Explicit identities for Lévy processes associated to symmetric stable processes.

Transcription

1 Explicit identities for Lévy processes associated to symmetric stable processes. M.E. Caballero 1, J.C. Pardo and J.L. Pérez 3 Abstract In this paper we introduce a new class of Lévy processes which we call hypergeometricstable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems. Key words and phrases: Symmetric stable Lévy processes, Positive self-similar Markov processes, Lamperti representation, first exit time, first hitting time. MSC subject classifications: 6 G 18, 6 G 51, 6 B 5. 1 Introduction and preliminaries. Let Z Z t {Z 1 t,... Z d t }, t be a symmetric stable Lévy process of index α, in R d d 1, that is, a process with stationary independent increments, whose sample paths are càdlàg and E exp{i < λ, Zt >} exp{ t λ α }, for all t and λ R d. Here P z denotes the law of the process Z started from z R d, the norm in R d and <, > the Euclidean inner product. The process Z k Z k t, t will be called the k-th coordinate process of Z. Of course, Z k is a real symmetric stable process whose characteristic exponent is given by E exp { iθz k } t exp{ t θ α }, 1,3 Instituto de Matemáticas, Universidad Nacional Autonoma de México, México D.F C.P marie@matem.unam.mx, 3 garmendia@matem.unam.mx Department of Mathematical Science, University of Bath. Bath BA 7AY. United Kingdom, jcpm@bath.ac.uk 1

2 for all t and θ R. According to Bertoin chapter I, the process Z is transient for α < d, that is lim Z t t a.s., and it oscillates otherwise, i.e. for α [1, and d 1, we have lim sup Z t and lim inf Z t a.s. t t When d, we have that single points are polar, i.e. for every x, z R d P x Z t z for some t >. In the one-dimensional case, points are polar for α, 1] and when α 1, the process Z makes infinitely many jumps across a point, say z, before the first hitting time of z see for instance Proposition VIII.8 in. One of the main properties of the process Z is that it satisfies the scaling property with index α, i.e. for every b > The law of bz b α t, t under P x is P bx. 1.1 This implies that the radial process R R t, t defined by R t Z t satisfies the same scaling property 1.1. Since Z is isotropic, its radial part R is a strong Markov process see Millar 1. When d, the radial process R hits points if and only if Z 1 hits points i.e. when α 1, see for instance Theorem 3.1 in 1. Finally, we note that when points are polar for Z the radial process R will never hit the point. In what follows we will assume that α d, so the radial process R will be a positive self-similar Markov process pssmp with index α and infinite lifetime. A natural question arises: can we characterize the Lévy process ξ associated to the pssmp R t, t via the Lamperti transformation? We briefly recall the main features of the Lamperti transfomation, between pssmp and Lévy processes. A positive self-similar Markov processes X, Q x, x >, is a strong Markov processes with càdlàg paths, which fulfills a scaling property. Well-known examples of this kind of processes are: Bessel processes, stable subordinators, stable processes conditioned to stay positive, etc. According to Lamperti 11, any pssmp up to its first hitting time of may be expressed as the exponential of a Lévy process, time changed by the inverse of its exponential functional. More formally, let X, Q x be a pssmp with index β >, starting from x >, set S inf{t > : X t } and write the canonical process X in the following form: X t x exp { } ξ τtx β t < S, 1. where for t < S, τt inf { s : s } exp {βξ u } du t. Then under Q x, ξ ξ t, t is a Lévy process started from whose law does not depend on x > and such that:

3 i if Q x S + 1, then ξ has an infinite lifetime and lim sup ξ t +, P x -a.s., t + ii if Q x S < +, XS 1, then ξ has an infinite lifetime and lim t ξ t, P x -a.s., iii if Q x S < +, XS > 1, then ξ is killed at an independent exponentially distributed random time with parameter λ >. As mentioned in 11, the probabilities Q x S +, Q x S < +, XS and Q x S < +, XS > are or 1 independently of x, so that the three classes presented above are exhaustive. Moreover, for any t < exp{βξ s } ds, τt x β t ds X s β, Q x a.s. 1.3 Therefore 1. is invertible and yields a one-to-one relation between the class of pssmp s killed at time S and the one of Lévy processes. Another important result of Lamperti 11 provides the explicit form of the generator of any pssmp X, Q y in terms of its underlying Lévy process. Let ξ be the underlying Lévy process associated to X, Q y via 1. and denote by L and M their respective infinitesimal generators. Let D L be the domain of the generator L and recall that it contains all the functions with continuous second derivatives on [, ], and that if f is such a function then L acts as follows for x R, where µ R and σ > : Lfx µf x + σ f x + R fx + y fx f xhy Πdy bfx. 1.4 The measure Πdx is the so-called Lévy measure of ξ, which satisfies Π{} and 1 x Πdx <. The function h is any bounded Borel function such that hy y as y. The positive constant b represents the killing rate of ξ b if ξ has infinite lifetime. It is important to note that in 1.4 the choice of the function h is arbitrary and the coefficient µ is the only one which depends on this choice. Lamperti establishes the following result in 11. Theorem 1. If g is such that g, yg and y g are continuous on [, ], then they belong to the domain, D M, of the infinitesimal generator of X, Q y, which acts as follows for a > Mga µa 1 β g a + σ a β g a ba β ga + a β gau ga ag ahlog u Gdu, where Gdu Πdu log u, for u >. This expression determines the law of the process X t, t T under Q y. R 3

4 Previous work on this subject appears in Carmona et al. 5 where the authors study the radial part of a Cauchy process C C t, t i.e. α d 1 and they obtain the infinitesimal generator of its associated Lévy process ξ ξ t, t via the Lamperti transformation. More precisely, the infinitesimal generator of ξ is given as follows Lgξ 1 π and its characteristic exponent satisfies E exp{iλξ t } cosh η sinh η gξ + η gξ ηg ξ1i η 1 dη, πλ iλ tanh e. As we will see in sections and 5 this example is a particular case of the results obtained in this paper by very different methods. As it is expected, the formulas obtained in both papers coincide for α d 1. It is important to point out that in Carmona et al. 5, it is announced that the authors will continue this line of research by studying the case of the norm of a multidimensional Cauchy process, but up to our knowledge this has not be done. The paper is organized as follows: In section, we compute the infinitesimal generator of the radial process R and using theorem 1 we obtain the characteristics of its associated Lévy process ξ. The Lévy measure obtained has a rather complicated form since it is expressed in terms of the Gauss hypergeometric function F 1. When d 1 we show that the process ξ can be expressed as the sum of a Lamperti stable process see Caballero et al.4 for a proper definition and an independent Poisson process. In section 3 we study one sided exit problems of the Lévy process ξ, using well known results of Blumenthal et al. 3 for the symmetric α-stable process Z. When α < d, a straightforward computations allows us to deduce the law of the random variable ξ inf t ξ t. In section 4, we study the special case 1 < α < d. Using the work of S. Port 13 on the radial processes of Z, we compute the probability that the Lévy process ξ hits points. Finally in section 5 we obtain the Wiener-Hopf factorization of ξ and deduce the explicit form of the characteristic exponent. Concluding remarks show in section 6 how to obtain n-tuple laws for ξ and R following Kyprianou et al. 1. The underlying Lévy process of R In this section, we compute the generator of the radial process R and the characteristics of the underlying Lévy process ξ in the Lamperti representation 1. of the latter. To this end, we will use the expression of Z as a subordinated Brownian motion. More precisely, if B B t, t is a d-dimensional Brownian motion started from x R d and let σ σ t, t be an independent stable subordinator with index α/ initiated from. Then the process B σt, t is a standard symmetric α-stable process. Let us define the so-called Pochhammer symbol by z α Γz + α, for z C, Γz 4

5 and the Gauss s hypergeometric function by where a, b, c >. F 1 a, b; c; z k z k a kb k, for z < 1, c k k! Theorem. Let g : R + R be such that g CR +. The infinitesimal generator of R R t, t, denoted by M, acts as follows for a >, Mga a α gya ga ag allog y y d 1 y 1 + y F dy, α+d/ 1 + y where F z α αd/ α/ Γ1 α/ F 1 α + d/4, α + d/4 + 1/; d/; z for z 1, 1,.5 and the function l is given by ly y 1 + y e1 dy 1 + e y α+d/ 1 1I{ y <1}..6 Remark 1. Following the original notation of Lamperti in 11, the generator can also be written as Mga a α gya ga ag log y 1 + log y a 1 + log y log dg dy + a 1 α νg a y where G is a finite measure given by and G dy ν y d y α+d/ log y 1 + log y F y dy 1 + y log y 1 + log y llog y 1 + log G dy y log y It is easy to prove that the integral defining ν converges. Proof: From Theorem 3.1 in 15 and the fact that Z can be seen as a subordinated Brownian motion, the infinitesimal generator M of R R t, t is given as follows Mh P s h hρds, where ρ is the Lévy measure of the stable subordinator σ and is given by ρds α/ 1 α Γ1 α/ s 1+α/ 1I {s>} ds, 5

6 P s is the semi-group of the d-dimensional Bessel process and h is any function in the domain of the infinitesimal generator of P t, t. Let g be as in the statement and recall that for a >, the semi-group for the d- dimensional Bessel process satisfies P s ga dy gy s y d/ 1 y exp a y + a s ay I d/ 1, s where I d/ 1 is the modified Bessel function of index d/ 1 see for instance 14. Therefore putting the pieces together, it follows Mga α/ 1 α Γ1 α/ y d/ 1 y gy ga a 1 exp a + y s+α/ s Now, recall the following identity for the modified Bessel function I d/ 1, ay I d/ 1 dyds. s.7 I d/ 1 x k x/ k+d/ 1 Γd/ + kk!, and note that for a y ds exp a + y ay I s+α/ d/ 1 s s ay k+d/ 1 s ds α/ k s Γd/ + kk! exp a + y s k+α+d/ 1+α/ 1 ay du u k+α+d/ 1 e u Γd/ + kk! a + y ay k 1+α/ ay d/ 1 k ay Γk + α + d/ a + y α+d/ a + y Γk + 1Γd/ + k..8 k Next, we consider the following property of the Gamma function, and deduce that Γz π 1/ z 1/ ΓzΓz + 1/,.9 Γk + α + d/ π 1/ k+α+d/ 1/ Γk + α + d/4γk + α + d/4 + 1/ k Γα + d/α + d/4 k α + d/4 + 1/ k. Therefore using the above identity, we see that.8 is equal to α/+1 ay d/ 1 Γα + d/ a + y α+d/ Γd/ k ay α + d/4k α + d/4 + 1/ k, a + y d/ k k! k 6

7 where the series above is the Gauss s hypergeometric function ay F 1 α + d/4, α + d/4 + 1/; d/;. a + y We remark that we cannot use Fubini s theorem in.7 because the expression inside the integral with respect to the product measure is not integrable. This is easily seen by noting that ay F 1 α + d/4, α + d/4 + 1/; d/; a + y y a α+1 as y a. So instead let us consider ε, a and we introduce the sets, A ε a, A εa a a + ε a + ε,, C a a,ε, a + ε a + ε a, we then study the integral y d/ 1 1 y gy ga exp a + y ay I a s+α/ d/ 1 dyds..1 s s We would like to use Fubini s Theorem in the expression above, to this end we now prove the integrability of the integrand in.1 with respect the product measure. For simplicity, we use the notation established in.5, and using Tonelli s theorem and.8 we have y d/ 1 1 y gy ga exp a + y A εa a s+α/ s y d 1 ay g a + y F dy, α+d/ a + y Ca,ε A εa I d/ 1 ay s dyds. which is finite. So now let us return to.1, then applying Fubini s theorem and.8 we obtain α/ 1 α y d/ 1 1 y gy ga exp a + y ay I Γ1 α/ A εa a s+α/ d/ 1 dyds. s s y d 1 ay gy ga A εa a + y F dy α+d/ a + y a α y d 1 y gay ga 1 + y F dy..11 α+d/ 1 + y In order to get the result, we first show that if Ba, ε Ca, ε 1/e, e, then log y 1 y 1 + log y 1 + y F dy y Ba,ε To do so, we note that the integral in.1 is equal to e log y 1 y a/a+ε 1+a 1 ε 1 + log y 1 + y F dy+ 1 + y 1/e 7 log y 1 + log y 1 y 1 + y F dy. 1 + y

8 Making the change of variable y z 1 in the first integral of above, we get that e log y 1 y 1+a 1 ε 1 + log y 1 + y F dy 1 + y a/a+ε log z 1 z 1/e 1 + log z 1 + z F dz, 1 + z and the identity.1 follows. It is easy to see using.8 the following equality: log y 1 y Ba,ε 1 + log y 1 + y F dy 1 + y aα α/ 1 α y d/ 1 yllog y/a 1IAεay Γ1 α/ a 1 exp s+α/ a + y where l is defined as in.6. Finally, we add the null term a 1 α g log y a 1 + log y y F s y 1I 1 + y Ba,ε ydy, to the identity.11 and after some calculations using.13 we obtain α/ 1 α Γ1 α/ a α Ca,ε A εa y d/ 1 y gy ga ag allogy/a a 1 exp a + y ay I s+α/ d/ 1 s s y 1 + y gya ga ag allog y y d y α+d/ F ay I d/ 1 dyds..13 s dyds So using the dominated convergence theorem and.14, we can conclude that Mga α/ 1 α Γ1 α/ α/ 1 α lim ε Γ1 α/ a α y d/ 1 y gy ga a A εa 1 exp a + y s+α/ s y gy ga ag allogy/a 1 exp a + y s+α/ s gya ga ag allog y y d y α+d/ F ay I d/ 1 s y d/ 1 a dy..14 dyds ay I d/ 1 s y 1 + y dyds dy. 8

9 Using Lamperti s result recalled in Theorem 1, we may now give the explicit form of the generator of the Lévy process ξ associated to the pssmp R. We will call this new class of Lévy processes hypergeometric-stable. Corollary 1. Let ξ be the Lévy process in the Lamperti representation 1. of the radial process R. The infinitesimal generator L, of ξ, with domain D L is given in the polar case by Lfx fx + y fx f xly Πdy,.15 for any f D A and x R, where Πdy R e dy 1 + e y α+d/ F 4e y e y + 1 dy. Equivalently, the characteristic exponent of ξ is given by Ψλ iλµ + 1 e iλy + iλy1i { y <1} Πdy where µ R R e ly dy 4e y y1i { y 1} 1 + e y F dy. α+d/ e y + 1 We finish this section with a remarkable result on the decomposition of the Lévy measure of the process ξ when the dimension is d 1 and α, 1] polar case. Such decomposition describes the structure of ξ in terms of two independent Lévy processes, each with different type of path behaviour. Recall in this case that the symmetric stable process Z is of bounded variation and so its the radial part R and the associated Lévy process ξ will also be of bounded variation. Hence, the characteristic exponent of ξ is given by Ψλ e iλy 1 Πdy. Proposition 1. Assume that d 1, then we have Ψλ e iλy 1 Π 1 dy + R R R e iλy 1 Π dy, where Π 1 is the Lévy measure of a Lamperti Lévy process with characteristics, 1, α see for instance 4, i.e. and Π 1 dy α 1 α1/ α/ Γ1 α/ e y e y 1 1 e y α+1 {y>} + 1 e y 1 α+1 {y<} dy, Π dy α 1 α1/ α/ Γ1 α/ is the Lévy measure of a compound Poisson process. e y dy, e y + 1 α+1 9

10 Proof: Let x [, 1. Using identity.9 twice, we deduce that F 1 α + 1/4, α + 1/4 + 1/; 1/; x x k α + 1/4 kα + 1/4 + 1/ k k!1/ k k Γ1/ 1/ α/ k Γα + 1/ + k x Γα + 1/4 + 1/ Γα + 1/4 Γk + 1 k 1/ α/ Γ1/ π 1/ 1/ α+1/ Γα + 1/ 1 k Γα + 1/ + k x + Γ1 + k 1 x k α + 1/ k + k! k k 1 1 x α+1/ x. α+1/ x k α + 1/ k k! k Γα + 1/ + k x Γ1 + k Now, from the identity above, we deduce that the Lévy measure of the process ξ satisfies Πdy α 1 α1/ α/ e y α+1 1 ey α ey dy Γ1 α/ 1 + e y α+1/ e y + 1 e y + 1 α 1 α1/ α/ 1 Γ1 α/ ey e y dy, α+1 e y + 1 α+1 and the statement follows. 3 Entrance laws for the process ξ: Intervals. In this section we will work in the case α < d, which is the transient case and we obtain some explicit identities for the one sided exit problem. In what follows, P will be a reference probability measure on D the Skorokhod space of R-valued càdlàg paths under which ξ is the hypergeometric-stable Lévy process described in Corollary 1 starting from. For any y R let and for any x > let T + y inf{t : ξ t > y} and T y inf{t : ξ t < y}, σ + x inf{t : R t > x} and σ x inf{t : R t < x}. Lemma 1. Fix < v < < u <. Suppose that A is any interval in [u, and B is any interval in, v]. Then, P ξ T + u A; T u + < P x R σ + ea ; σ + e u e < u and where x is such that x 1. P ξ T v B; Tv < P x R σ eb ; σ e v e <, v 1

11 The proof is a consequence of the Lamperti representation and is left as an exercise. Although somewhat obvious, this lemma indicates that in order to understand the exit problem for the process ξ, we need to study how the radial process R exits a positive interval around x >. Fortunately this is possible thanks to a result of Blumenthal et al. 3 who established the following for the symmetric α-stable process Z. Define, d πα 1 fy, z π d/+1 Γ sin y α/ 1 z α/ y z d. Theorem 3 Blumenthal et al. 3. Suppose that α < d and that Z, P x is a symmetric α-stable process with values in R d, started from x. For y < 1 and z 1, we have P y Z σ + dz; σ+ 1 1 < fy, zdz Similarly for y > 1 and z 1, we have P y Z σ dz; σ 1 1 < fy, zdz The one-side exit problem for ξ can be solved using Lemma 1 and Theorem 3 as follows. Theorem 4. Suppose that α < d and fix θ and < v < < u <. Then P ξ T + u u dθ, T u + < πα π sin e u+θ 1 e u α/ e θ 1 α/ e θ+u 1 1 dθ, 3.18 and P v ξ T v dθ, T v < π sin πα e dv θ e v 1 α/ 1 e θ α/ 1 e v θ 1 dθ Proof: Since Z is a symmetric α-stable process, we have for any x R d and b > P x b 1 Z σ + dy; σ + b b < P x/b Z σ + dy; σ+ 1 1 <, which implies that P x R σ + [eu, e u+θ ]; σ + e u e < P u e u x R σ + [1, eθ ]; σ <. 3. We first study the case d 1. Here, we assume that x 1. From 3.16, 3. and Lemma 1, we have for u, θ P ξ T + u u + θ; T u + < P e u R σ + [1, eθ ]; σ < 1 πα π sin 1 e u α/ 1 y α/ e u y 1 dy, 1 y e θ from which 3.18 follows. Now, we study the case d. To this end, we fix x R d such that x 1, and 11

12 Γd/ 1. w d π d/ Hence using identity 3.16 and polar coordinates in R d, we have for u, θ P e u x R σ + [1, eθ ]; σ < d πα π d/+1 Γ sin 1 e u α/ 1 y α/ e u x y d dy 1 y e θ d πα e θ π d/+1 Γ sin 1 e u α/ r d 1 dr r 1 α/ 1 π w d 1 sin d φ dφ r re u cos φ + e u. d/ On the other hand, from formula in 9 we get for r > 1 π sin d φ dφ r re u cos φ + e u π1/ Γ d 1/ e u r d r e u 1 1, d/ Γd/ which implies that P e u x R σ + [1, eθ ]; σ < πα e θ π sin 1 e u α/ e u dr rr 1 α/ r 1 1. Therefore from Lemma 1 and 3., we conclude P ξ T + u u + θ; T u + < 1 π sin πα e θ 1 e u α/ e u dr rr 1 α/ r 1 1, 1 which proves 3.18 for the case d. The second part of the theorem can be proved in a similar way. Indeed from the scaling property of Z, we have for θ and v P x R σ [ev θ, e v ]; σ e v e < P v e v x R σ [e θ, 1]; σ 1 1 <. 3.1 Assume that d 1 and take x 1. From 3.17, 3.1 and Lemma 1, we have P ξ T v θ v; Tv < P e v R σ [e θ, 1]; σ 1 1 < 1 πα π sin e v 1 α/ 1 y α/ e v y 1 dy, e θ y 1 from which 3.19 follows. Now, we study the case d. To this end, we fix x R d such that x 1, and set w d π d/ Γd/ 1. Hence using 3.17, polar coordinates and formula in 9, we 1

13 get for θ and v P e v x R σ [e θ, 1]; σ 1 1 < d πα π d/+1 Γ sin e v 1 α/ 1 y α/ e v x y d dy e θ < y 1 d πα 1 π d/+1 Γ sin e v 1 α/ r d 1 dr e 1 r θ α/ π w d sin d θ dθ r + e v re v cos θ d/ πα 1 π sin e v 1 α/ e dv dr r d 1 1 r α/ e v r 1 e θ Therefore from Lemma 1 and 3.1, we conclude P v ξ T v θ; Tu < πα 1 π sin e v 1 α/ e dv dr r d 1 1 r α/ e v r 1. e θ This complete the proof. Additional computations yield the following corollary. Corollary. Suppose that α < d and let ξ inf t ξ t. For z, Γd/ P ξ dz Γ d α/ Γα/ e d z e z 1 α/ 1 dz. Proof: We first note that r and that for u [, 1] and z > 1 u Thus, we have 1 u d α 1 r u α / du rd dyy α/ e z 1 + y 1 1 y u α/ 1 dr r d 1 1 r α/ e z r 1 Γd/ Γα/Γ d α/ Γd/ Γα/Γ d α/ 1 1 Γd/ Γα/Γ d α/ π sinπα/ ez 1 α/ Γα/Γ d α/ Γd/ π e z u α/ 1. sinπα/ e z 1 α/ r dr r1 r α/ e z r 1 u d α 1 r u α / du 1 u duu d α 1 dyy α/ e z 1 + y 1 1 y u α/ 1 1 Γd/ Γα/Γ d α/ π sinπα/ ez 1 α/ e d z 13 du u d α 1 e z u α/ 1 e z 1 dr, r α/ 1 r + 1 d/.

14 Therefore, from the above computations and 3.19 we get for z > P ξ z P T z < πα π sin e dz e z 1 α/ e dθ 1 e θ α/ 1 e z+θ 1 dθ πα π sin e d z e z 1 1 α/ dr r d 1 1 r α/ e z r 1 Γd/ Γα/Γ d α/ r α/ 1 dr e z 1 r + 1. d/ This complete the proof. 4 Entrance laws: points For any y R and r >, let T y inf{t > : ξ t y} and σ r inf{t > : R t r}. We also introduce P µ νz 1 Γ1 µ µ/ z + 1 F 1 ν, ν + 1; 1 µ; 1 z z 1 z > 1 the so called Legendre function of the first kind. The purpose of this section is to explicitly compute the probability that the process ξ hits a point i.e. P T r <, as well as some related quantities. Our study is based on the work of Port 13, where the author computes the probability that the radial process R hits a given point when α 1,. We recall that the radial process R only hits points when α 1,. The one-point hitting probability for R, presented in Port 13 is given by the formula P x σ r < α π 1/ Γ d + α/ 1 r d/+1 α 1 r α/ 1 P 1 d/ 1 + r α/, 4. Γ α 1/ 1 r where r > and x R d such that x 1. From the Lamperti representation 1. and identity 4., we obtain the one-point hitting problem for ξ as follows. Theorem 5. Let 1 < α < d. Then for y R P T y < α π 1/ Γ d + α/ 1 e d/ 1y e y 1 α/ 1 P 1 d/ 1 + e y α/. Γ α 1/ 1 e y Now, we explore more elaborate hitting probabilities n-point hitting problem for the Lévy process ξ when 1 < α < d. This is possible thanks to a result of Port 13 and the Lamperti representation 1. of the process R. Let B {r 1, r,, r n } where r 1 < r < < r n. 14

15 Recall from 13, that the potential density u, of the radial process R which is specified by 1 E z 1I {Rt A}dt dy y d u z, y, for z R d, A BR d/ +, Γd/ + 1 A satisfies see Lemmas.1 and. in 13, for x, y > ux, y d/ α Γd/Γ d α/ xy 1 d/ x y α/ 1 P 1 d/ x + y α/, Γα/ x y and ux, x π 1/ d/ Γα 1/ Γ α + d Γd/Γ d α/ x α d, / 1 Γα/ [ ] and that the matrix U ur i, r j is invertible. Let us denote its inverse by K B [ ] n n K B i, j and set σ B inf{t > : R t B}. n n According to Port, the probability that the process R hits the set B at a finite time is given by n n P z σ B < u z, r j K B i, j, 4.3 and the probability that it first hits the point r j is given by i1 j1 P z R σb r j ; σ B < n u z, r i K B i, j. 4.4 i1 For a two point set B {r 1, r } we have that K B 1 U U 1 U 1 U 11, where U 11 U U 1. Then from 4.3 and 4.4, we have P z σ B < u z, r 1ur, r + u z, r ur 1, r 1 ur 1, r 1 ur, r ur 1, r ur 1, r [u z, r 1 + u z, r ] ur 1, r 1 ur, r ur 1, r, and P z σ r1 < σ r u z, r 1ur, r u z, r ur, r 1 ur 1, r 1 ur, r ur 1, r, P z σ r < σ r1 u z, r ur 1, r 1 u z, r 1 ur 1, r ur 1, r 1 ur, r ur 1, r. Hence the two-point hitting probabilities for the Lévy process ξ are as follows. Theorem 6. Suppose that 1 < α < d and fix < v < < u <. Define T {v,u} inf{t > : ξ t {v, u}}. 15

16 We have P T {v,u} < P ξ T{v,u} v u1, ev ue u, e u + u1, e u ue v, e v uev, e u [u1, e v + u1, e u ] ue v, e v ue u, e u ue v, e u ue v, e v ue u, e u ue v, e u, f1, e v, e u and P ξ T{v,u} u f1, e u, e v, where fx, a, b ux,a ux,b ub,a ub,b ua,a ua,b ub,a ub,b. 5 Wiener-Hopf factorization. In this section we work in the polar case recall that this happens when d or when d 1 and α, 1 and compute explicitly the characteristic exponent of the process ξ using its Wiener-Hopf factorization. Denote by {L 1 t, H t : t } and { L 1 t, Ĥt : t } the possibly killed bivariate subordinators representing the ascending and descending ladder processes of ξ see for a proper definition. Write κθ, λ and κθ, λ for their joint Laplace exponents for θ, λ. For convenience we will write κ, λ q + ĉλ + 1 e λx ΠĤdx,, where q is the killing rate of Ĥ so that q > if and only if lim t ξ t, ĉ is the drift of Ĥ and Π Ĥ is its jump measure. Similar notation will also be used for κ, λ by replacing q, ξ, ĉ and ΠĤ by q, ξ, c and Π H. Note that necessarily q since lim t ξ t. Associated with the ascending and descending ladder processes are the bivariate renewal functions V and V. The former is defined by V ds, dx dt P L 1 t and taking double Laplace transforms shows that ds, H t dx e θs λx V ds, dx 1 κθ, λ for θ, λ. 5.5 A similar definition and relation holds for V. These bivariate renewal measures are essentially the Green s measures of the ascending and descending ladder processes. With an abuse of notation we shall also write V dx and V dx for the marginal measures V [,, dx and V [,, dx respectively. Since we shall never use the marginals V ds, [, and V ds, [, there should be no confusion. Note that local time at the maximum is defined only up to a multiplicative constant. For this reason, the exponent κ can only be defined up to a multiplicative constant and hence the same is true of the measure V and then obviously this argument applies to V. The main result of this section is the Wiener-Hopf factorization of the characteristic exponent of the Lévy process ξ. 16

17 Theorem 7. Let α < d and ξ be the hypergeometric-stable Lévy process. Then its characteristic exponent Ψ enjoys the following Wiener-Hopf factorization α Γ iλ + α/ Γiλ + d/ Ψλ Γ iλ/ Γiλ + d α/ α Γd/Γ iλ + α/ Γd α/γiλ + d/ Γd α/γ iλ/ Γd/Γiλ + d α/ 5.6 where the first equality hold up to a multiplicative constant. The proof of Theorem 7 relies on the computation of the Laplace exponents of the ascending ladder height and the descending ladder height processes of ξ. Lemma. Let α < d and ξ be the hypergeometric-stable Lévy process. exponent of its descending ladder height process Ĥ is given by The Laplace ˆκ, λ Γd + λ/γd α/ Γd/Γd α + λ/. 5.7 Proof: Recall from the proof of Corollary that P inf ξ t z t Γd/ Γd α/γα/ e z 1 u + 1 d/ u α/ 1 du. Also recall that V denotes the renewal function associated with Ĥ. From Proposition VI.17 in, we know that V z : V [, z] V [, P inf ξ t z for all z. t As we mentioned before, it is well known that V is unique up to a multiplicative constant which depends on the normalization of local time of ξ at its infimum. Without loss of generality we may therefore assume in the forthcoming analysis that V may be taken identically equal to 1. Hence V z Γd/ Γd α/γα/ e z 1 Now, let Kα, d Γd/ Γd α/γα/ 1 and note λ u + 1 d/ u α/ 1 du. e x 1 e λx V xdx λkα, d dx e λx du u + 1 d/ u α/ 1 Kα, d Kα, d 1 u + 1 d+λ/ u α/ 1 du u d α+λ/ 1 1 u α/ 1 du Γd/Γd + λ α/ Γd + λ/γd α/. 17

18 Finally, from 5.5 we deduce that This completes the proof. ˆκ, λ Γd + λ/γd α/ Γd/Γd α + λ/. For the computation of the Laplace exponent of the ascending ladder height process H, we will make use of an important identity obtained by Vigon 16 that relates Π H, the Lévy measure of the ascending ladder height process H, with that of the Lévy process ξ and V, the potential measure of the descending ladder height process Ĥ. Specifically, defining Π H x Π H x,, the identity states that Π H r V dlπ + l + r r >, 5.8 where Π + u Πu, for u >. Now, recall the following property of the hypergeometric function F 1 see for instance identity in 1 F 1 a, b; a b + 1; x 1 + x a 4x F 1 a/, a + 1/; a b + 1;, x and note that the Lévy measure of the process ξ can be written as follows e αy 4e y Πdy 1 + e y F 1I α+d/ 1 + e y {y>} dy e dy 4e y e y F 1I α+d/ 1 + e y {y<} dy. Therefore Πdy α αd/ α/ Γ1 α/ e αy F 1 α + d/, α/ + 1; d/; e y 1I {y>} dy + α αd/ α/ Γ1 α/ edy F 1 α + d/, α/ + 1; d/; e y 1I {y<} dy. Lemma 3. Let α < d and ξ be the hypergeometric-stable Lévy process. exponent of its ascending ladder height process H is given by 5.3 The Laplace κ, λ α Γd/Γλ + α/ Γd α/γλ/ Proof: We first note from the proof of Lemma, that the renewal measure V dy associated with Ĥ satisfies V dy Γd/ Γd α/γα/ e dy e y 1 α/ 1 dy. 5.3 We also recall the following property of the Gamma function, Γ1 α/γα/ 18 π sinπα/.

19 From Vigon s formula 5.8 and identity 5.3, we have Π H x α+1 α sinαπ/ Γd + α/ π Γd α/ x+y On the other hand from the definition of F 1, we get Set x+y dy e dy e y 1 α/ 1 du e αu F 1 α + d/, α/ + 1; d/; e u 1 e x+y e αx+y α Hence putting the pieces together, we obtain du e αu F 1 α + d/, α/ + 1; d/; e u. dz z α/ 1 F 1 α + d/, α/ + 1; d/; z F 1 d + α/, α/; d/; e x+y. Cα, d α+1 sinαπ/ Γd + α/ π Γd α/. Π H x Cα, de αx F 1 d + α/, α/; d/; e x+y e y d α e y 1 α/ 1 dy Cα, d Cα, d Cα, d k e xα/+k d + α/ kα/ k d/ k k! Cα, d Γd/Γα/ Γd + α/ e αx α sinαπ/ π e xα/+k d + α/ kα/ k d/ k k! k e xα/+k d + α/ kα/ k d/ k k! k e kx α/ k k! k 1 Γd/Γα/ Γd α/ e αx 1 e x α/. e yd/+k 1 e y α/ 1 dy u d/+k 1 1 u α/ 1 du Γd/ + kγα/ Γd + α/ + k From Theorem 3, we deduce that the process ξ does not creep upwards. Hence by Theorem VI.19 of the ascending ladder height process H has no drift. Also recall that the process ξ drift to which implies that the process H has no killing term. Therefore the Laplace exponent κ, λ of H is given by κ, λ λ α sinαπ/ Γd/Γα/ π Γd α/ By integrating by parts and a change of variable, we get κ, λ αα sinαπ/ Γd/Γα/ π Γd α/ 19 e λx e αx 1 e x α/ dx. 1 e λ/x e x dx. e x 1 α/+1

20 According to Theorem 3.1 of 4 see Theorem 3.1, the previous integral satisfies 1 e λ/x e x + α/ dx Γ α/γλ, e x 1 α/+1 Γλ/ where Γ α/ α 1 Γ1 α/. Therefore, This completes the proof. κ, λ α Γd/Γλ + α/ Γd α/γλ/ Proof of Theorem 7: From the fluctuation theory of Lévy processes, it is known that Wiener-Hopf factorization of the characteristic exponent of ξ is given by ψλ κ, iλ ˆκ, iλ up to a multiplicative constant. Hence, the result follows from Lemmas and 3. Remark. We have obtained the characteristic exponent for the process ξ in the case where α < d using the Wiener-Hopf factorization. We will now see that the same formula holds true in the example studied in 5: α d 1. Recall that they obtained the following characteristic exponent of ξ: We have E ψλ λ tanh [ ] exp{iλξ t } πλ exp { tλ tanh π coshπλ/ π λ/ sinhπλ/ πλ }, t, λ R. Γ iλ+1 iλ + 1 Γ iλ 1/ iλ. 1/ Recall that the characteristic exponent in the case α < d is given by 5.6. From the above computation we note that this formula still holds for the case α d 1. From the uniqueness of the Wiener-Hopf factorization, we deduce that the characteristic exponent of the subordinators Ĥ and H are: iλ + 1 ˆκ, iλ 1/ κ, iλ iλ. 1/ 6 n-tuple laws at first and last passage times. Recall that the renewal measure V dy associated with Ĥ satisfies V dy Γd/ Γd α/γα/ e dy e y 1 α/ 1 dy. From the form of the Laplace exponent of H and 5.5, we get that the renewal measure V dy associated with H satisfies V dy Γd α/ α 1 Γd/Γα/ 1 e y α/ 1 dy.

21 Since we have explicit expressions for the renewal functions V and V, we can get, from the main results of Doney and Kyprianou 8 and Kyprianou et al. 1, n-tuple laws at first and last passage times for the Lévy process ξ and the radial part of the symmetric stable Lévy process Z. Marginalizing the quintuple law at first passage of Doney and Kyprianou 8 see Theorem 3 and by the Lamperti representation 1., we now obtain the following new identities. Proposition. Let ξ t sup s t ξ s. For y [, x], v y and u >, P ξ T x du, x ξ +x T +x dv, x ξ T +x dy 4αΓα + d/ Γd/Γα/ sinαπ/ 1 e x y α/ 1 e dv y e v y 1 α/ 1 π e αu+v F 1 α + d/, α/ + 1; d/; e dydvdu. u+v For z [x, 1], w [, z] and θ > 1 P x sup R s dz, R σ + 1 dw, R σ + dθ 1 s<σ + 1 4αΓα + d/ Γd/Γα/ sinαπ/ z 3 d α w d 1 θ α z x α/ 1 π z w α/ 1 F 1 α + d/, α/ + 1; d/; w/θ dzdwdθ. Note that the normalizing constant above is chosen to make the densities on the righthand side distributions. It is also important to remark that the triple law for the Lévy process ξ extends the identity in Let us define the last passage time and the future infimum for the processes ξ and R, respectively U x sup{t : ξ t < x}, L x sup{t : R t < x}, J t inf s t ξ s and F t inf s t R s. From Proposition.3 in Millar 1, we know that if z > the radial process R of the symmetric stable Lévy process is regular for both z, and [, z. Hence from the Lamperti representation 1., we deduce that the Lévy process ξ is regular for both, and,. Now, applying corollaries and 5 in Kyprianou et al. 1, we obtain quadruple laws at last passage times for ξ and R. Proposition 3. For x, v >, y < x + v and w v >, P J dv, J Ux x du, x ξ Ux dy, ξ Ux x dw 8αΓα + d/ Γd α/γ α/ sinαπ/ e dv+w u e v 1 α/ 1 e w u 1 α/ 1 π 1 e x+v y α/ 1 e αw+y F 1 α + d/, α/ + 1; d/; e w+y dwdydudv. 1

22 For x, b >, we have on v x 1 b 1, v 1 < y < b and b < u w < P x 1/F dv, R Lb dy, R Lb dw, F Lb du 8αΓα + d/ Γd α/γ α/ sinαπ/ b d α v 1 d yw 1 d α u d α 1 v 1 y bv α/ 1 π w bu α/ 1 F 1 α + d/, α/ + 1; d/; y/bw dvdydwdu. We conclude this section with a nice formula for the potential kernel of the Lévy process ξ killed as it enters,, that follows from Theorem VI. in Bertoin. Proposition 4. There exist a constant k > such that for every measurable function f : [, [, and x, one has T E x fξ t dt k α Γ α/ x dy1 e y α/ 1 dze dz e z 1 α/ 1 fx + y z. In particular, the potential measure of the Lévy process ξ killed as it enters, has a density which is given by α rx, u k Γ α/ u u x 1 e y α/ 1 e dx+y u e x+y u 1 α/ 1 dy. Note that from the previous proposition, we can obtain the potential kernel of the radial process R killed as it enters, 1. Let x > 1, then σ 1 T E x fr t dt E log x fe ξt e αξt dt k α Γ α/ In particular, T E x σ 1 Elog x k α Γ α/ k xα Γα log x dy1 e y α/ 1 dze dz e z 1 α/ 1 x α e αy z fxe y z. 1 e αξt dt log x dy1 e y α/ 1 dze dz e z 1 α/ 1 x α e αy z x du u d/ 1 1 u α/ 1. Acknowledgements. This research was supported by EPSRC grant EP/D4546/1, CONACYT grant 419, and the project PAPIITT-IN165. We are much indebted to Andreas Kyprianou for many fruitful discussions on Lévy processes and fluctuation theory and to Marc Yor for pointing out the relationship with their pioneering work 5 as well as for many enlighting related conversations. Finally we want to thank the anonymous referee for his/her valuable comments which allowed us to improve this work.

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