Some explicit identities associated with positive self-similar Markov processes

Transcription

1 Stochastic Processes and their Applications Some explicit identities associated with positive self-similar Markov processes L. Chaumont a, A.E. Kyprianou b, J.C. Pardo b,c, a LAREMA, Département de Mathématiques, Université d Angers. 2, Bd Lavoisier , Angers Cedex, France b Department of Mathematical Science, University of Bath, Bath BA2 7AY, United Kingdom c Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu PARIS CEDEX 5, France Received 7 August 27; received in revised form 3 April 28; accepted May 28 Available online 7 May 28 Abstract We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type dx e γ x νe x dx, where ν is the density of the stable Lévy measure and γ is a positive parameter which depends on its characteristics. These processes were introduced in [M. E. Caballero, L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab ] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points. c 28 Elsevier B.V. All rights reserved. MSC: 6 G 8; 6 G 5; 6 B 52 Keywords: Positive self-similar Markov processes; Lamperti representation; Conditioned stable Lévy processes; First exit time; First hitting time; Exponential functional. Introduction In recent years there has been a general recognition that Lévy processes play an ever more important role in various domains of applied probability theory such as financial mathematics, Corresponding author at: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu PARIS CEDEX 5, France. addresses: loic.chaumont@univ-angers.fr L. Chaumont, ak257@bath.ac.uk A.E. Kyprianou, jcpm2@bath.ac.uk, pardomil@ccr.jussieu.fr J.C. Pardo /$ - see front matter c 28 Elsevier B.V. All rights reserved. doi:.6/j.spa.28.5.

2 L. Chaumont et al. / Stochastic Processes and their Applications insurance risk, queueing theory, statistical physics or mathematical biology. In many instances there is a need for explicit examples of Lévy processes where tractable mathematical expressions in terms of the characteristics of the underlying Lévy process may be used for the purpose of numerical simulation. Depending on the problem at hand, particular functionals are involved such as the first entrance times and overshoot distributions. In this paper, we exhibit some special classes of Lévy processes for which we can compute explicitly the law of the position at the first exit time of an interval, the two point-hitting probability and the exponential functional. Moreover, two new, concrete examples of scale functions for spectrally one-sided processes will fall out of our analysis. Known examples of overshoot distributions concern essentially some particular classes of strictly stable processes and processes whose jumps are of a compound Poisson nature with exponential jumps or slightly more generally whose jump distribution has a rational Fourier transform. For example, let us state the solution of the two-sided exit problem for completely asymmetric stable processes. In that case we take X, P x, x R, to be a spectrally positive Lévy stable process with index α, 2 starting from x. Let σ a + inf{t > : X t > a} and σ inf{t > : X t < }. It is known cf. Rogozin [27] that for y >, P x X σ + a a dy; σ a + < σ sin α x a x y α dy y + ay + a x. For the case of processes whose jumps are of a compound Poisson nature with exponential jumps, the overshoot distribution is again exponentially distributed; see Kou and Wang [8]. See also Lewis and Mordecki [2] and Pistorius [24] for the more general case of a jump distribution with a rational Fourier transform and for which the overshoot distribution belongs to the same class as the respective jump distribution of the underlying Lévy process. The exponential functional of a Lévy process, ξ, i.e. exp{ξ s } ds, also appears in various aspects of probability theory, such as: self-similar Markov processes, random processes in random environment, fragmentation processes, mathematical finance, Brownian motion on hyperbolic spaces, to name but a few. In general, the distribution of exponential functionals can be rather complicated. Nonetheless, it is known for the case that ξ is either: a standard Poisson processes, Brownian motion with drift and a particular class of spectrally negative Lévy processes of bounded variation whose Laplace exponent is of the form ψq qq + a, q, b + q where < a < < a + b. See Bertoin and Yor [6] for an overview on this topic. The class of Lévy processes that we consider in this paper do not fulfill a scaling property and may have two-sided jumps. Moreover, they have no Gaussian component and their Lévy measure is of the type dx e γ x νe x dx, where ν is the density of the stable Lévy measure with index α, 2 and γ is a positive parameter which depends on its characteristics. It is not difficult to see that the latter Lévy measure has a density which is asymptotically equivalent to that of an α-stable process for small x and has exponential decay for large x. This implies that such processes have paths which are of bounded or unbounded variation accordingly as α, and α [, 2 respectively. Further, they also have exponential moments. Special families of tempered stable processes, also known as CGMY processes, are classes of Lévy

3 982 L. Chaumont et al. / Stochastic Processes and their Applications processes with similar properties to the aforementioned which have enjoyed much exposure in the mathematical finance literature as instruments for modelling risky assets. See for example Carr et al. [], Boyarchenko and Levendorskii [8], Cont [4] or Schoutens [29]. Although the Lévy processes presented in this paper are not tempered stable processes, it is intriguing to note that they possess properties which have proved to be popular for financial models but now with the additional luxury that they come with a number of explicit fluctuation identities. We conclude the introduction with a brief outline of the remainder of the paper. The next section introduces the classes of processes which are concerned in this study. In Section 3, we give the law of the position at the first exit time from a semi-finite interval. In Section 4 we compute explicitly the two-point hitting probability and in Section 5, we study the law of the exponential functional of Lévy Lamperti processes. 2. Preliminaries on Lévy Lamperti processes Denote by D the Skorokhod space of R-valued càdlàg paths and by X the canonical process of the coordinates on D. Positive R + -valued, self-similar Markov processes X, P x, x >, are strong Markov processes with paths in D, which fulfill a scaling property, i.e. there exists a constant α > such that for any b > : The law of bx b α t, t under P x is P bx. 2. We shall refer to these processes as pssmp. According to Lamperti [2], 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, P x be a pssmp with index α >, 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 α t < S, 2.2 where for t < S, τt inf { s : s } exp {αξ u } du t. Then under P x, ξ ξ t, t is a Lévy process started from whose law does not depend on x > and such that: i if P x S +, then ξ has an infinite lifetime and lim sup t + ξ t +, P x -a.s., ii if P x S < +, X S, then ξ has an infinite lifetime and lim t ξ t, P x -a.s., iii if P x S < +, X S >, then ξ is killed at an independent exponentially distributed random time with parameter λ >. As mentioned in [2], the probabilities P x S +, P x S < +, X S and P x S < +, X S > are or independently of x, so that the three classes presented above are exhaustive. Moreover, for any t < exp{αξ s } ds, τt x α t ds X s α, P x a.s. 2.3

4 L. Chaumont et al. / Stochastic Processes and their Applications Therefore 2.2 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. Now let us consider three particular classes of pssmp. We refer to [] for more details in what follows. The first one is identified as a stable Lévy processes killed when it first exits from the positive half-line. In particular, if P x is the law of a stable Lévy process with index α or α-stable process for short initiated from x > with α, 2], then with T inf{t : X t }, under P x, the process X t {t<t } is a pssmp which satisfies condition ii if it has no negative jumps or iii if it has negative jumps. We call ξ the Lévy process with finite or infinite lifetime resulting from the Lamperti representation of the killed stable process. The characteristic exponent of ξ has been computed in [] and is given by Φ λ ia λ + [e iλx iλe x { e x <}] x dx c α, λ R, 2.4 R where a is a constant, x c + e x e x α+ c e x {x>} + e x α+ {x<}, and c, c + are nonnegative constants such that c + c + >. Note that the Lévy measure of ξ satisfies x e x νe x, where ν is the density of the stable Lévy measure with index α and symmetry parameters c and c +. The second class is that of stable processes conditioned to stay positive. See for instance in [2] for an overview of such processes. Processes in this class are the result of a Doob h- transforming stable processes killed on exiting, with hx x αρ and ρ P X <. More precisely, h is an invariant function for the killed process mentioned above, X t {t<t }, P x, and the law P x defined on each σ -field F t generated by the canonical process up to time t by dp x X t αρ dp x x αρ {t<t } 2.5 Ft is that of a pssmp which drifts toward + in particular it satisfies condition i. Then the underlying Lévy process, which will be denoted by ξ, is such that lim ξ t +, a.s., t + and from [] its characteristic exponent is Φ λ ia λ + [e iλx iλe x { e x <}] x dx, λ R, 2.6 R where a is a real constant and x c +e αρ+x e x α+ {x>} + c e αρ+x e x α+ {x<}. The third class of pssmp that we will consider is that of stable processes conditioned to hit continuously. Processes in this class are again defined as a Doob h-transform with respect to the

5 984 L. Chaumont et al. / Stochastic Processes and their Applications function h x αρx αρ which is excessive for the killed process X t {t<t }, P x. Its law P x which is defined on each σ -field F t by dp x X t αρ dp x x αρ {t T } 2.7 Ft is that of a pssmp who hits in a continuous way, i.e. X, P x satisfies condition ii. Let ξ by the underlying Lévy process in the Lamperti representation of this process, then lim ξ t a.s., t + and the characteristic exponent of ξ is given by Φ λ ia λ + [e iλx iλe x { e x <}] x dx, λ R, 2.8 R where a is a constant and x c +e αρx e x α+ {x>} + c e αρx e x α+ {x<}. Note that the constants a, a and a are computed explicitly in [] in terms of α, ρ, c and c +. Actually the process ξ corresponds to ξ conditioned to drift toward or equivalently ξ is ξ conditioned to drift to +. We will sometime use this relationship which is stated in a more formal way in the next proposition. In what follows, P will be a reference probability measure on D under which ξ, ξ and ξ are Lévy processes whose respective laws are described above. Proposition. For every t, and every bounded measurable function f, E[ f ξ t ] E[expξ t f ξ t ]. In particular, processes ξ and ξ satisfy Cramer s condition: Eexp ξ and Eexp ξ. Proof. Let f be as in the statement. From 2.5 and 2.7, we deduce that for every P x -a.s. finite F u -stopping time., xe x [ f X U ] E x [X U f X U ]. 2.9 Let t. By applying 2.9 to the F u -stopping time x α inf {u : τu > t}, which is P x -a.s. finite, and using 2.2 note that τu is continuous and increasing, we obtain E[ f ξ t ] E[expξ t f ξ t ], which is the desired result. We refer to Rivero [25], IV.6. for a similar discussion on conditioned stable processes considered as pssmp. In what follows we call ξ, ξ and ξ the Lévy Lamperti processes. We now compute the law of some of their functionals.

6 L. Chaumont et al. / Stochastic Processes and their Applications Entrance laws for Lévy Lamperti processes: Intervals In this section, by studying the two-sided exit problems for ξ, ξ and ξ, we shall obtain a variety of new identities including the identification of two new scale functions in the case of one-sided jumps. To this end, we shall start with a generic result pertaining to any positive self-similar Markov process X, P x, for x >. We denote by ξ the Lévy process starting from associated to X through the Lamperti transformation 2.2 with reference measure P on D. For any y R let T + y inf{t : ξ t y} and T y inf{t : ξ t y}, and for any y > let σ + y inf{t : X t y} and σ y inf{t : X t y}. Lemma. 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 + < T v P X σ + ea ; σ + e u e u < σ e v and P ξ T v B; T u + > T v P X σ e v eb ; σ e + u > σ e v. The proof is a straightforward consequence of the Lamperti representation 2.2 and is left as an exercise. Although somewhat obvious, this lemma indicates that for the three processes ξ, ξ and ξ, we need to understand how, respectively, an α-stable process conditioned to stay positive, an α-stable process killed when it exits the positive half-line and an α-stable process conditioned to hit the origin continuously, exit a positive interval around x >. Fortunately this is possible thanks to a result of Rogozin [27] who established the following for α-stable processes. Theorem Rogozin [27]. Suppose that X, P x is an α-stable process, initiated from x, which has two-sided jumps. Denoting ρ P X < we have for a > and x, a, P x X σ + a a dy; σ a + < σ sin α ρ a x αρ x αρ y αρ y + a αρ y + a x dy. Note that an expression for P x X σ dy; σ + a > σ can be derived from the above expression by replacing x by a x and ρ by ρ. In what follows, with an abuse of notation, we will denote by T y + and Ty for the first passage times above and below y R, respectively, of the processes ξ, ξ or ξ depending on the case that we are studying. We now proceed to split the remainder of this section into three subsections dealing with the two-sided exit problem and its ramifications for the three processes ξ, ξ and ξ respectively. 3.. Calculations for ξ The two-sided exit problem for ξ can be obtained from Lemma and Theorem as follows. We give the case for two-sided jumps. Note that this is more for convenience than a restriction.

7 986 L. Chaumont et al. / Stochastic Processes and their Applications By taking limits as α ρ or αρ in the given expressions we deduce identities for the case that ξ is spectrally negative and spectrally positive respectively. Note that necessarily in the spectrally one-sided case α, 2. Theorem 2. Fix θ and < v < < u <. P ξ T u + u dθ; T u + < T v sin α ρ e u αρ e v αρ e u+θ αρ+ e u+θ e u αρ e u+θ e v αρ e u+θ dθ and P v ξ Tv dθ; T u + > T v sin αρ e v αρ e u αρ e vθ αρ+ e v e vθ αρ e u e vθ αρ e vθ dθ. Proof. Recall that X, P denotes an α-stable process initiated from and that X, P is an α-stable process conditioned to stay positive initiated from. From Lemma, we have for θ ξ T + u P u + θ; T u + < T v P X σ + [eu, e u+θ ]; σ + e u e u < σ e v e u+θ e u e u+θ e u y + e u αρ P X σ + e u eu dy; σ + e u < σ e v y + e u αρ P e v sin α ρ e u αρ e v αρ e u+θ e u X σ + e u e v dy; σ + e u e v e u e v < σ y + e u αρ y αρ y + e u e v αρ y + e u dy from which the first part of the theorem follows. The second part of the theorem can be proved in a similar way. Indeed for θ P ξ Tv v θ; T u + > T v P X σ e v [evθ, e v ]; σ e + u > σ e v e v e vθ e v y αρ P e v X σ dy; σ + e v e u > σ e v e v e vθ e v y αρ P e v sin αρ e v αρ e u αρ X σ dy; σ + e u e v > σ

8 L. Chaumont et al. / Stochastic Processes and their Applications e v e vθ This completes the proof. e v y αρ y αρ y + e u e v αρ y + e v dy. Note that since the process X, P x, x >, is an α-stable process conditioned to stay positive, it follows that, in the case that there are two-sided jumps, there is no creeping out of the interval v, u with probability one. That is to say, P ξ T + u u; T + u < T v P ξ T v v; T + u > T v. Taking v in the first part of the above theorem and u in the second part we obtain the solution to the one-sided exit problem as follows. Corollary. Fix θ and < v < < u <. P ξ T u + u dθ, T u + < sin α ρ e u αρ e u+θ e u+θ e u αρ e u+θ dθ and P v ξ Tv dθ; T v < sin αρ e v αρ e vθ αρ+ e v e vθ αρ e vθ dθ. To give some credibility to these identities, and for future reference, let us check that we may recover the identity in Caballero and Chaumont [] for the law of the minimum. Corollary 2. Let ξ inf t ξt. For z, P ξ z e z αρ. Proof. The required probability may be identified as equal to PTz and hence, since there is no probability of creeping over the level z, P ξ z sin αρ e z αρ e zθ αρ+ e z e zθ αρ e zθ dθ sin αρ e z αρ e θ αρ+ e θ αρ e z e θ dθ. Next note that the integral on the right-hand side satisfies e θ αρ+ e θ αρ e z e θ dθ e z yy αρ y e z dy

9 988 L. Chaumont et al. / Stochastic Processes and their Applications e z u + u αρ u + e z du { } u αρ u + e z u + u αρ du e z αρ v αρ v + dv du u + uαρ [ e z αρ ] du 3. u + uαρ where in the first equality we have applied the change of variable y e θ, in the second equality y u + and in the fourth equality u e z v. Note also that by writing w u + we also discover that du u + uαρ In conclusion we deduce that w αρ w αρ dw Γ αργ αρ e θ αρ+ e θ αρ e z e θ dθ [ e z αρ ] sin αρ and hence the required identity holds. sin αρ. 3. Finally, to complete this subsection, when X, P is a spectrally negative process we also gain some information concerning the scale function, W,n, of the underlying Lévy process, denoted here by ξ,n. Specifically, in that case it is known that ρ /α and α, 2 and that Pξ,n x mw,n x, where m Eξ,n see for instance Theorem 8. in [9] or identity in [5]. This implies W,n x m ex αρ m ex α. Recall that for a given spectrally negative Lévy process it is known that the Laplace transform of the scale function is given by the inverse of the associated Laplace exponent see for instance Theorem VII.8 in Bertoin [2]. We can therefore compute the Laplace exponent ψ θ log Ee θξ,n for θ, as follows: ψ θ m e θ x e x α dx m u θ u α Γ θ + α du m Γ θγ α. Knowledge of the scale function allow us to write a stronger result than the one given in Corollary as follows. Lemma 2. Let ξ,n inf t s t ξs,n. For v <, θ, φ η and η [, v] we have P v ξ,n Tv dθ, ξ,n,n Tv v dφ, ξ Tv v dη K e v+η α2 e v+η e θφ α e θφ α dθdφdη,

10 where L. Chaumont et al. / Stochastic Processes and their Applications eα2v K αα e v e v y yy α dy ev α αα sin α. Proof. First recall that the process ξ,n drifts towards + a.s. Taking this into account, we have from Example 8 of Doney and Kyprianou [6] that the required probability is proportional to W,n v dη θ φdθdφ. Hence the triple law of interest has a density with respect to dθdφdη which is proportional to e v+η α2 e v+η e θφ α e θφ α. For convenience let us write the constant of proportionality as K. As X, P is derived from a spectrally negative stable process, it cannot creep downwards cf. p75 of Bertoin [2]. This allows us to compute the unknown constant via the total probability formula and after a straightforward computation, we have K v eα2v αα e v and the proof is complete Calculations for ξ e v+η α2 e v+η e θφ α e θφ α dθdφdη e v y yy α dy ev α αα sin α Henceforth we shall assume that X, P x is an α-stable process killed on first exit of the positive half-line starting from x >. As before, unless otherwise stated, we shall assume that there are two-sided jumps, moreover, spectrally one-sided results may be considered as limiting cases of the two-sided jumps case. We start with the two- and one-sided exit problems with the latter as a limiting case of the former. We offer no proof as the calculations are essentially the same. Theorem 3. Fix θ and < v < < u <. P ξ T u + u dθ; T u + < T v sin α ρ e u αρ e v αρ e u+θ e u+θ e u αρ e u+θ e v αρ e u+θ dθ and P v ξ Tv dθ; T u + > T v sin αρ e v αρ e u αρ e vθ e v e vθ αρ e u e vθ αρ e vθ dθ.

11 99 L. Chaumont et al. / Stochastic Processes and their Applications Corollary 3. Fix θ and < v < < u <. and Pξ T + u u dθ, T + u < sin α ρ e u αρ e u+θ αρ e u+θ e u αρ e u+θ dθ P v ξ Tv dθ; T v < sin αρ e v αρ e vθ e v e vθ αρ e vθ dθ. One may also think of computing the distribution of the maximum, ξ, and the minimum, ξ, of ξ in a similar way to the previous section by integrating out u and v in the above corollary. The law of the minimum was already computed in [9,] and we refrain from producing the alternative computations here. For the maximum, an easier approach is at hand. Since ξ is derived from a stable process killed on first exit of the positive half-line one may write P ξ z P exp{ξ } ez P σ e + z > σ P e z σ + > σ. The probability on the right-hand side above may be obtained from Theorem by a straightforward integration. The latter calculation has already been performed however in Rogozin [27] and is equal to Γ α Γ αργ α ρ e z y αρ y αρ dy. Hence, together with the result for the minimum from [9,] which we include for completeness, we have the following corollary. Corollary 4. For z we have that P ξ dz Γ α Γ αργ α ρ ez αρ e z αρ dz and P ξ dz Γ αργ αρ ez αρ dz. In the case that ξ is spectrally one sided, it seems difficult to use the above result to extract information about any underlying scale functions. The reason for this is that the process ξ is exponentially killed at a rate which is intimately linked to its underlying parameters and not at a rate which can be independently varied Calculations for ξ Henceforth we shall assume that X, P x is an α-stable process conditioned to hit zero continuously starting from x >. Again, unless otherwise stated, we shall assume that there are two-sided jumps, and spectrally one-sided results may be considered as limiting cases of the two-sided jumps case. We follow the same programme as the previous two sections dealing with

12 L. Chaumont et al. / Stochastic Processes and their Applications the two- and one-sided exit problems without offering proofs since they follow in a similar spirit to the calculations for ξ and Proposition. Theorem 4. Fix θ. P ξ T u + u dθ; T u + sin α ρ and < T v e u αρ e v αρ e u+θ αρ e u+θ e u αρ e u+θ e v αρ e u+θ dθ P v ξ Tv dθ; T u + > T v sin α ρ e u αρ e v αρ e vθ αρ e v e vθ αρ e u e vθ αρ e vθ dθ. Corollary 5. Fix θ. P ξ T u + and u dθ; T + u < sin α ρ e u αρ e u+θ e u αρ e u+θ dθ P v ξ Tv dθ; T v < sin αρ e v αρ e vθ αρ e v e vθ αρ e vθ dθ. From the above corollary we proceed to obtain the law of the maximum of ξ recalling that it is a process with drift to. Corollary 6. For z P ξ z e z αρ. Proof. Similarly to the calculations in Section 3. we make use of the fact that P ξ z PT z +. Hence P ξ z sin α ρ e z αρ e z+θ e z αρ e z+θ dθ sin α ρ e z αρ e θ αρ+ e θ αρ e z e θ dθ.

13 992 L. Chaumont et al. / Stochastic Processes and their Applications Next note that the integral on the right-hand side above has been seen before in 3. except for the case that ρ is replaced by ρ. We thus obtain from 3. and 3. e θ αρ+ e θ αρ e z e θ dθ [ e z αρ ] sin α ρ and hence P ξ z e z αρ as required. Now, we suppose that X, P has only positive jumps, in which case ρ /α. We denote by ξ,p its underlying Lévy process in this particular case. The associated scale function of ξ,p can be identified as Eξ,p W,p x P ξ,p x e x α W,n xeξ,n, 3.2 where W,n is the scale function of the spectrally negative Lévy process ξ,n. This observation reflects the duality property for positive self-similar Markov processes in this particular case see Section 2 in Bertoin and Yor [4]. More precisely, we have the duality property between the resolvent operators of X, P x, with underlying Lévy process ξ,n, and X, P x, for x >, with underlying Lévy process ξ,p. From the identification of the scale function in 3.2 and Lemma 2, it is possible to give a triple law for the process ξ,p at first passage over the level x >. Lemma 3. Let ξ,p t sup s t ξs,p. For x >, θ, φ η and η [, x] we have P ξ,p T x + x dθ, x ξ,p,p T x + dφ, x ξ T x + dη K e x+η α2 e x+η e θ+φ e θ+φ α dθdφdη, where eα2v K αα e v e v y yy α dy ev α αα sin α. In the remainder of this section, we assume that X, P x is spectrally negative and we denote its underlying Lévy process by ξ,n. The identification of the scale function of the Lévy process ξ,n and Proposition inspire the following result which identifies the scale function of the Lévy process ξ,n. We emphasized that ξ,n is in fact ξ,n conditioned to drift to. Lemma 4. The Laplace exponent of ξ,n satisfies ψ Γ θ + α θ m Γ θ Γ α for θ, and where m Eξ,n. Moreover, its associated scale function may be identified as W,n x m ex α e x.

14 L. Chaumont et al. / Stochastic Processes and their Applications Proof. Recall that the Laplace exponent of ξ,n is given by ψ Γ θ + α θ m Γ θγ α for θ. Hence from Proposition, we get that the Laplace transform of ξ,n, here denoted by ψ, satisfies ψ θ ψ Γ θ + α θ m Γ θ Γ α for θ which is also valid for θ, since any spectrally negative Lévy process has finite exponential moments see for instance [2,28]. Now, recall that the Laplace transform of the scale function of ξ,n, here denoted by W,n, is given by the inverse of ψ. Since, for θ >, we have that and hence Γ θ Γ α Γ θ + α W,n x m ex α e x, u θ u α du e θ x e x e x α dx, as required. One may also write down a triple law for the first passage problem of ξ,n as we have seen before for ξ,n. Lemma 5. For v <, θ, φ η and η [, v] we have P v ξ,n Tv dθ, ξ,n,n Tv v dφ, ξ Tv v dη K e qφη e v+η + α 2 e v+η α2 e θφ α e θφ α dθdφdη, where q > and K α v η e qηφ e vη + α 2 e v+η α2 e φ α dφdη. Proof. First recall that the process ξ,n drifts towards a.s. Again, we have from Example 8 of Doney and Kyprianou [6] that the required probability is proportional to e qφη W,n v dη θ φdθdφ, where q > is the killing rate of the descending ladder height process see for instance Chapter VI in Bertoin [2] for a proper definition of ξ,n. Hence the triple law of interest has a density with respect to dθdφdη which is proportional to e qφη e v+η α2 e v+η + α 2e θφ α e θφ α. As X, P is derived from a spectrally negative stable process, it cannot creep downwards cf. p. 75 of Bertoin [2]. This allows us to compute the unknown constant of proportionality K via the total probability formula and after a straightforward computation, we have

15 994 L. Chaumont et al. / Stochastic Processes and their Applications v K e qφη e v+η α2 e v+η e θφ α e θφ α dθdφdη v e qηφ e vη + α 2 e v+η α2 e φ α dφdη. α η and the proof is complete. 4. Entrance laws for Lévy Lamperti processes: Points In this section we explore the two-point hitting problem for the Lévy Lamperti processes ξ and ξ. There has been little work dedicated to this theme in the past with the paper of Getoor [7] being our principle reference. Henceforth we shall denote by X, P x a symmetric α-stable process issued from x > where α, 2. An important quantity in the forthcoming analysis is the resolvent density of the process X, P x killed on exiting,. The latter is known to have a density ux, ydy dt P x X t dy, t < σ for x, y >. From Blumenthal et al. [7] we know that { dt P x X t dy, t < σ a + σ x y α where sx, y, a 4xy a xa y x y 2 a 2. 2 α Γ α/2 sx,y,a It now follows taking limits as a that 4xy/xy 2 u α/2 ux, y 2 α x y α du. Γ α/2 u + /2 According to the method presented in Getoor [7] one may compute P x X σ{a,b} a; σ {a,b} < σ } u α/2 du dy u + /2 where σ {a,b} inf{t > : X t a or b} and a, b > using the following technique. The two-point hitting probability in Getoor [7] is given by the formula P x X σ{a,b} a; σ {a,b} < σ qx, a qx, x where the {x, a, b} {x, a, b}-matrix Q is defined by Q U and the {x, a, b} {x, a, b}-matrix is given by ux, x ux, a ux, b U ua, x ua, a ua, b. ub, x ub, a ub, b

16 L. Chaumont et al. / Stochastic Processes and their Applications In particular an easy computation shows that P x X σ{a,b} a; σ {a,b} < σ ux,a ub,a ux,b ub,b ua,a ub,a ua,b ub,b. 4.3 Recalling the definitions of ξ and ξ as the Lévy Lamperti processes associated now with our symmetric stable process conditioned to stay positive and conditioned to be killed continuously at the origin respectively we obtain the following result. Theorem 5. Fix α, 2 and < v < < u <. Define T {v,u} inf{t > : ξ t {v, u}} where ξ plays the role of either ξ or ξ. We have P ξ T {v,u} v e v α/2 f, e v, e u and P ξ T {v,u} v e v α/2 f, e v, e u where f x, a, b ux,a ub,a ux,b ub,b ua,a ub,a ua,b ub,b. 5. Exponential functionals of Lévy Lamperti processes We begin this section by recalling a crucial expression for the entrance law at of pssmp s. In [3,4], the authors proved that if a non-arithmetic Lévy process ξ satisfies E ξ < and < Eξ < +, then its corresponding pssmp X, P x in the Lamperti representation converges weakly as x tends to, in the sense of finite-dimensional distributions towards a nondegenerated probability law P. Under these conditions, the entrance law under P is described as follows: for every t > and every measurable function f : R + R +, E f X t where I ξ is the exponential functional: I ξ αeξ E I ξ f t I ξ, 5.4 exp{αξ s } ds. Necessary and sufficient conditions for the weak convergence of X, P x on Skorokhod s space were given in [9]. Recall that X, P x denotes a stable Lévy process conditioned to stay positive as it has been defined in Section 2. One easily checks that ξ satisfies conditions for the weak convergence of X, P x given in [3,4,9]. Note also that in this particular case, the weak convergence of X, P x had been proved in a direct way in [2]. We denote the limiting law by P. We first investigate the tail behaviour of the law of I ξ.

17 996 L. Chaumont et al. / Stochastic Processes and their Applications Theorem 6. The law of I ξ is absolutely continuous with respect to the Lebesgue measure. The density of I ξ is given by: P I ξ dy αeξ yαρ q ydy, 5.5 where q t is the density of the entrance law of the excursion measure of the reflected process X X, P, where X t inf s t X s. Moreover, the law of I ξ behaves as PI ξ x C x α, as x If X has positive jumps, then PI ξ x C 2 x αρ, as x. 5.7 The constants C and C 2 depend only on α and ρ. In the case where the process has no positive jumps, the law of I ξ is given explicitly in the next theorem. Proof. Let n be the measure of the excursions away form of the reflected process X X under P. It is proved in [23] that the entrance law of n is absolutely continuous with respect to the Lebesgue measure. Let us denote by q t its density. Then from [2], the entrance law of X, P is related to q by P X dy y αρ q ydy, y. We readily derive 5.5 from identity 5.4. Moreover from 3.8 in [23]: x q y dy Cx αρ+, as x, and from 3.2 of the same paper, if X has positive jumps, then: x q y dy C x α, as x +. This together with 5.4 imply 5.6 and 5.7. The constants C and C depend only on α and ρ. Another way to prove part i of this theorem is to use a result due to Méjane [22] and Rivero [26] which asserts that for a non-arithmetic Lévy process ξ, if Cramer s condition is satisfied for θ >, i.e. Eexp θξ and Eξ + exp θξ <, then PI ξ x Cx αθ. These arguments and Proposition allow us to obtain the asymptotic behaviour at + of PI ξ x: Proposition 2. The law of I ξ behaves as PI ξ x C 3 x α, as x The constant C 3 depends only on α and ρ. Now we consider the exponential functional I ξ exp{αξ s } ds.

18 L. Chaumont et al. / Stochastic Processes and their Applications Recall from Section 2 that ξ is the Lévy process which is associated to the pssmp X t {t<t }, P x by the Lamperti representation. From this representation, we may check path by path the equality x α I ξ T. Moreover, it follows from Lemma in [] that when X, P x has negative jumps, P x T t c t αx α, as t tends to. This result leads to: Proposition 3. Suppose that ξ has negative jumps, then the law of I ξ behaves as PI ξ x c x, as x. 5.9 α In the remainder of this section, we assume that X, P x has no positive jumps and recall that m Eξ,n. Theorem 7. The law of exponential functional I ξ,n exp{αξs,n } ds is absolutely continuous with respect to the Lebesgue measure and has a continuous density p,n which has the following representation by power series p,n x c Γ + n n x /α n sin, for x >, x α α n! n where c c Γ 2 αα α >. Moreover the positive entire moments of X, P, for t >, are given by the identity E X t k k Γ αk + mt Γ α k+, k, 5.2 k! and its law is absolutely continuous with respect to the Lebesgue measure and has a continuous density p t which has the following representation by power series p t x t/α α Γ + n n x /α n sin, for x >. Γ 2 αc m α α n! n Proof. From Bertoin and Yor [5], we know that the distribution of the exponential functional of a spectrally negative Lévy process is determined by its negative entire moments. In particular, the exponential functional for ξ,n satisfies E I ξ,n k αm k Γ kα Γ α k k!, 5.2 with the convention that the right-hand side equals αm for k. In particular, from 5.4 we have that E X t k k Γ αk + mt Γ α k+ k!, which proves the identity 5.2. Now, from the time reversal property of Theorem VII.8 in [2], we deduce that the last passage time of X, P, defined by U x sup {t : X t x} for x,

19 998 L. Chaumont et al. / Stochastic Processes and their Applications is a stable subordinator of index /α. More precisely, its Laplace exponent is given by Φλ λ/c /α, where c c Γ 2αα α. According to Zolotarev [3], stable subordinators have continuous densities with respect to the Lebesgue measure which may be represented by power series. More precisely, the density of a normalized stable subordinator of index β,, i.e. Φλ λ β, is given by ρ t x, β t/β x Γ + nβ sinβn xβ n, for x > n! n On the other hand from Proposition in [3], we know that U x has the same law as x α I ξ,n. Hence, I ξ,n satisfies that { } E exp λi ξ,n e λ/c/α, λ, and its density p,n x is given by ρ c /αx, /α, which proves the first part of the theorem. Finally from 5.4, we deduce that the density of X is given by p x c αm Γ n The proof is now complete. + n sin α n α x /α n, for x >. n! Theorem 8. The exponential functional I ξ exp{αξs } ds has a continuous density p with respect to the Lebesgue measure which has the following representation by power series p x c /α αn Γ αnγ n + + /α xα2nα, for x >, n where c c + Γ 2 αα α >. Proof. First, let us define ˆX X and denote by ˆP for its law starting from. Note that the process X, ˆP is a stable Lévy process with no negative jumps of index α, 2 starting from. From the Lamperti representation, it is clear that T I ξ and from the self-similar property, we have P T > t P σ > t P inf X s > s t P t /α inf X s > ˆP t /α sup X s < s s ˆP sup s X s α > t. Hence the exponential functional I ξ, under P, has the same law as sup s X α, under ˆP. Recently, Bernyk, Dalang and Peskir [] computed the density of the supremum of a stable Lévy process with no negative jumps of index α, 2. More precisely with our notation, the

20 L. Chaumont et al. / Stochastic Processes and their Applications density f of sup s X, under ˆP, is described as follows f x c /α n Therefore the density p of I is given by p x c /α n which completes the proof. αn Γ αnγ n + + /α xnα2, for x >. αn Γ αnγ n + + /α xα2nα, for x >, Acknowledgement The research was supported by EPSRC grant EP/D4546/. References [] V. Bernyk, R.C. Dalang, G. Peskir, The law of the supremum of a stable Lévy process with no negative jumps, Ann. Probab. 28 in press. [2] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 996. [3] J. Bertoin, M.E. Caballero, Entrance from + for increasing semi-stable Markov processes, Bernoulli [4] J. Bertoin, M. Yor, The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes, Potential Anal [5] J. Bertoin, M. Yor, On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes, Ann. Fac. Sci. Toulouse VI Ser. Math [6] J. Bertoin, M. Yor, Exponential functionals of Lévy processes, Probab. Surv [7] R. Blumenthal, R.K. Getoor, D.B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc [8] S.I. Boyarchenko, S.Z. Levendorskii, Non-Gaussian Merton Black Scholes Theory, World Scientific, Singapore, 22. [9] M.E. Caballero, L. Chaumont, Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes, Ann. Probab [] M.E. Caballero, L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab [] P. Carr, H. Geman, D.B. Madan, M. Yor, The fine structure of asset returns: An empirical investigation, J. Business [2] L. Chaumont, Conditionings and path decompositions for Lévy processes, Stochastic Process. Appl [3] L. Chaumont, J.C. Pardo, The lower envelope of positive self-similar Markov processes, Electron J. Probab [4] R. Cont, P. Tankov, Financial Modeling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL, 24. [5] R.A. Doney, Fluctuation theory for Lévy processes, in: Ecole d été de Probabilités de Saint-Flour, in: Lecture Notes in Mathematics, vol. 897, Springer, 27. [6] R.A. Doney, A.E. Kyprianou, Overshoots and undershoots of Lévy processes, Ann. Appl. Probab [7] R.K. Getoor, Continuous additive functionals of a Markov process with applications to processes with independent increments, J. Math. Anal. Appl [8] S.G. Kou, H. Wang, First passage times of a jump diffusion process, Adv. in Appl. Probab [9] A.E. Kyprianou, Introductory Lectures of Fluctuations of Lévy Processes with Applications, Springer, 26. [2] J.W. Lamperti, Semi-stable Markov processes, Z. Wahrscheinlichkeitstheor. Verwandte Geb [2] A. Lewis, E. Mordecki, Wiener-Hopf factorization for Lévy processes having negative jumps with rational transforms. 25 preprint.

21 L. Chaumont et al. / Stochastic Processes and their Applications [22] O. Méjane, Upper bound of a volume exponent for directed polymers in a random environment, Ann. Inst. H. Poincaré Probab. Statist [23] D. Monrad, M.L. Silverstein, Stable processes: Sample function growth at a local minimum, Z. Wahrscheinlichkeitstheor. Verwandte Geb [24] M.R. Pistorius, On maxima and ladder processes for a dense class of Lévy processes, J. Appl. Probab [25] V. Rivero, Recouvrements aléatoires et processus de Markov auto-similaires. Thèse de doctorat de l université Paris VI, 24. [26] V. Rivero, Recurrent extensions of self-similar Markov processes and Cramér s condition, Bernoulli [27] B.A. Rogozin, The distribution of the first hit for stable and asymptotically stable random walks on an interval, Theory Probab. Appl [28] K.I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 999. [29] W. Schoutens, Lévy Processes in Finance. Pricing Finance Derivatives, Wiley, New York, 23. [3] V.M. Zolotarev, One-Dimensional Stable Distributions, in: Translations of Mathematical Monographs, vol. 65, American Mathematical Society, Providence, RI, 986.