (xx ɛ > where µ is the tail of µ the measure associate to. ( If µ has a ecreasing ensity g an either lim inf b g b (bb > or (bln(b = o ( g b as b. 4. Finally, if < z = H φ + π 2 H φ + < π then the mapping z p Ψ z is analytical in the sector arg z < z. This is, for instance, true if the conition (3a, (3b or the first conition in (3 above hols. Note that the conition on Π in (3b is very natural if we have a process of paths of unboune variation. Since Π (r r 1 at worst we see that the ratio is justifie if we assume a bit heavier tails. We conclue this note by pointing out that if Π, which correspons to the Brownian motion case with positive rift m, i.e. Ψ(z = z2 2 + mz, then H = H φ+ = π 2 an hence z = π 2 in (4. Inee, it is well known that ( in this case p 1 Ψ z = z m Γ(m e z an the Mellin transform of I Ψ is Γ(m+1 z Γ(m whose exponential rate of ecay along complex lines of the type a + ib is precisely of the rate of ( π 2 b. However, the rate of ecay of the Mellin transform oes not escribe the sector of analyticity of p 1 Ψ z which is obviously the whole complex plane in the case m is an integer. Such phenomena coul also be observe for any Ψ N with Π an 1 rπ(r = for which we also know that p Ψ is an entire function, see [12]. z References [1] Bertoin, J. Lévy Processes Cambrige University Press, [2] Bertoin, J. an Linner, A. an Maller, R. On continuity properties of the law of integrals of Lévy processes Séminaire e probabilités XLI, v of Lecture Notes in Math., , Springer, 28. [3] Bertoin, J. an Yor, M. On suborinators, self-similar Markov processes an some factorizations of the exponential variable Electron. Comm. Probab., 6, (electronic, 21. [4] Bertoin, J. an Yor, M. Exponential functionals of Lévy processes Probab. Surv., v.2, , 25. 4

5 [5] Carmona, Ph. an Petit, F. an Yor, M. Beta-gamma ranom variables an intertwining relations between certain Markov processes Rev. Mat. Iberoamericana, v. 14(2, , [6] Haas, B. an Rivero, V. Quasi-stationary istributions an Yaglom limits of self-similar Markov processes Stochastic Processes an their Applications, v. 122 (12, , 212. [7] Hirsch, F. an Yor, M. On the Mellin transforms of the perpetuity an the remainer variables associate to a suborinator Preprint Université Evry, 211. [8] Kuznetsov, A. an Paro, J.C. Fluctuations of stable processes an exponential functionals of hypergeometric Lévy processes Acta Aplicanae Mathematicae, to appear. [9] Lebeev, N.N. Special Functions an their Applications Dover Publications, New York, [1] Maulik, K. an Zwart, B. Tail Asymptotics for Exponential Functionals of Lévy Processes Stochastic Processes an their Applications, v. 116, , 26. [11] Paro, J.C. an Patie, P. an Savov, M. A Wiener-Hopf type factorization for the exponential functional of Lévy processes J. Lonon Math. Soc., v.86(3, , 212. [12] Patie, P. Law of the absorption time of some positive self-similar Markov processes Ann. Probab., v.4(2, , 212. [13] Patie, P. an Savov, M. Spectral theory for positive invariant Lamperti-Feller semigroups: The iscrete spectrum case via intertwining Working Paper, Available upon request, 212. [14] Patie, P. an Savov, M. Extene factorizations of exponential functionals of Lévy processes Electron. J. Probab.,v. 17 (38, 22 pages, 212. [15] Sato, K. Lévy Processes an Infinitely Divisible Distributions Cambrige University Press, Cambrige, [16] Yor, M. Exponential functionals of Brownian motion an relate processes Springer Finance, Berlin, 21. School of Operations Research an Information Engineering, Cornell University, Ithaca, NY 14853, USA. aress: pp396@cornell.eu University of Reaing, Department of Mathematics an Statistics, Whiteknights, PO Box 22, Reaing RG6 6AX, UK. aress: m.savov@reaing.ac.uk 5