D Gruyer
email: diego.gruyer@fi.infn.it

E Bonnet

A Chbihi

J D Frankland

S Barlini

B Borderie

R Bougault

J A Dueñas

E Galichet

A Kordyasz

T Kozik

N Le Neindre

O Lopez

M Pârlog

G Pastore

S Piantelli

S Valdré

G Verde

E Vient

Keywords: Silicon Detector, Computer Data Analysis, Charged Particle Identification

New semi-automatic method for reaction product charge and mass identification in heavy-ion collisions at Fermi energies

New semi-automatic method for reaction product charge and mass identification in heavy-ion collisions at Fermi energies 1 Introduction

In the intermediate energy regime, violent heavy-ion collisions produce many nuclear species with a large range of charge (Z), mass (A), and kinetic energy (E k ) [1,2]. Studying this kind of reactions requires detectors with almost 4π solid angle coverage, high granularity, low energy thresholds, large dynamic range in energy and capable of characterizing reaction products on an event by event basis. The first generation of 4π multi-detectors focused on complete collection of charged particles produced in a reaction [3,4,5,6,7], providing little isotopic information for heavy fragments (Z > 5). More recently detectors have evolved to provide isotopic resolution for a broader range of products [8,9,10], by improving existing detectors and identification techniques, or developing new methods such as the Pulse Shape Analysis (PSA) in silicon detectors [11,12,13].

Such multi-detectors are generally made of telescopes (stacks of detector material layers) measuring the energy lost by charged particles in the different stages. Several combinations of detectors have been used for this purpose, such as ionization chambers (IC), silicon detectors (Si), plastic scintillators, and thallium-activated cesium-iodide scintillators (CsI(Tl)). When a charged particle passes through such a telescope, its charge, mass, and kinetic energy determine the number of detectors it can cross before stopping, and the energy loss in the different layers. Charged particles are then identified by plotting the energy loss in one or several layers of the telescope (∆E) versus the residual energy released in the detector in which the particle is stopped (E). Within this representation, called ∆E -E matrix, different particles populate identification lines characteristic of their charge and mass (see for example Fig.

1(a)).

Three main methods are then used to identify such particles:

(i) Interactive drawing of lines in order to discriminate between the ridges corresponding to a given charge and/or mass. Particles are then identified from their relative distance between pairs of ridge lines. (ii) Fit of a limited set of ridge lines with a functional describing the relation between ∆E and E, in which Z and A enter as parameters [14,15,16,17]. In this case, particle identification is obtained by inversion of the functional for given ∆E and E, in order to extract Z and possibly A. (iii) Fit of detector calibration parameters on a limited set of ridge lines using energy loss table to generate all identification lines [20,27,10,21].

Particles are then identified from their relative distance between pairs of ridge lines.

The method (i) is probably more powerful and allows to face any situation, such as complex detector responses, but it suffers from two main limitations: it does not provide any extrapolation in regions of low statistics, and it is time consuming because each line has to be accurately drawn.

Methods (ii) and (iii) allow extrapolation and suffer less from this latter inconvenience, since only a subset of ridge lines have to be drawn by hand. It may still become problematic when using multidetectors composed of thousands of identification telescopes. In both cases, identification performances rely on the precise knowledge of the detector response, which can be very complex in the case of CsI(Tl) scintillators light output-energy conversion [18] or in presence of pulse-height defect in silicon detectors [19]. These two methods are very efficient in reproducing Z lines but are generally not accurate enough to reproduce isotopic lines over a large range of elements.

With increasing numbers of identification matrices to treat which include information on increasing numbers of individual isotopes of different elements, it becomes essential to develop automatic or semi-automatic methods to extract identification lines in ∆E -E matrices. The need for automation was already evident with the advent of the first large charged particle arrays, and some methods were developed at that time, using for example image processing [22,25,26], or artificial neural networks [23]. More recently, neural networks have been applied directly to digital current signal [13] for isotopic identification. Cluster algorithms have also been used for proton-γ discrimination [24].

The evolution of computer resources, and the availability of powerful libraries dedicated to large scale data analysis [28,29] allow us to consider new types of algorithms. In this article we present a new method, called SPIDER1 identification (for Spider Particle Identification in ∆E -E Representation) for semiautomatic ridge line determination in two-dimensional matrices. This method has been developed avoiding as much as possible the use of a priori information on the exact form of identification lines, in order to be applicable to a large variety of identification matrices. Particular attention has been payed to the implementation in a suitable graphical environment. The extracted lines can then be directly used to identify charged particles (i), or set as an input of a functional fit (ii) or energy loss fit (iii).

SPIDER identification

Algorithm

Determining ridge lines in two-dimensional matrices (x, y) is a hard task, whereas powerful algorithms for peak localization in N -dimensional matrices are available [30]. The main idea of the present method is then to transform our problem into a problem of peak localization in one-dimensional histograms.

To do so, we have to project a part of the matrix onto a relevant axis. It is the shape of the identification lines and their relative population that guided the choice of this projection. The one-dimensional histogram shown in Fig. 1(b) is obtained by projecting all points between D(θ -α/2) and D(θ + α/2) onto the straight line D(θ), passing through (x 0 , y 0 ) and making an angle θ with respect to Ox (see Fig. 1(a) where x stands for E and y for ∆E). Each peak appearing on this projection corresponds to the intersection between D(θ) and a ridge line of a given Z and A2 . The angle of the first projection, θ 0 , and the pedestal coordinates (x 0 , y 0 ) are input parameters of the algorithm.

To localize the peaks in Fig. 1(b), the binning of the histogram should be chosen carefully. The number of bins n b of the projection is defined as:

n b = d θ × ρ(θ) × β, (1)
with d θ the length of the projection, β a binning parameter to be provided by the user, and ρ(θ) an internal parameter given by:

ρ(θ) =              √ 2 if θ > θ 0 1 if θ = θ 0 5/4 if θ < θ 0 . (2)
The choice of ρ(θ) is purely phenomenological.

Maxima are then located using the algorithm described in [30], and their position in the two-dimensional matrix (Fig. 1(c)) is used as starting point for all subsequently generated identification lines, making crucial the choice of θ 0 .

Linear abscissa [a. The operation of projection/localization is then repeated in order to cover the full matrix, varying θ from θ 0 to 90°, and then from θ 0 to 0°by steps of δθ. In practice, δθ and α slightly depend on θ, and can be modified by the user in order to adapt the algorithm to a specific situation (very low statistics for example). Each new point P (x p , y p ) is associated to the line Z, so far containing n Z points and whose end point coordinates are (x Z , y Z ), if:

     |y p -y Z | < δy, for n Z < 10, |y p -f Z (x p )| < δy, for n Z ≥ 10, (3)
with f Z (x) = a 0 Z × (x + a 1 Z ) -a 2
Z a function fitted to the n Z points already associated to the line Z, and δy = y Z × Z -1 . The choices of f Z (x) and δy are purely phenomenological.

Once the spiderweb is woven (see Fig. 1(d)), identification lines that do not fulfill the following criteria:

             n Z > 10 a 1 Z < 3000 0.35 < a 2 Z < 1 (4)
are rejected, where n Z is the final number of points associated to the line Z, and (a 0 Z , a 1 Z , a 2 Z ) are the parameters of f Z (x). This procedure aims at eliminating lines with a form completely incoherent with the Bethe-Bloch formula, without being too restrictive in order for this method to be applicable to different types of identification matrices.

It is then possible to build the identification grid from each of the individual functions f Z (x), either limited to the range where peaks were found (see Fig. 2 for example), or extrapolated over the whole residual energy range (see Fig. 4(a) for example).

Implementation

In order to run the algorithm presented above, several input parameters should be provided by the user: the pedestal coordinates (x 0 , y 0 ), the first projection angle θ 0 , and the binning parameter β. These parameters are generally difficult to estimate, which makes our method unusable without an implementation in a suitable graphical environment. It has therefore been included in the identification grid editor of KaliVeda [29], which is a graphical user interface dedicated to the creation and editing of identification grids developed initially within the INDRA Collaboration [4].

The pedestal coordinates (x 0 , y 0 ) can be set by the user with a simple click on the ∆E -E matrix. Our algorithm needs also another point, (X 0 , Y 0 ), situated approximately in the middle of a high-Z line; and the knowledge of the charge Z 0 associated to this line. The values of θ 0 and β are then calculated as follows :

tan

(θ 0 ) = Y 0 -y 0 X 0 -x 0 (5)
β = 1 20Z 0 (X 0 -x 0 ) 2 + (Y 0 -y 0 ) 2 . ( 6
)
Thanks to the implementation in a "user friendly" graphical environment, our method needs only two mouse-clicks from the user to calculate all initialization parameters, making it very easy to use.

Examples of use

The present method has been initially developed to treat INDRA Si-CsI(Tl) matrices [4]. Since it uses as little information as possible on the exact form of Z lines, it can be applied to different types of identification telescopes.

Here are several examples of use on Si-CsI(Tl), IC-Si, Si-Si matrices; and also matrices from Pulse Shape Analysis of the charge signal in silicon detectors. Fig. 2 presents the result of the SPIDER identification on a ∆E -E matrix coming from a Si-CsI(Tl) telescope of the INDRA multidetector. In this example, Z lines up to Z = 29 were found by our algorithm. Since extrapolation using individual functions is not safe in the case of Si-CsI(Tl) matrix, because of the strong curvature of Z lines at low residual energy (E), identification lines have been generated on the range where points were found by the algorithm. The obtained identification grid cannot be used directly. It has to be completed by hand, or can be used as input to constrain a fit using some functional. This point is discussed in Section 2.4. generally broad due to the poorer energy resolution of such large-area ionization chambers operated at low pressure, and rarely homogeneously populated. Nevertheless, our algorithm has extracted ridge lines from Z = 2 to Z = 16 (Fig. 3(a)), providing a good charge identification (Fig. 3(b)). It can be noted that, in this example, the Z = 4 line was not generated because it did not satisfy the criteria of Eq. ( 4).

(in channel) E

The result of the SPIDER identification on a ∆E -E matrix coming from a Si-Si telescope of FAZIA [9] is presented in Fig. 4. In this kind of matrix, ridge lines corresponding to different elements are clearly separated thanks to the very good quality of FAZIA silicon detectors. In addition, the line curvature is quite slight, making the SPIDER identification very efficient (Fig. 4(a)). Even Fig. 4. Same as Fig. 3 for Si-Si matrix of FAZIA (θ ∼ 7°). Data come from the Kr+Sn at 35 MeV/A reaction measured at LNS [31,32].

if a few lines (Z = 7 and Z = 9) have not been generated, the grid provides a very good charge identification, up to Z = 34 (Fig. 4(b)). For each integer value of Z, several peaks appear on the identification spectrum up to Z ∼ 20. These peaks correspond to different isotopes of each element, and allow to discriminate particles of different masses. The charge-identification grid can then be used as a starting point to generate a mass-identification grid. This point is discussed in sec.

Previous examples all concern the ∆E -E method. Identification matrices obtained by Pulse Shape Analysis of the charge signal in FAZIA silicon detectors [9] present a form quite similar to that obtained with the ∆E -E method. Since the SPIDER identification method uses little a priori information on the ridge line form, it can also be applied to this type of matrix. In the example shown in Fig. 5, the generated grid provides a satisfying charge 3 for energy versus rise time of the charge signal in the first silicon layer matrix of FAZIA (θ ∼ 7°). Data come from the Kr+Sn at 35 MeV/A reaction measured at LNS [31,32].

identification from Z = 3 up to Z = 30 (Fig. 5(b)), even if identification lines do not cover the full matrix range (Fig. 5(a)).

Coupling with a fitting procedure

We have shown in previous examples that Z-lines generated by the SPIDER identification, possibly extrapolated using individual functions, can be directly used to identify particles. This procedure is efficient but does not allow to extrapolate the identification to higher Z. These lines, without extrapolation, can also be used as input to fit functional parameters. In the present example, we used the functional proposed in [16]. Fig. 6. Example of coupling between the SPIDER identification method and a functional fit [16]. The identification grid used as input is shown in Fig. 2.

Raw Z-lines obtained with the SPIDER method on an INDRA Si-CsI(Tl) matrix are those previously presented in Fig. 2. These lines, which do not cover the whole residual energy range, are used to fit the 9 parameters of the functional [16]. The result of the fit is shown in Fig. 6. It can be seen that satisfactory agreement is obtained for all charges and over the full matrix. The quality of the charge identification can be checked in Fig. 6: a good charge identification is achieved up to Z ∼ 36, even if the statistics for high Z is very poor. The coupling between the SPIDER method and a functional fit allows to obtain a full charge identification in a very short time. This procedure was used during the data reduction of the INDRA experiment presented in [33].

Isotopic identification

The method presented above, and its possible coupling with a fitting procedure, facilitate a lot the extraction of Z-identification lines. As illustrated in Fig. 4(b), the good quality of FAZIA silicon detectors allows also isotopic identification of charged particles up to Z ∼ 25. In order to carry out an isotopic identification, ridge lines corresponding to each (Z, A) couple have to be drawn. This is again a very fastidious task. We propose here a method to extract these lines in a fully automatic way. The only input of the algorithm is a charge-identification grid that can be easily generated using the SPIDER identification method, which is particularly efficient for the case of Si-Si matrices (see Fig. 4).

In ∆E -E matrices, lines corresponding to a given atomic number Z are regularly spaced and populated. The scheme for isotopic lines is much more complex: the relative population of each isotope depends strongly on the considered element, on the studied reaction, and isotopes with short lifetimes (typically lower than 1 ns) are never detected. The extraction of isotopic (Z, A)lines from the raw ∆E -E matrix is therefore very complex. To simplify the treatment, we have to transform the matrix in order to extract masses Z by Z. The first stage of the algorithm is then to linearize the ∆E -E matrix according to the Z-identification grid provided by the user, in order to obtain a PID-E matrix (Fig. 7). If the quality of the grid is good enough, A-lines associated to a given Z are almost horizontal and lie in the PID range [Z-0.5, Z+0.5].

All points contained in the range PID = Z ± 0.5 and E = E i ± δE (rectangular box in Fig. 7) are projected on the PID axis. Peaks corresponding to the intersection of A-lines and the vertical line E = E i appear in the projection histogram. The binning of this projection is set to 60 bins whatever the Z considered. Peaks are located and replaced in the PID-E matrix. Each new point is simply associated to the closest A-line. The operation is then repeated by varying E i in order to cover the whole residual energy range, and for all Z (Fig. 7). The obtained identification grid in the PID-E plane is finally transformed in the ∆E -E plane (Fig. 8). The corresponding isotopic identification matrix is presented in Fig. 8.

This algorithm has also been implemented in the identification grid editor of KaliVeda [29].

Conclusion

In this article, we proposed a new method (SPIDER identification) for generating Z and A identification grids in two-dimensional matrices. This method has been developed avoiding as much as possible the use of a priori information on the exact form of identification lines, in order to be applicable to a large variety of identification matrices. It has been successfully tested on various types of matrices obtained with ∆E -E and Pulse Shape Analysis techniques. Particular attention has been paid to the implementation in a suitable graphical environment, so it needs only two mouse-clicks from the user in order to calculate all initialization parameters. Extracted lines can then be directly used to identify charged particles, set as an input of a functional fit, or used to extract isotopic lines in a fully automatic way.

Fig. 1 .

 1 Fig. 1. Illustration of the different steps of the SPIDER identification: (a) definition of the projection; (b) projection on line D(θ) and localization of maxima; (c) positioning of the maxima on the ∆E -E matrix; (d) weaving of the spiderweb.

Fig. 2 .

 2 Fig.2. Application of the SPIDER identification method on a Si-CsI(Tl) matrix of INDRA (θ ∼ 17°). Data come from the Ta+Zn at 39 MeV/A reaction measured at GANIL.

Fig. 3 Fig. 3 .

 33 Fig. 3 presents the results of the SPIDER identification on a ∆E -E matrix coming from an INDRA IC-Si telescope. In this kind of matrix, Z lines are

Fig. 5 .

 5 Fig.5. Same as Fig.3for energy versus rise time of the charge signal in the first silicon layer matrix of FAZIA (θ ∼ 7°). Data come from the Kr+Sn at 35 MeV/A reaction measured at LNS[31,32].

Fig. 7 .

 7 Fig. 7. Illustration of the isotopic identification method in PID-E matrices.

Fig. 8 .

 8 Fig. 8. Result of the automatic isotopic line extraction applied to a Si-Si matrix of FAZIA (θ ∼ 7°) in the Ar+Sn at 25 MeV/A reaction: isotopic identification grid, isotopic and neutron number distribution after linearization.

not to be confused with the "SpiderWeb" surface construction algorithm

The ridge lines of individual isotopes are indistinguishable in these data, due to insufficient resolution of the ∆E detectors. In this case ions of different Z populate broad ridges around the mean value < A > of their isotopic distribution.

Acknowledgements

The authors express their acknowledgement to the GANIL and LNS accelerator staff for their continued support during the experiments. We would especially like to thank the INDRA and FAZIA collaborations for allowing us to use their unpublished data. One of us (J.D.F.) would like to thank L. Tassan-Got for many fruitful discussions and for providing his code which is used in KaliVeda both for linearization of the identification grids and for fitting to his identification functional(s). This work was supported by the Polish National Science Centre under Contract No. UMO-2014/14/M/ST2/00738 (COPIN-INFN Collaboration) and Contract No. UMO-2013/08/M/ST2/0025 (LEA-COPIGAL).


Bernard Roynette
email: roynette@iecn.u-nancy.frvallois@iecn.u-nancy.frvolpi@esstin.uhp-nancy.fr

Pierre P Vallois

Agnès Volpi

Asymptotic Behavior of the Hitting Time, Overshoot and Undershoot for some Lévy Processes

Keywords: Lévy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation AMS 2000 Subject classification: 60E10, 60F05, 60G17, 60G40, 60G51, 60J65, 60J75, 60J80, 60K05

published or not. The documents may come

Asymptotic behavior of the hitting time, overshoot and

undershoot for some Lévy processes

Introduction

1. Let (X t , t ≥ 0) be a Lévy process, which is right continuous with left limits and starts at 0. Let X t = σB tc 0 t + J t t ≥ 0 , (0.1) be the canonical decomposition, when c 0 ∈ R, σ > 0 and (B t , t ≥ 0) is a onedimensional Brownian motion started at 0. (J t , t ≥ 0) is a pure jump Lévy process which is independent from (B t , t ≥ 0). In addition J 0 = 0. Recall (see for instance Theorem 2.1, Chap. 2, [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF]) that (J t , t ≥ 0) is the sum of a compound Poisson process and a square integrable martingale whose jumps are of magnitude less than 1.

For simplicity, we may assume that σ = 1.

2. We are interested in the first hitting time of level x > 0

T x := inf {t ≥ 0; X t > x} . (0.2)
We also consider the overshoot K x and the undershoot L x :

K x := X Tx -x , (0.3)
L x := x -X T x -. (0.4)
The aim of this paper is to study the joint distribution of the triple (T x , K x , L x ).

3. In the usual theory of risk in continuous time the surplus of an insurance company is modelled by a stochastic process (Z t , t ≥ 0). The real number x = Z 0 denotes the initial surplus, and the random time T x := inf {t ≥ 0; Z t < 0} may be interpreted as the ruin time. Historically, the first model (called classical or the Cramér-Lundberg one) was initiated by F. Lundberg [START_REF] Lundberg | I-Approximerad Framställning av Sannolikhetsfunktionen. II-Aterförsäkering av Kollectivrisker[END_REF] and H. Cramér [START_REF] Cramér | On the mathematical Theory of Risk[END_REF], [START_REF] Cramér | Collective risk theory: A survey of the theory from the point of view of the theory of stochastic processes[END_REF]. It refers to the case when (Z t , t ≥ 0) is the sum of a drift and a compound Poisson process. The later represents the aggregate claims. In [START_REF] Dufresne | Risk theory for the compound Poisson process that is perturbed by diffusion[END_REF], Dufresne and Gerber have added a Brownian perturbation in the surplus process. Thus, the process (x -Z t , t ≥ 0) corresponds to a particular case of our process (X t , t ≥ 0). A lot of authors have developed extensions and have considered a great variety of processes (Z t , t ≥ 0). They have mainly focused on the choice of more adapted processes (Z t , t ≥ 0) to take into account the complex reality. It is not our purpose to present here all these developments. For more information one should refer to Rolski, Schmidli, Schmidt and J. Teugels' book [START_REF] Rolski | Stochastic processes for insurance and finance[END_REF], in which a large panel of models can be found.

Here are a few papers which are closely connected to the limit distribution of (T x , K x , L x ), as x → ∞.

(a) J. Bertoin and R. Doney in [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF] proved that the ruin probability P(T x < ∞) is equivalent to Ce -αx , as x → ∞. The authors has given in their paper an expression of the constant C with the ascending ladder height process associated with (X t , t ≥ 0). When the Lévy process (X t , t ≥ 0) has no negative jumps, then C and α may be calculated explicitly. (b) In the discrete time model, i.e. when (X t , t ≥ 0) is replaced by a random walk, A. Gut [START_REF] Gut | Stopped random walks, volume 5 of Applied Probability[END_REF] has considered the limit distribution of normalized passage times.

(c) In both [START_REF] Griffin | On the rate of growth of the overshoot and the maximum partial sum[END_REF] and [START_REF] Doney | Stability of the overshoot for Lévy processes[END_REF] analytical conditions are given to ensure that the ratio X Tx /x almost surely converges to 1, as x → ∞, in the discrete time model as well as in the case of Lévy processes. (d) R. Doney and A. Kyprianou [START_REF] Doney | Overshoots and undershoots of Lévy processes[END_REF] have showed that (K x , L x ) converges in the distribution sense as x goes to infinity.

Our study presents the following features :

(a) original analytic arguments of complex analysis are used, and especially meromorphic and holomorphic functions (see Theorem 2.8); (b) a decorrelation phenomenon : the couple (K x , L x ) and a relevant normalization of T x become asymptotically independent, (x → ∞) (cf Theorems 2.1 and 2.3); (c) new functional equations (cf Theorems 2.4 and 2.5).

4. Let us briefly describe the organization of the paper. In section 1, we will set up notation and assumptions. In Section 2, we will suppose that the Lévy measure ν of (X t , t ≥ 0) satisfies the condition (H), which is defined in item 1.4 of Section 1.

In Section 2 we will list the main results of the paper. The two major theorems (cf Theorems 2.1 and 2.3) are related to the convergence in distribution of the triple ( T x , K x , L x ) with x → ∞. In addition T x is expressed in terms of T x and x and it depends on the sign of E(X 1 ). In subsection 2.2 we present the important theorems which permit to demonstrate Theorems 2.1 and 2.3. Our approach is based on the study of the Laplace transform F of (T x , K x , L x ) :

F (θ, µ, ρ, x) := E e -θTx-µKx-ρLx 1l {T x <∞} , θ, µ, ρ ≥ 0. (0.5)

When ν(R) is finite, it is shown in Theorem 2.4 that F (θ, µ, ρ, •) satisfies some kind of integral equation. Introducing the Laplace transform F (θ, µ, ρ, •) of F (θ, µ, ρ, •) :

F (θ, µ, ρ, q) := +∞ 0 e -qy F (θ, µ, ρ, y)dy , (0.6) we proved in Theorem 2.5 that, under (H), the function F (θ, µ, ρ, •) solves an equation which looks like an integral equation. Then an asymptotic development of F (θ, µ, ρ, x) ( x → ∞) with a finite number of terms like C(θ, µ, ρ)e -α(θ)x is given in Theorem 2.8. In the third subsection 2.3 we will determine the behavior of the ruin probability P(T x < ∞), when x runs to infinity. Theorem 2.10 asserts that the ruin probability has a polynomial rate of decay at ∞ as soon as ν admits polynomial moments. All the proofs of results stated in Section 2 are postponed in Section 3.

Finally, in the last Section 4 we will give some complements and comments.

1 Characteristic exponent and elementary properties 1.1 Let ψ be the characteristic exponent of a Lévy process (X t , t ≥ 0) with canonical decomposition (0.1), i.e. E(e iqX t ) = e tψ(q) (q ∈ R). With the Lévy-Khintchine formula, we get :

ψ(q) = - q 2 2 -icq + R (e iqy -1 -iqy1l {|y|<1} )ν(dy), q ∈ R. (1.7)
where ν is the Lévy measure which satisfies

R y 2 ∧ 1 ν(dy) < ∞.
It is well known (cf [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], example 25.12) that as soon as

R |y|1l {|y|≥1} ν(dy) < ∞ then E[|X 1 |] < ∞, (1.8)
and

E(X 1 ) = -iψ ′ (0) = -c + R y1l {|y|≥1} ν(dy).
(1.9)

When (J t , t ≥ 0) is a compound Poisson process, there exists a relation between the drift term c 0 in (0.1) and c :

c = c 0 - R y1l {|y|<1} ν(dy).
(1.10)

1.2

The following assumptions will be needed throughout the paper : [START_REF] Gut | Stopped random walks, volume 5 of Applied Probability[END_REF] and ∞ 1 e sy ν(dy) < ∞, for some s > 0.

-1 -∞ e sy ν(dy) < ∞, ∀s ∈] -∞, 0[ (1.
(1.12)

It is convenient to introduce :

r ν := sup s ≥ 0; ∞ 1 e sy ν(dy) < ∞ . (1.13) It is clear that (1.12) implies that r ν ∈]0, ∞].
Under (1.11) and (1.12), the function ψ may be extended to the half-space {z ∈ C; Im(z) > -r ν }. Let ϕ denote the function : ϕ(q) := ψ(iq), q > -r ν . If we consider the definition of ψ and the identity (1.7), we can infer that :

E(e -qX t ) = e tϕ(q) , (1.14)
and

ϕ(q) = q 2 2 + cq + R (e -qy -1 + qy1l {|y|<1} )ν(dy), (1.15)
for any q ∈]r ν , ∞[. Note that :

ϕ ′ (0) = -E(X 1 ). (1.16)
Assumptions (1.11) and (1.12) imply that ϕ is a function defined on ]r ν , ∞[ and :

ϕ ′′ (q) = 1 + R y 2 e -qy ν(dy), for any q ∈]r ν , ∞[.

Consequently, ϕ is convex on ] -r ν , ∞[. It can be easily shown that ϕ(∞) := lim x→∞ ϕ(x) = ∞. Let us discuss the behavior of ϕ in the vicinity of -r ν . 1. When r ν = ∞ then ϕ(-r ν ) := lim q→∞ ϕ(q) = ∞. 2. When r ν < ∞ and ∞ 1 e rν y ν(dy) = ∞ then ϕ(-r ν ) := lim q→-rν ϕ(q) = ∞.
3. In the case :

r ν < ∞ and ∞ 1 e rν y ν(dy) < ∞, (1.17)
then ϕ(-r ν ) is a real number and :

ϕ(-r ν ) = r 2 ν 2 -cr ν + R (e rν y -1 -r ν y1l {|y|<1} )ν(dy). (1.18)
Let c ν be the real number defined as :

r 2 ν 2 -c ν r ν + R (e rν y -1 -r ν y1l {|y|<1} )ν(dy) = 0. (1.19) Hence : ϕ(-r ν ) > 0 ⇔ c < c ν . (1.20)
In the rest of the paper we assume

ϕ(-r ν ) ∈ ]0, ∞] (1.21)
holds in any case.

1.3

In the sequel of the paper it will be convenient to deal with the parameter E(X 1 ). Note that this expectation may be expressed in terms of c and ν, via (1.9). The zeros of ϕ θ (.

) := ϕ(.) -θ (1.22)
are important parameters of our study. We can plot two graphs :

1. one represents ϕ and corresponds respectively to the three cases : E(X 1 ) < 0, E(X 1 ) > 0 and E(X 1 ) = 0;

2. another which represents ϕ θ .

(see the Figures 1 and2 for illustration).

-γ 0 (0)

u -r ν ϕ(u) 0 = γ * 0 (0) a) E(X 1 ) = -ϕ ′ (0) < 0 -r ν -γ 0 (0) = 0 u γ * 0 (0) ϕ(u) b) E(X 1 ) = -ϕ ′ (0) > 0 -r ν u ϕ(u) -γ 0 (0) = 0 = γ * 0 (0) c) E(X 1 ) = -ϕ ′ (0) = 0 Figure 1: Graph of ϕ 0 γ * 0 (θ) -γ 0 (θ) -θ u -r ν ϕ θ (u) Figure 2: Graph of ϕ θ
From the Figures 1 and2, we can easily infer the existence of κ > 0 so that :

1. there is a unique γ * 0 (θ) ≥ 0 which satisfies :

ϕ γ * 0 (θ) = θ, ∀ θ ∈ [0, κ] (1.23) and γ * 0 (θ) > 0 if θ > 0, or θ = 0 and E(X 1 ) > 0 = 0 if θ = 0 and E(X 1 ) ≤ 0. (1.24)
2. there is a unique γ 0 (θ) ≥ 0 which satisfies :

ϕ -γ 0 (θ) = θ, ∀ θ ∈ [0, κ] (1.25)
and

-γ 0 (θ) < 0 if θ > 0, or θ = 0 and E(X 1 ) < 0 = 0 if θ = 0 and E(X 1 ) ≥ 0. (1.26)
So, when θ > 0 is rather small, the positive (resp. negative) zero of ϕ θ is γ * 0 (θ) (resp. -γ 0 (θ)). 1.4 In the rest of the paper, excepted in Section 2.3, we will require that ν and c satisfy (1.11), (1.12) and (1.21). On principle, let us consider these three conditions as assumption (H). Note that, under (H), there is κ > 0 so that (1.23)-(1.26) holds.

The results

We keep notation given in Section 1.

2.1 Normalized limit distribution of (T x , K x , L x ), as x → ∞

In this section we will investigate the limit behavior of the triple (T x , K x , L x ), as x → ∞. Recall that K x and L x are defined by (0.3), resp. (0.4). Here as three cases : either E(X 1 ) > 0, or E(X 1 ) < 0 or E(X 1 ) = 0. First, it can be assumed that E(X 1 ) < 0.

Theorem 2.1 Under (H) and E(X 1 ) < 0 then, conditionally on {T x < ∞}, the

triple 1 √ x T x + x ϕ ′ (-γ 0 (0)) , K x , L x converges in distribution when x → ∞ to the 3-dimensional law N 0; - ϕ ′′ (-γ 0 (0)) ϕ ′3 (-γ 0 (0)) ⊗ w -. In addition, w -is the probability measure on R + × R + : w -(dk, dl) = -1 E(X 1 ) γ 0 (0) 2 δ 0,0 (dk, dl) + (e γ 0 (0)l -1)1l {k≥0;l≥0} ν l (dk) dl + R - -y 0 (1 -e γ 0 (0)(b+y) )P(T b < ∞)n(b, dk, dl)db 1l {k≥0,l≥0} ν(dy) , (2.1)
ν l (dk) is the image of ν(dk) by the map y → yl, and n(b, dk, dl) is the distribution of (K b , L b ) conditionally on {T b < ∞}.

If moreover the support of ν is included in [0; ∞[, w -(dk, dl) is given explicitly :

w -(dk, dl) = -1 E(X 1 ) γ 0 (0) 2 δ 0,0 (dk, dl) + (e γ 0 (0)l -1)1l {k≥0;l≥0} ν l (dk) dl (2.2)
Remark 2.2 1. N (0; σ 2 ) denotes the Gaussian distribution with mean 0 and variance σ 2 .

a)

We may observe that time and positions become asymptotically independent.

However the two components of the positions are not independent. In subsection 4.2, we give a stochastic interpretation of the limit distribution w -(dk, dl) defined by (2.2). b) Obviously K x + L x = X Tx -X Tx -is the jump size of (X t , t ≥ 0) at T x . It is easy to infer from Theorem 2.1 that

1 √ x T x + x ϕ ′ (-γ 0 (0)) , X Tx -X Tx - converges in distribution, as x → ∞, to N 0; - ϕ ′′ (-γ 0 (0)) ϕ ′3 (-γ 0 (0))
⊗ w where w is the probability measure on R + :

w(ds) = -1 E(X 1 ) γ 0 (0) 2 δ 0 (ds) + e γ 0 (0)s -1 -γ 0 (0)s γ 0 (0) 1l {s≥0} ν(ds) + R - -y 0 (1 -e γ 0 (0)(b+y) )P(T b < ∞)n(b, ds)db ν(dy)1l {s≥0} , (2.3)
and n(b, ds) is the distribution of

X T b -X T b -conditionally on {T b < ∞}.
If the jumps of (X t , t ≥ 0) are positive, w(ds) can be simplified :

w(ds) = -1 E(X 1 ) γ 0 (0) 2 δ 0 (ds) + e γ 0 (0)s -1 -γ 0 (0)s γ 0 (0) 1l {s≥0} ν(ds) . (2.4)
Let list the results related to the two other cases : E(X 1 ) > 0 and E(X 1 ) = 0.

Theorem 2.3 Assume (H).

1. When E(X 1 ) > 0, then T x < ∞ a.s. and the triple 1

√ x T x + x ϕ ′ (0) , K x , L x converges in distribution, as x → ∞, to N 0; - ϕ ′′ (0) ϕ ′3 (0) ⊗ w +
, where w + is defined by the relation obtained after replacing γ 0 (0) by -γ * 0 (0) in (2.1). In particular if (X t , t ≥ 0) has only positive jumps :

w + (dk, dl) = 1 E(X 1 )
γ * 0 (0) 2 δ 0,0 (dk, dl) + (1e -γ * 0 (0)l )1l {k≥0, l≥0} ν l (dk) dl .

(2.5)

2. When E(X 1 ) = 0, then T x x 2 , K x , L x converges in distribution, as x → ∞, to ̺⊗w 0 . ̺ denotes the law of the first hitting time of level 1 ϕ ′′ (0) by a standard

Brownian motion started at 0. Let us recall that

∞ 0 e -θx ̺(dx) = e - 2θ ϕ ′′ (0) . Moreover ϕ ′′ (0) = 1 + R y 2 ν(dy).
The probability measure w 0 on R + × R + is defined as follows :

w 0 (dk, dl) = 1 ϕ ′′ (0) δ 0,0 (dk, dl) + 2l1l {k≥0, l≥0} ν l (dk) dl -2 R - -y 0 (b + y)n(b, dk, dl)db 1l {k≥0, l≥0} ν(dy) , (2.6)
when n(b, dk, dl) is the distribution of (K b , L b ). This expression may be simplified if (X t , t ≥ 0) has only positive jumps (i.e.

ν(] -∞, 0[) = 0):

w 0 (dk, dl) = 1 ϕ ′′ (0) δ 0,0 (dk, dl) + 2l1l {k≥0, l≥0} ν k (dl) dk (2.7)
We will present two complements of Theorems 2.1 and 2.3.

1. Let us introduce :

T x =              1 √ x T x + x ϕ ′ (-γ 0 (0)) when E(X 1 ) < 0 1 √ x T x + x ϕ ′ (0) when E(X 1 ) > 0 T x x 2 when E(X 1 ) = 0 (2.8)
In subsection 4.1, a rate of convergence of T x to the associated Gaussian distribution, as x → ∞ is given.

2. In subsection 4.2, we will provide a stochastic realization of the probability measure w -, resp. w + defined by (2.2), resp. (2.5).

Auxiliary results

To study the joint distribution of (T x , K x , L x ), the Laplace transform F of this three dimensional r.v. is used :

F (θ, µ, ρ, x) = E e -θTx-µKx-ρLx 1l {Tx<∞} , (2.9)
for any θ ≥ 0, µ ≥ 0, ρ ≥ 0. a) When ν(R) < ∞ (i.e. (J t , t ≥ 0) is a compound Poisson process), we first prove (see Theorem 2.4 below) that F (θ, µ, ρ, •) satisfies a kind of integral equation.

Theorem 2.4 Assume λ = ν(R) < ∞. For any θ ≥ 0, µ ≥ 0 and ρ ≥ 0, the function F (θ, µ, ρ, .) is solution of the following integral equation :

G(x) = F 0 (θ, µ, ρ, x) + F 1 (θ, µ, ρ, x) + Λ θ G(x) ∀x ≥ 0 (2.10)
where

α θ = c 2 0 + 2(λ + θ) , (2.11)
F 0 (θ, µ, ρ, x) = e -(c 0 +α θ )x , (2.12)

F 1 (θ, µ, ρ, x) = e -(c 0 +α θ )x α θ (µ -ρ + c 0 + α θ ) [0,x]
e (-ρ+c 0 +α θ )ye -µy ν(dy)

+ e -ρx α θ (µ -ρ + c 0 -α θ ) ]x,∞[
e -(ρ+α θ -c 0 )(y-x)e -µ(y-x) ν(dy)

+ e (µ-ρ)x -e -(c 0 +α θ )x α θ (µ -ρ + c 0 + α θ ) ]x,∞[ e -µy ν(dy) - e -(c 0 +α θ )x α θ (µ -ρ + c 0 -α θ ) ∞ 0
e -(ρ+α θ -c 0 )ye -µy ν(dy) , (2.13) and Λ θ is the operator :

Λ θ G(x) = 1 α θ ∞ -∞ ν(dy) (x-y)∧x -∞ e -c 0 a e -α θ |a| -e -(2x-a)α θ G(x -a -y)da . (2.14)
The positive operator Λ θ will be studied in details in subsection 4.3. In particular, it is proved that, under suitable assumptions, F (θ, µ, ρ, •) is the unique function G which solves the equation (2.10), and can be strongly approximated by a series. b) However the operator Λ θ cannot be defined if ν is not a finite measure. The formula (2.10) does not permit to consider Lévy processes which are not reduced to a Brownian motion with drift plus a compound Poisson process. To avoid this difficulty, we introduce the Laplace transform F (θ, µ, ρ, •) of F (θ, µ, ρ, •) :

F (θ, µ, ρ, q) = ∞ 0 e -qy F (θ, µ, ρ, y)dy. (2.15)
This definition is meaningful for any q such that Re (q) > 0, since F (θ, µ, ρ, .) is a bounded function on [0, ∞[. Taking the Laplace transform in (2.10), proves (see Theorem 2.5 below) that under (H), F (θ, µ, ρ, .) verifies some kind of integral equation which remains valid when ν is a Lévy measure with no necessary finite mass. Before stating this result, let us denote by R the operator :

Rh(q) := 0 -∞ ν(dy) -y 0 e -q(b+y) -1 h(b)db , (2.16)
where q ∈ C, Re(q) > 0 and h ∈ L ∞ (R + ). Note that the identity (1.11) implies that Rh(q) is well defined.

Theorem 2.5 Let us assume that (H) holds. Let θ, µ, ρ ≥ 0, q ∈ C, Re(q) > 0.

We get :

F (θ, µ, ρ, q) = 1 ϕ(q) -θ q -γ * 0 (θ) 2 + ∞ 0 e -(q+ρ)y -e -µy q + ρ -µ - e -(γ * 0 (θ)+ρ)y -e -µy γ * 0 (θ) + ρ -µ ν(dy) +RF (θ, µ, ρ, .)(q) -RF (θ, µ, ρ, .)(γ * 0 (θ)) (2.17)
where γ * 0 (θ) is defined in (1.23) and (1.24).

Remark 2.6 1. If ν(]-∞, 0[) = 0 then RF (θ, µ, ρ, .
) is cancelled, and F (θ, µ, ρ, q) is given by the following explicit formula :

F (θ, µ, ρ, q) = 1 ϕ(q) -θ q-γ * 0 (θ) 2 + ∞ 0 e -(q+ρ)y -e -µy q + ρ -µ - e -(γ * 0 (θ)+ρ)y -e -µy γ * 0 (θ) + ρ -µ ν(dy) (2.18) 2. If ν(]0, ∞[) = 0 (2.17
) is reduced to :

F (θ, µ, ρ, q) = 1 ϕ(q)-θ q-γ * 0 (θ) 2 +RF (θ, µ, ρ, .)(q)-RF (θ, µ, ρ, .)(γ * 0 (θ)) (2.19)
3. Let us briefly detail the case θ = µ = ρ = 0 (i.e. F (0, 0, 0, x) is the ruin probability). If E(X 1 ) ≥ 0, it is easy to check that f : x → 1 satisfies (2.17).

In the more interesting case : E(X 1 ) < 0, since γ * 0 (0) = 0, then the relation (2.17) may be simplified. Suppose moreover that ν(] -∞, 0[) = 0, then (2.18) is reduced to :

F (0, 0, 0, q) = ∞ 0 e -qy P(T y < ∞)dy = 1 q + E(X 1 ) ϕ(q) . (2.20)
Therefore (2.20) generalizes identity (3.3) in [START_REF] Dufresne | Risk theory for the compound Poisson process that is perturbed by diffusion[END_REF].

Since {T y < ∞} = { X∞ > y}, where X∞ := sup t≥0 X t , then (2.20) is equivalent to : E e -q X∞ = -E(X 1 ) q ϕ(q)
, q > 0.

(2.21)

Note that in the case ν(]0, ∞[) = 0, then F (0, 0, 0, x) = P(T x < ∞) = e -γ 0 (0)x .
c) Let us briefly indicate how Theorem 2.5 enables to obtain the asymptotic behavior of T x , K x , L x as x → ∞. Using the Mellin-Fourier inverse transformation, it is actually possible to recover F (θ, µ, ρ, •) (see the formula (3.70) in Section 3.4). It implies that the asymptotic behavior of F (θ, µ, ρ, x), x → ∞ depends on the poles of F (θ, µ, ρ, •). According to (2.17), the complex zeros of ϕ θ are the poles of F (θ, µ, ρ, •). Under (H) and (2.23), we may determine in Proposition 2.7 the zeros of ϕ θ . Hence, we may obtain an expansion of F (θ, µ, ρ, x) as x → ∞ (see Theorem 2.8 below). Finally that allows to determine the limit distribution of the random triple considered above as x goes to infinity. Let

B ν := sup {b > 0 ; q → ∞ 1
e -qy ν(dy) admits a meromorphic extension to D -b } ,

(2.22) where D -b := {q ∈ C; Re q > -b} Note that B ν ≥ r ν and B ν may be equal to ∞. Then ϕ has a meromorphic extension to D -Bν . For simplicity, this extension will be denoted ϕ . Our last assumption on ν is :

∀B ∈]0, B ν [, ∃K, R 0 > 0, so that : ∞ 1 e -qy ν(dy) ≤ K|q| , for any q so that -B ≤ Re q ≤ 0 |Im q| ≥ R 0 (2.23)
A large class of measures ν satisfying (H) and (2.23) will be given in subsection 4.4.

First, we will concentrate on the complex zeros of the function ϕ θ defined by (1.22). Then, we will give in Theorem 2.8 an asymptotic expansion of F (θ, µ, ρ, x) as x → ∞.

Proposition 2.7 Let us suppose that (H) and (2.23) hold. Then for any θ ∈ [0, κ], there exists β θ > 0 such that for any B ∈]0, B ν [, ϕ θ admits a finite number of conjugated zeros in the strip

D -B,β θ := {q ∈ C ; -B ≤ Re q ≤ β θ }. This set of zeros of ϕ θ in D -B,β θ is equal to (cf. Figure 3 below) : 1. {-γ 0 (θ), -γ 1 (θ), -γ 1 (θ), • • • , -γ p (θ), -γ p (θ)} if θ > 0 , 2. {0, -γ 0 (0), -γ 1 (0), -γ 1 (0), • • • , -γ p (0), -γ p (0)} if θ = 0 and E(X 1 ) < 0, 3. {-γ 0 (0) = 0, -γ 1 (0), -γ 1 (0), • • • , -γ p (0), -γ p (0)} if θ = 0 and E(X 1 ) ≥ 0,
where

-B < Re (-γ p (θ)) ≤ • • • ≤ Re (-γ 1 (θ)) < -γ 0 (θ) ≤ 0 . (2.24)
4. -γ 0 (θ) is a simple (resp. double) zero of ϕ θ , if θ > 0, or θ = 0 and E(X 1 ) = 0 (resp. otherwise, i.e. θ = 0 and E(X 1 ) = 0).

The zeros of ϕ θ which are the poles of F (θ, µ, ρ, •) will play an important role in our approach, as shown below (see the proof of Theorem 2.8 in Section 3.4, and properties in Section 4.5).

Theorem 2.8

Let us suppose (H) and (2.23). Then for any θ ∈ [0, κ], µ ≥ 0 and ρ ≥ 0, there exists a positive number C 0 (θ, µ, ρ) > 0 and complex x-polynomial functions C 1 (θ, µ, ρ, x), • • • , C p (θ, µ, ρ, x) so that F (θ, µ, ρ, x) has the following asymptotic expansion as x → ∞ :

F (θ, µ, ρ, x) =C 0 (θ, µ, ρ)e -γ 0 (θ)x + p i=1 a i C i (θ, µ, ρ, x)e -γ i (θ)x + C i (θ, µ, ρ, x)e -γ i (θ)x + O e -Bx , (2.25)
where

a i = 1 2 if γ i (θ) is real and a i = 1 otherwise. The degree of C i (θ, µ, ρ, .) is n i -1,
where n i is the order of multiplicity of -γ i (θ). In addition, O is uniform with respect to µ ≥ 0, ρ ≥ 0 and θ ∈ [0, κ].

Remark 2.9 1. In (2.25), it is understood that B may be chosen in ]0, B ν [ closest as possible to B ν .

2. In Section 4.5 we study the coefficients C i (θ, µ, ρ, x).

3. Obviously, (2.25) and (2.24) imply that :

lim x→∞ F (θ, µ, ρ, x)e γ 0 (θ)x = C 0 (θ, µ, ρ).
(2.26)

This property has already been reached in [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF] in the particular case : θ = µ = ρ = 0 :

F (0, 0, 0, x) = P(T x < ∞)∼C 0 (0, 0, 0)e -γ 0 (0)x as x → ∞. (2.27)
4. Heuristically, no assumption on the negative jumps is required to get :

F (θ, µ, ρ, x) ≤ Ce -γ 0 (θ)x .
(2.28)

However to obtain an equivalent, or an asymptotic development of F (θ, µ, ρ, x) when x goes to infinity, it is natural to suppose that the negative and the positive parts of the jumps of (X t , t ≥ 0) are controlled. Our asymptotic development looks like a perturbation theorem around the case of Brownian motion with negative drift.

Polynomial decay

According to item 3. of Remark 2.9, under (H) and (2.23), the ruin probability goes to 0, with exponential rate. The aim is to prove that under weaker assumptions,

F (x) := F (0, 0, 0, x) = P(T x < ∞) (2.29)
has a polynomial type rate of decay, as x → ∞.

In this section we suppose neither (H) nor (2.23).

Theorem 2.10 Let us assume that

R |y|1l {|y|>1} ν(dy) < ∞ and E(X 1 ) = -c + R y1l {|y|≥1} ν(dy) < 0 . (2.30) and ∞ 0 y p ν(dy) < ∞ , for some p ≥ 2 .
(2.31)

Let n be the integer part of p -2, then

∀x ∈ R + P(T x < ∞) ≤ C n 1 + x n , (2.32)
where C n > 0.

3 Proofs

Proof of Theorems 2.1 and 2.3

Our approach is only based on the following estimate :

F (θ, µ, ρ, x)∼C 0 (θ, µ, ρ)e -γ 0 (θ)x as x → ∞ , (3.1)
where F (θ, µ, ρ, x) (resp. γ 0 (θ)) is defined by (2.9) (resp. (1.25) and (1.26)). The coefficient C 0 (θ, µ, ρ) comes from (2.25) included in Theorem 2.8. Theorems 2.1 and 2.3 related to the three cases E(X 1 ) > 0, E(X 1 ) < 0 and E(X 1 ) = 0 may be proved by using the same technique. Therefore we will only consider the case E(X 1 ) = 0 and the case E(X 1 ) < 0 (see Section 4.6 too).

3.1.1

Let us start with the case E(X 1 ) = 0.

(i) Since ϕ(-γ 0 (θ)) = θ, ϕ(0) = 0 and ϕ ′ (0) = 0, we may use the asymptotic expansion of ϕ at 0 at order 2 :

θ = ϕ(-γ 0 (θ)) = ϕ(-h) = h 2 2 ϕ ′′ (0) + o(h 2 ) . (3.2)
Therefore :

γ 0 (θ) = h = 2θ ϕ ′′ (0) + o( √ θ) . (3.3)
Recall that (3.1) is uniform with respect to θ ∈ [0, κ]. Then replacing θ by θ x 2 in (3.1) brings to :

F θ x 2 , µ, ρ, x = E e -θ Tx x 2 -µKx-ρLx 1l {T x <∞} ∼ C 0 θ x 2 , µ, ρ e - 2θ ϕ ′′ (0) as x → ∞, ∼ C 0 (0, µ, ρ) e - 2θ ϕ ′′ (0) as x → ∞ . (3.4)
(ii) We would like to demonstrate in this item that C 0 (0, µ, ρ) and e As for e Then, we obtain :

C 0 (0, µ, ρ) = 1 ϕ ′′ (0) 1 -2 ∞ 0 e -ρy y 0 (z -y)e (ρ-µ)z dz ν(dy) -2 0 -∞ -y 0 (y + b)E e -µK b -ρL b 1l {T b <∞} db ν(dy) . (3.5)
Let n(b, du, dz) be the distribution of the couple (K b , L b ). Consequently, C 0 (0, µ, ρ) may be written as follows :

C 0 (0, µ, ρ) = 1 ϕ ′′ (0) 1 + 2 ∞ 0 e -µz [z,∞[ (y -z)e -ρ(y-z) ν(dy) dz -2 ∞ 0 ∞ 0 e -µu e -ρz 0 -∞ -y 0 (y + b)n(b, du, dz)db ν(dy) = ∞ 0 ∞ 0 e -µu e -ρz w 0 (du, dz) . (3.6)
with w 0 defined in (2.6).

⊓ ⊔

3.1.2

Next we study the case E(X 1 ) < 0.

(i) On one hand, from (3.1), we have :

E e -θT x -µK x -ρL x |T x < ∞ = F (θ, µ, ρ, x) P(T x < ∞) = F (θ, µ, ρ, x) F (0, 0, 0, x) ∼ C 0 (θ, µ, ρ) C 0 (0, 0, 0) e -(γ 0 (θ)-γ 0 (0))x as x → ∞. (3.7)
On the other hand, the behavior of γ 0 (θ) in a neighborhood of 0 is the following :

γ 0 (θ) -γ 0 (0) = - 1 ϕ ′ -γ 0 (0) θ + ϕ ′′ -γ 0 (0) 2ϕ ′3 -γ 0 (0) θ 2 + o(θ 2 ) (θ → 0). (3.8)
Combining (3.7) and (3.8) leads after easy calculations to :

E e -θ √ x Tx+ x ϕ ′ (-γ 0 (0)) -µKx-ρLx |T x < ∞ ∼ C 0 (0, µ, ρ) C 0 (0, 0, 0) e -1 2 ϕ ′′ (-γ 0 (0)) ϕ ′3 (-γ 0 (0)) θ 2 as x → ∞. (3.9)
Let us observe that e

-1 2 ϕ ′′ (-γ 0 (0)) ϕ ′3 (-γ 0 (0)) θ 2
is the Laplace transform (with respect to the θ variable) of the Gaussian distribution with mean 0 and variance ϕ ′′ (-γ 0 (0)) ϕ ′3 (-γ 0 (0)) . As a result Theorem 2.1 will be proved as soon as we demonstrate that C 0 (0, µ, ρ) C 0 (0, 0, 0) is the Laplace transform of a probability measure on [0, ∞[×[0, ∞[. We proceed as in the case 3.1.1 above. We begin by using the relation (4.34) :

C 0 (θ, µ, ρ) C 0 (0, 0, 0) = 1 C 0 (0, 0, 0)ϕ ′ (-γ 0 (0)) - γ 0 (0) 2 + ∞ 0 e -(-γ 0 (0)+ρ)y -e -µy -γ 0 (0) + ρ -µ ν(dy) - ∞ 0 e -ρy -e -µy ρ -µ ν(dy) + RF (0, µ, ρ, .)(-γ 0 (0)) . (3.10)
It follows : Using the Fubini theorem we get :

C 0 (θ, µ, ρ) C 0 (0, 0, 0) = 1 C 0 (0, 0, 0)ϕ ′ (-γ 0 (0)) - γ 0 (0) 2 + ∞ 0 e -µy
C 0 (θ, µ, ρ) C 0 (0, 0, 0) = 1 C 0 (0, 0, 0)ϕ ′ (-γ 0 (0)) - γ 0 (0) 2 - ∞ 0 (e γ 0 (0)z -1)e -ρz [z;∞[ e -µ(y-z) ν(dy) dz + 0 -∞ -y 0 (e γ 0 (0)z -1)E e -µK b -ρL b 1l {T b <∞} db ν(dy) . (3.12)
Let n(b, du, dz) be the distribution of (K b , L b ) conditionally on {T b < ∞}. Then :

C 0 (θ, µ, ρ) C 0 (0, 0, 0) = 1 C 0 (0, 0, 0)ϕ ′ (-γ 0 (0)) - γ 0 (0) 2 - ∞ 0 (e γ 0 (0)z -1)e -ρz [z;∞[ e -µ(y-z) ν(dy) dz + ∞ 0 ∞ 0 e -µu e -ρz 0 -∞ -y 0 (e γ 0 (0)z -1)P(T b < ∞)n(b, du, dz)db ν(dy) = ∞ 0 ∞ 0 e -µu e -ρz w -(du, dz) . (3.13)
with w -defined in (2.1). ⊓ ⊔

Proof of Theorem 2.4

In this subsection it is assumed that : λ := ν(R) < ∞. Then (J t , t ≥ 0) is a compound Poisson process. As a result, it admits a first jump time τ 1 , exponentially distributed with parameter ν(R) (cf [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], theorem 21.1). The process (X t+τ 1 -X τ 1 ; t ≥ 0) is again a Lévy process distributed as (X t , t ≥ 0). This property is the key of our approach that we will briefly describe. Let us consider three cases :

• T x = inf {t ≥ 0 ; B t -c 0 t > x} < τ 1 if sup 0≤t≤τ 1 (B t -c 0 t) > x, • T x = τ 1 if sup 0≤t≤τ 1 (B t -c 0 t) < x and J τ 1 + B τ 1 -c 0 τ 1 > x,
• T x > τ 1 otherwise. Then, conditionally on {T x > τ 1 }, T xτ 1 is distributed as T x-Xτ 1 where ( T x ; x > 0) is an independent copy of (T x ; x > 0). ( T x ; x > 0) is independent from (X t , t ≥ 0) as well. This "renewal" part gives rise to the integral kernel Λ θ defined in (2.14).

This leads us to decompose F (θ, µ, ρ, .) defined by (2.9), as follows :

F (θ, µ, ρ, x) =E e -θTx-µKx-ρLx 1l {Tx<τ 1 } + E e -θTx-µKx-ρLx 1l {Tx=τ 1 }
+ E e -θTx-µKx-ρLx 1l {τ 1 <Tx<∞} .

(3.14)

We will calculate the two first terms of (3.14) in Lemmas 3.1, 3.2. The third one will be determined in Lemma 3.3.

Lemma 3.1 Let α θ be the real number, defined in (2.11), then :

E e -θTx-µKx-ρLx 1l {Tx<τ 1 } = e -(c 0 +α θ )x . (3.15)
Proof of Lemma 3.1 Let ( B t , t ≥ 0) be the Brownian motion with drift -c 0 :

B t := B t -c 0 t ∀t ≥ 0 . (3.16)
We set

T x := inf {t ≥ 0; B t > x}, x ≥ 0. Then, {T x < τ 1 } = { T x < τ 1 } a.s. and K x = L x = 0 on {T x < τ 1 }.
Since τ 1 is exponentially distributed with parameter λ and is independent from T x , we have :

E e -θTx-µKx-ρLx 1l {Tx<τ 1 } = E e -(λ+θ) Tx . ( 3
E e -θT x -µK x -ρL x 1l {Tx=τ 1 } = e -(c 0 +α θ )x α θ (µ -ρ + c 0 + α θ ) [0,x]
e (-ρ+c 0 +α θ )ye -µy ν(dy)

+ e -ρx α θ (µ -ρ + c 0 -α θ ) ]x,∞[
e -(ρ+α θ -c 0 )(y-x)e -µ(y-x) ν(dy)

+ e (µ-ρ)x -e -(c 0 +α θ )x α θ (µ -ρ + c 0 + α θ ) ]x,∞[ e -µy ν(dy) - e -(c 0 +α θ )x α θ (µ -ρ + c 0 -α θ ) ∞ 0
e -(ρ+α θ -c 0 )ye -µy ν(dy) .

(3.18)

Proof of Lemma 3.2 Let us write : Y 1 := J τ 1 . We observe that on {T x = τ 1 }, Y 1 > 0. Morever :

{T x = τ 1 } = {sup t≤τ 1 B t < x, B τ 1 + Y 1 > x} , (3.19)
and

K x = B τ 1 + Y 1 -x , L x = x -B τ 1 , (3.20)
where ( B t , t ≥ 0) is defined by the relation (3.16). Conditioning by (τ 1 , Y 1 ) implies that :

∆ :=E e -θTx-µKx-ρLx 1l {Tx=τ 1 } =e -ρx ∞ 0 dt e -(λ+θ)t ∞ 0 ν(dy) E e -(µ-ρ) Bt-µ(y-x) 1l {sup u≤t B u <x ; x-y≤ B t } (3.21) since the distribution of Y 1 is 1 λ ν.
The density function of (sup u≤t B u , B t ) is given in ( [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] page 95), i.e. :

P B t ∈ da; sup u≤t B u ∈ db = 2(2b -a) √ 2πt 3 e -(2b-a) 2 2t 1l {a<b; b>0} dadb . (3.22)
Let us apply Girsanov's formula : Let us recall the classical identities (cf. [START_REF] Lebedev | Special functions and their applications[END_REF] p 118, or [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] sections 8.432 6 page 959, and 8.469 3 page 967) :

P B t ∈ da; sup u≤t B u ∈ db = 2(2b -a) √ 2πt 3 e -c 0 a-c 2 0 2 t e -(2b
K 1 2 (δ) := 1 2 ∞ 0 1 √ t e -δ 2 (t+ 1 t ) dt = π 2δ e -δ ∀δ > 0 , (3.25)
and

∞ 0 1 √ t 3 e -1 2 (βt+ γ t ) dt = 2π γ e - √ βγ ∀β > 0, ∀γ > 0 . ( 3
E e -θTx-µKx-ρLx 1l {τ 1 <Tx<∞} = 1 α θ ∞ -∞ ν(dy) (x-y)∧x -∞
e -c 0 a e -α θ |a|e (2x-a)α θ F (θ, µ, ρ, xay)da .

(3.30)

Morever :

e -α θ |a| -e -(2x-a)α θ ≥ 0 if a ≤ (x -y) ∧ x , (3.31)
so Λ θ defined by (2.14) is an non-negative operator.

Proof of Lemma 3.3 Formula (3.30) may be proved proceeding analogously to the proof of previous Lemma. ⊓ ⊔ Remark 3.4 By introducing adapted functional Banach spaces, we may study in subsection 4.3 (cf. Theorem 4.2 and Proposition 4.3) the operator Λ θ . We can prove that F (θ, µ, ρ, .) is the unique solution of (2.10). Moreover, if θ > 0, or θ = 0 and E(X 1 ) < 0, then F (θ, µ, ρ, .) has a sub-exponential rate of decay :

F (θ, µ, ρ, x) ≤ Ce -γx , ∀x ≥ 0, for some C > 0, γ > 0 . (3.32)
The optimal value of γ has be given in Remark 2.9. Note that if θ = 0 and E(X 1 ) ≥ 0, then F (0, 0, 0, x) = 1, hence there is no hope to obtain a sub-exponential rate of decay.

Proof of Theorem 2.5

Assume that ν satisfies the assumptions given in Theorem 2.5. The proof of Theorem 2.5 will be divided into two steps. We will first prove (2.17) when ν(R) < ∞.

In a second step, we will approximate ν by a sequence of finite measures (ν n ). Then by replacing ν by ν n , and by taking the limit n → ∞, it will be proved that (2.17) remains valid.

Step 1 Let us suppose that ν(R) < ∞. a) Taking the Laplace transform in functional equation (2.10) leads to :

F (θ, µ, ρ, q) = F 0 (θ, µ, ρ, q) + F 1 (θ, µ, ρ, q) + Λ θ F (θ, µ, ρ, .)(q) q ∈ D 0 , (3.33) with D 0 := {z ∈ C, Re(z) > 0}.
The identity (2.12) implies :

F 0 (θ, µ, ρ, q) = 1 c + α θ + q , ( 3.34)
As for F 1 (θ, µ, ρ, x), starting from (2.13), we split the integral in four parts :

F 1 (θ, µ, ρ, q) = ∞ 0 e -qx F 1 (θ, µ, ρ, x)dx
= I 1 (θ, µ, ρ, q) + I 2 (θ, µ, ρ, q) + I 3 (θ, µ, ρ, q) + I 4 (θ, µ, ρ, q) (3.35)

where

I 1 (θ, µ, ρ, q) := ∞ 0 e -(q+c+α θ )x α θ (µ + c + α θ -ρ) [0,x]
e (c+α θ -ρ)ye -µy ν(dy) dx )ye -µy q + ρµ ν(dy) ,

= 1 α θ (q + c + α θ )(µ + c + α θ -ρ) ∞ 0 e -(q+ρ)y -e -(q+µ+c+α θ )y ν(dy) , (3.36) I 2 (θ, µ, ρ, q) := ∞ 0 e -(q+ρ)x α θ (µ + c -α θ -ρ) ]x,∞[ e -(ρ+α θ -c)(y-x) -e -µ(y-x) ν(dy) dx = 1 α θ (µ + c -α θ -ρ) ∞ 0 e -(ρ+α θ -c)y -e -(q+ρ)y q + c -α θ + e -(q+ρ
(3.37)

I 3 (θ, µ, ρ, q) := ∞ 0 e -(q+ρ-µ)x -e -(q+c+α θ )x α θ (µ -ρ + c + α θ ) ]x,∞[ e -µy ν(dy) dx = - 1 α θ (µ -ρ + c + α θ ) ∞ 0 ν(dy)
e -(q+ρ)ye -µy q + ρµ -e -(q+µ+c+α θ )ye -µy q + c + α θ , (3.38)

I 4 (θ, µ, ρ, q) := - ∞ 0 e -(q+c+α θ )x α θ (-ρ + µ + c -α θ ) dx ν + (ρ + α θ -c) -ν + (µ) = ν + (µ) -ν + (ρ + α θ -c) α θ (q + c + α θ )(-ρ + µ + c -α θ ) , (3.39)
and ν + denotes the Laplace transform of ν | ]0,∞] :

ν + (q) := ν | ]0,∞] (q) = ∞ 0 e -qy ν(dy), q ∈] -r ν , ∞[ . (3.40)
Therefore :

F 1 (θ, µ, ρ, q) = - ν + (q + ρ) α θ (q + c + α θ )(q + ρ -µ) + ν + (q + ρ) α θ (q + c -α θ )(q + ρ -µ) + 2 ν + (ρ + α θ -c) (µ + c -α θ -ρ)(q + c -α θ )(q + c + α θ ) - 2 ν + (µ) (µ + c -α θ -ρ)(q + ρ -µ)(q + c + α θ ) . (3.41)
Let us introduce :

C θ (q) := (q + c + α θ )(q + c -α θ ) = q 2 + 2cq -2(λ + θ) . ( 3

.42)

A straight calculation gives :

F 1 (θ, µ, ρ, q) = 2 C θ (q) ν + (q + ρ) -ν + (µ) q + ρ -µ - ν + (µ) -ν + (ρ + α θ -c) µ -ρ + c -α θ . (3.43) b)
Let us now compute Λ θ F (θ, µ, ρ, .)(q). Setting b = xay in (2.14) leads to :

Λ θ F (θ, µ, ρ, .)(x) = 1 α θ ∞ -∞ ν(dy) ∞ 0 e -c(x-b-y) e -α θ |x-y-b| -e -(x+y+b)α θ F (θ, µ, ρ, b)db - 1 α θ 0 -∞ ν(dy) -y 0 e -c(x-b-y) e -α θ (x-y-b) -e -(x+y+b)α θ F (θ, µ, ρ, b)db = H 1 F (θ, µ, ρ, .)(x) + I(θ, µ, ρ, x) + e -(c+α θ )x α θ (RF (θ, µ, ρ, .)(α θ -c) -RF (θ, µ, ρ, .)(-α θ -c)) , (3.44)
where we recall that R has been defined by (2.16), and

H 1 F (θ, µ, ρ, .)(x) := 1 α θ ∞ -∞ ν(dy) ∞ 0 e -c(x-b-y) e -α θ |x-y-b| F (θ, µ, ρ, b)db , (3.45)
I(θ, µ, ρ, x) := - 1 α θ ∞ -∞ ν(dy) ∞ 0 e -c(x-b-y) e -(x+y+b)α θ F (θ, µ, ρ, b)db = 1 α θ e -(c+α θ )x ν(α θ -c) F (θ, µ, ρ, α θ -c) .
(3.46) and ν stands for the Laplace transform of ν :

ν(q) := ∞ -∞ e -qy ν(dy), q ∈] -r ν , ∞[ . (3.47)
We multiply both sides of (3.44) by e -qx and we integrate with respect to the Lebesgue measure restricted to [0, ∞[. We get :

Λ θ F (θ, µ, ρ, .)(q) = H 1 F (θ, µ, ρ, .)(q) - ν(α θ -c) F (θ, µ, ρ, α θ -c) α θ (q + c + α θ ) + RF (θ, µ, ρ, .)(α θ -c) -RF (θ, µ, ρ, .)(-α θ -c) α θ (q + c + α θ ) . ( 3

.48)

We now focus on the calculation of H 1 F (θ, µ, ρ, .)(q). By using (3.45), we come to : The x-integrals can be computed :

H 1 F (θ, µ, ρ, .)(q) = 1 α θ ∞ -∞ e cy ν(
H 1 F (θ, µ, ρ, .)(q) = 1 α θ 0 -∞ e (c+α θ )y ν(dy) -y 0 e (c+α θ )b F (θ, µ, ρ, b)db 1 q + c + α θ + 1 α θ ∞ -∞ e (c-α θ )y ν(dy) ∞ 0∨(-y) e (c-α θ )b F (θ, µ, ρ, b) 1 -e -(q+c-α θ )(b+y) q + c -α θ db + 1 α θ ∞ -∞ e (c+α θ )y ν(dy) ∞ 0∨(-y) e (c+α θ )b F (θ, µ, ρ, b) e -(q+c+α θ )(b+y) q + c + α θ db .
By using (2.16) we may obtain :

H 1 F (θ, µ, ρ, .)(q) = 1 α θ (q + c + α θ ) RF(θ, µ, ρ, .)(-α θ -c) + 0 -∞ ν(dy) -y 0 F (θ, µ, ρ, b)db + 1 α θ (q + c -α θ ) ∞ -∞ e (c-α θ )y ν(dy) ∞ 0∨(-y) e (c-α θ )b F (θ, µ, ρ, b)db - 1 α θ (q + c -α θ ) ∞ -∞ e -qy ν(dy) ∞ 0∨(-y) e -qb F (θ, µ, ρ, b)db + 1 α θ (q + c + α θ ) ∞ -∞ e -qy ν(dy) ∞ 0∨(-y) e -qb F (θ, µ, ρ, b)db . Since 1 α θ (q + c + α θ ) - 1 α θ (q + c -α θ ) = - 2 C θ (q) , (3.49)
we get :

H 1 F (θ, µ, ρ, .)(q) = RF (θ, µ, ρ, .)(-α θ -c) α θ (q + c + α θ ) + ν(α θ -c) F (θ, µ, ρ, α θ -c) -RF (θ, µ, ρ, .)(α θ -c) α θ (q + c -α θ ) - 2 C θ (q)
ν(q) F (θ, µ, ρ, q) -RF (θ, µ, ρ, .)(q) . (3.50)

Combining (3.33) and (3.48) gives :

F (θ, µ, ρ, q) 1 + 2 ν(q) C θ (q) = F 0 (θ, µ, ρ, q) + F 1 (θ, µ, ρ, q) + 2 C θ (q) RF (θ, µ, ρ, .)(q) + 2 C θ (q) ν(α θ -c) F (θ, µ, ρ, α θ -c) -RF (θ, µ, ρ, .)(α θ -c) . (3.51)
As C θ (q) + 2 ν(q) = 2(ϕ(q)θ), it is easy to check :

(ϕ(q) -θ) F (θ, µ, ρ, q) = 1 2 C θ (q) F 0 (θ, µ, ρ, q) + F 1 (θ, µ, ρ, q) + RF (θ, µ, ρ, .)(q) + ν(α θ -c) F (θ, µ, ρ, α θ -c) -RF (θ, µ, ρ, .)(α θ -c) .
(3.52)

Let us recall that (H) implies the existence of γ * 0 (θ) ≥ 0 such that ϕ γ * 0 (θ) = θ (cf (1.23) and (1.24)). Therefore taking q = γ * 0 (θ) in (3.52) brings to :

ν(α θ -c) F (θ, µ, ρ, α θ -c) -RF (θ, µ, ρ, .)(α θ -c) = - 1 2 C θ (γ * 0 (θ)) F 0 (θ, µ, ρ, γ * 0 (θ)) + F 1 (θ, µ, ρ, γ * 0 (θ)) -RF (θ, µ, ρ, .)(γ * 0 (θ)) . (3.53)
First we determine F 0 (θ, µ, ρ, γ * 0 (θ)) resp. F 1 (θ, µ, ρ, γ * 0 (θ)) by using (3.34) resp. (3.43). Then relations (3.53) and (3.52) imply directly (2.17).

Step 2 Let ν n be the finite measure on R :

ν n (dy) := ν | R ]-1 n , 1 n [ (dy) ∀n ≥ 1 . (3.54)
Let F n be the Laplace transform of (T n x , K n x ) :

F n (θ, µ, ρ, x) = E e -θT n x -µK n x -ρL n x 1l {T n x <∞} , (3.55)
where X n t = B tc 0 + J n t , (J n t , t ≥ 0) a compound Poisson process with Lévy measure ν n , T n x = inf {t ≥ 0 , X n t > x} and K n x = X n T n

xx. Since lim n→∞ T n x = T x and lim n→∞ K n x = K x , we can therefore infer that (2.17) will result from a limit procedure. ⊓ ⊔

Proof of Theorem 2.8

Let us recall that F (θ, µ, ρ, .) resp. F (θ, µ, ρ, .) have been defined by (2.9) resp. (0.6). In the sequel, B ∈]0, B ν [ is supposed to be as close as possible to B ν .

Step 1 : We replace F (θ, µ, ρ, .) by F (θ, µ, ρ, .); an equation associated with F (θ, µ, ρ, .) We extend continuously F (θ, µ, ρ, .) to the whole line as follows :

F (θ, µ, ρ, x) := F (θ, µ, ρ, x)1l [0;∞[ (x) + (1 + x)1l [-1;+0] (x), ∀x ∈ R . (3.56)
Let F (θ, µ, ρ, .) be the Laplace transform of F (θ, µ, ρ, .) :

F (θ, µ, ρ, q) := ∞ -∞ e -qx F (θ, µ, ρ, x)dx . (3.57)
The advantage of using F (θ, µ, ρ, .) instead of F (θ, µ, ρ, .) lies in the fact that we shall prove below in Lemma 3.5 , that t → F (θ, µ, ρ, q 1 + it) is an integrable function on R, for any q 1 in ]0, β θ [ (β θ being defined in Proposition 2.7).

Since 0 -1

(1 + x)e -qx dx = e q -1q q 2 , (2.17) implies :

F (θ, µ, ρ, q) = e q -1 -q q 2 + 1 ϕ(q) -θ q -γ * 0 (θ) 2 + ∞ 0 e -(q+ρ)y -e -µy q + ρ -µ - e -(γ * 0 (θ)+ρ)y -e -µy γ * 0 (θ) + ρ -µ ν(dy) + RF (θ, µ, ρ, .)(q) -RF (θ, µ, ρ, .)(γ * 0 (θ)) (3.58)
We may observe that

ϕ(q) -θ q + γ * 0 (θ) = ϕ(q) -ϕ(γ * 0 (θ)) q + γ * 0 (θ) = q -γ * 0 (θ) 2 + c q -γ * 0 (θ) q + γ * 0 (θ) + ∞ -∞ e -qy -e -γ * 0 (θ)y + (q -γ * 0 (θ))y1l {|y|<1} q + γ * 0 (θ) ν(dy) , (3.59)
and

e q -1 -q q 2 + 1 q + γ * 0 (θ) = e q -1 q 2 - γ * 0 (θ) q(q + γ * 0 (θ))
.

(3.60)

Consequently, for any q, so that Re q > 0 :

F (θ, µ, ρ, q) = e q -1 q 2 - γ * 0 (θ) q(q + γ * 0 (θ)) + 1 ϕ(q) -θ -c q -γ * 0 (θ) q + γ * 0 (θ) - ∞ -∞ e -qy -e -γ * 0 (θ)y + (q -γ * 0 (θ))y1l {|y|<1} q + γ * 0 (θ) ν(dy) + ∞ 0 e -(q+ρ)y -e -µy q + ρ -µ ν(dy) - e -(γ * 0 (θ)+ρ)y -e -µy γ * 0 (θ) + ρ -µ ν(dy) +RF (θ, µ, ρ, .)(q) -RF (θ, µ, ρ, .)(γ * 0 (θ)) . (3.61)
It is clear that (1.11) implies that RF (θ, µ, ρ, .) (defined by (2.16)) is an entire function on C and all the integrals in (3.61) are holomorphic at least in D -Bν . Therefore, the right hand-side of (3.61) is the meromorphic extension of F (θ, µ, ρ, .) to D -Bν . Before ending step 1, we may remark that if θ > 0 or θ = 0 and E(X 1 ) = 0, F (θ, µ, ρ, .) and F (θ, µ, ρ, .) are holomorphic in a neighborhood of q = γ * 0 (θ). Although γ * 0 (θ) is a zero of ϕ θ , this value is a false singularity for F (θ, µ, ρ, .). But if θ = 0 and E(X 1 ) = 0, then γ * 0 (0) = 0 is a pole for F (0, µ, ρ, .).

Step 2 : t → F (θ, µ, ρ, q 1 + it) belongs to L 1 (R)

We may consider θ as a fixed element in [0, κ] and β θ > 0 as being given by Proposition 2.7. In addition, let q 1 ∈]0, β θ [. Lemma 3.5 Under (H) and (2.23), the function t → F (θ, µ, ρ, q 1 + it) belongs to L 1 (R).

So that to prove Lemma 3.5, we may begin with enumerating a few technical inequalities. These relations will be also used in the sequel.

Lemma 3.6 Let θ 1 < θ 2 . Let us suppose (2.23). Then :

(i) ∃k > 0 so that for any q which satisfies Re q ∈ [θ 1 , θ 2 ], we have : (iii) ∀A > 0 ∃k 1 > 0 such that ∀q which satisfies :

-B ≤ Re q ≤ A |Im q| ≥ R 0 , ∞ -∞ e -qy -1 + qy1l {|y|<1} ν(dy) ≤ k 1 (1 + |q|), (3.66) 1 k 1 |q| 2 ≤ |ϕ(q)| ≤ k 1 |q| 2 , (3.67) (iv) ∀h ∈ R, sup Re q≤h |RF (θ, µ, ρ, q)| < ∞ . (3.68)
Proof of Lemma 3.6

The proof of Lemma 3.6 may be directly infered from (2.23).

⊓ ⊔

Proof of Lemma 3.5 Proposition 2.7 tells us that ϕ θ has no zero in the strip {q ∈ C ; 0 < Re q < β θ }. So, if q 1 ∈]0, β θ [, ϕ(q 1 +it)-θ never cancels and (3.61) implies that t → F (θ, µ, ρ, q 1 + it) is a continuous function. Let us focus on (3.61). Lemma 3.6 implies that all the numerators in (3.61) are bounded on the line {q 1 +it ; t ∈ R}, and the denominators are smaller than C|q 2 | for some C > 0 and |q| large . This proves t → F (θ, µ, ρ, q 1 + it) belongs to L 1 (R).

⊓ ⊔

Step 3 : Proof of the asymptotic development (2.25), through the Mellin Fourier inverse transform Proposition 2.7 provides the existence of κ > 0 and B so that for any θ ∈ [0, κ], ϕ θ does not cancel on {-B + it ; t ∈ R} and B = γ * 0 (θ). Let 0 < q 1 < β θ . Since t → F (θ, µ, ρ, q 1 + it) belongs to L 1 (R) (cf. Lemma 3.5), we may use the Mellin Fourier inverse transform (see for instance [START_REF] Sveshnikov | The theory of functions of a complex variable[END_REF], pages 232-239). So, for any x ≥ 0 :

e -q 1 x F (θ, µ, ρ, x) = e -q 1 x F (θ, µ, ρ, x) = 1 2π ∞ -∞ e itx F (θ, µ, ρ, q 1 + it)dt , (3.69)
hence

F (θ, µ, ρ, x) = 1 2π ∞ -∞ e (q 1 +it)x F (θ, µ, ρ, q 1 + it)dt = - i 2π Γq 1 e zx F (θ, µ, ρ, z)dz , (3.70
) where Γ q 1 is the path :

Γ q 1 := {z = q 1 + it such that t ∈ R, t increasing} . (3.71)
In Proposition 2.7, it has been proved that there is R 1 > R 0 , so that ϕ θ has no zero in the two half-strips {q ∈ C ; -B ≤ Re q < β θ and |Im q| > R 1 }. In particular

F (θ, µ, ρ, .
) is holomorphic in this domain. Let Γ -B,q 1 ,R be the rectangular path (see Figure 3) :

Γ -B,q 1 ,R := Γ q 1 ,R ∪ Γ R ∪ Γ -B,R ∪ Γ -R , (3.72)
where :

Γ q 1 ,R := {q 1 + it ; |t| ≤ R, t growing} , (3.73)
Γ R := {t + iR ; -B ≤ t ≤ q 1 , t decreasing} (3.74) Γ -B,R := {-B + it ; |t| ≤ R, t decreasing} , (3.75)
Γ -R := {t -iR ; -B ≤ t ≤ q 1 , t growing} . (3.76)
If the residual theorem is applied to the meromorphic extension of z → e zx F (θ, µ, ρ, z) to D -Bν , the result is for any R > R 1 :

Γ -B,q 1 ,R e zx F (θ, µ, ρ, z)dz = 2iπ C 0 (θ, µ, ρ, )e -γ 0 (θ)x + p i=1 a i C i (θ, µ, ρ, x)e -γ i (θ)x + C i (θ, µ, ρ, x)e -γ i (θ)x (3.77)
where

a i = 1 2 if -γ i (θ)
is a real number and a i = 1 otherwise, and :

C 0 (θ, µ, ρ) := Res F (θ, µ, ρ, z); -γ 0 (θ) , (3.78)
C i (θ, µ, ρ, x) := e γ i (θ)x Res e zx F (θ, µ, ρ, z); -γ i (θ) , (3.79)
-γ 0 (θ)

β θ γ * 0 (θ) 0 -r ν -γ 1 (θ) -γ 2 (θ) -γ p (θ) -γ p (θ) -γ 1 (θ) -γ 2 (θ) q 1 R 0 -R 0 R -R -B ν -B Figure 3: The path Γ -B,q 1 ,R
where Res(f (z); γ) denotes the residual of f at point γ. Let us notice that (3.78) is valid since -γ 0 (θ) is a simple pole of F (θ, µ, ρ, .).

Since z →

e z -1z z 2 has an holomorphic extension to the whole plan C, the identity (3.56) implies that :

C 0 (θ, µ, ρ) = Res F (θ, µ, ρ, z); -γ 0 (θ) , (3.80)
C i (θ, µ, ρ, x) = e γ i (θ) Res e zx F (θ, µ, ρ, z); -γ i (θ) ∀i ∈ {1, • • • , p} . (3.81) We observe that C i (θ, µ, ρ, x) = e γ i (θ)
x Res e zx F (θ, µ, ρ, z); -γ i (θ) . We will prove below in Section 4.5, that x → C i (θ, µ, ρ, x) is a polynomial function (see the formula (4.30)). Since z → e zx F (θ, µ, ρ, z) belongs to L 1 (R), we come to (cf. (3.69)) :

F (θ, µ, ρ, x) = - i 2π lim R→∞ Γ q 1 ,R e zx F (θ, µ, ρ, z)dz = - i 2π lim R→∞ Γ -B,q 1 ,R e zx F (θ, µ, ρ, z)dz - Γ R e zx F (θ, µ, ρ, z)dz - Γ -B,R e zx F (θ, µ, ρ, z)dz - Γ -R e zx F (θ, µ, ρ, z)dz . (3.82)
We claim that in the right hand-side of (3.82), the limits of the second term and the fourth one are null. As for the third limit, we have :

lim R→∞ Γ -B,R e zx F (θ, µ, ρ, z)dz = O(e -Bx ) , (3.83)
where O is uniform with respect to θ ∈ [0, κ], µ ∈ R + and ρ ∈ R + . Hence, as x → ∞ :

F (θ, µ, ρ, x) = C 0 (θ, µ, ρ)e -γ 0 (θ) + p i=1 a i C i (θ, µ, ρ, x)e -γ i (θ)x + C i (θ, µ, ρ, x)e -γ i (θ)x + O(e -Bx ) . (3.84)
⊓ ⊔ Remark 3.7 Let us determine the sign of C 0 (θ, µ, ρ). The asymptotic expansion (2.25) implies that :

lim x→∞ e γ 0 (θ)x F (θ, µ, ρ, x) = C 0 (θ, µ, ρ) ≥ 0 . (3.85)
The residual C 0 (θ, µ, ρ) at -γ 0 (θ) of F (θ, µ, ρ, .) cannot cancel because -γ 0 (θ) is a single pole of F (θ, µ, ρ, .) when θ > 0 or θ = 0 and E(X 1 ) = 0.

If θ = 0 and c = E(J 1 ), it is easy to see that C 0 (0, µ, ρ) = 0 (cf. item c) iii) in subsection 4.5 below). We will give in subsection 4.5 some complements related to the calculation of the coefficients C i (θ, µ, ρ, x).

Proof of Theorem 2.10

Let us recall that the ruin probability is the function :

F (x) := F (0, 0, 0, x) = P(T x < ∞) , (3.86)
Let us denote F the Laplace transform of F :

F (q) := ∞ 0 e -qx F (x)dx ∀q ∈ C , Re (q) > 0 . (3.87)
In this section, it is only assumed that the Lévy measure ν satisfies (2.30) and (2.31). The proof will be divided into five parts.

Step 1 We will prove that we only need to consider a Lévy measure

ν whose support is included in [-k, ∞[, for some (finite) k ≥ 0.
The assumption (2.30) implies that there is k > 0 such that

∞ -k 1l {|y|≥1} yν(dy) < c . (3.88)
Let (X k t , t ≥ 0) be a Lévy process with decomposition :

X k t := B t -c 0 t + J k t ∀t ≥ 0 , (3.89)
where (J k t , t ≥ 0) is a pure jump process independent from (B t , t ≥ 0) and with Lévy measure :

ν k := ν | [-k,∞[
. In addition, it can be supposed that the processes (J k t , t ≥ 0) and (J t , t ≥ 0) are defined on the same probability space. Since νν k is a non-negative measure whose support is included in ] -∞, -k], then J t ≤ J k t , ∀t ≥ 0. Consequently X t ≤ X k t , ∀t ≥ 0 a.s., and T k x ≤ T x , ∀x ≥ 0 where

T k x := inf {t ≥ 0 / X k t > x} . (3.90) As a result, ∀x ≥ 0 F (x) ≤ F k (x) := P(T k x < ∞) . (3.91)
Note that the relation (3.88) implies that ν k satisfies (2.30) and (2.31). This proves the claim.

Step 2 The aim is to prove that we can suppose that the support of ν is included in

[-k, -1] ∪ [0, ∞[.
It is clear that the relation (2.30) implies that there is ε > 0 so that

-c + ε + [-k,∞[ y1l {|y|≥1} ν(dy) < 0. (3.92)
Let ν (1) be the Lévy measure : ν (1) (dy) := 1l ]-1,0[ (y)ν(dy). Consider X (1) the Lévy process whose function ϕ (1) is defined as follows :

ϕ (1) (q) := εq + R e -qy -1 + qy1l {|y|<1} ν (1) (dy) = εq + ]-1,0[ e -qy -1 + qy ν(dy).

Consequently E(X

1 ) = -ε < 0. Since ϕ (1) (q) ≥ -qε for any q ∈] -∞, 0], then lim q→-∞ ϕ (1) (q) = ∞. Note that the graph of ϕ (1) corresponds to the Figure 1, case a). As a result, there exist γ (1) > 0 such that ϕ (1) γ (1) = 0. According to [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF] there exists C (1) > 0 such that P(T (1) x < ∞) ≤ C (1) e -γ (1) x , x ≥ 0 (3.93)

where T

x := inf{t ≥ 0, X

> x}. Next let ν (2) (dy) := 1l [-k,-1]∪]0,∞[ (y)ν(dy), and X (2) the Lévy process associated with

ϕ (2) (q) := q 2 2 + (c -ε)q + R e -qy -1 + qy1l {|y|≥1} ν (2) (dy).
Consequently :

E(X (2) 1 ) = -(c -ε) + R y1l {|y|≥1} ν (2) (dy) = -c + ε + R y1l {|y|≥1} ν(dy) < 0.
In addition suppose that X (1) and X (2) are independent. Since ϕ = ϕ (1) + ϕ (2) , therefore X is distributed as X (1) + X (2) . Let us note that :

T

x/3 = ∞ ∩ T

x/3 = ∞ ⊂ X

t + X

(2) t < x , x > 0.

Applying moreover (3.93) we get :

P(T x < ∞) ≤ P T (1)
x/3 < ∞ + P T

(2)

x/3 < ∞ ≤ C (1) e -γ (1) x/3 + P T

(2)

x/3 < ∞ .

This inequality implies that the rate of decay of x → P(T x < ∞) is polynomial as soon as x → P(T

x < ∞) enjoys the same asymptotic behavior. In the sequel we can suppose that (2.30) and (2.31) hold and the support of ν is included in [-k, -1] ∪ [0, ∞[, for some k > 0.

Step 3 F belongs to L 1 (R + ).

We will first prove that :

sup 0<q≤q 0 | F (q)| < ∞ , for some q 0 > 0 .
(3.94)

By taking the limit θ, µ, ρ → 0 in (2.17), we may easily obtain :

F (q) = 1 ϕ(q) q 2 + 1 q ∞ 0
(e -qy -1 + qy)ν(dy) + RF (q) . (3.95)

Let us determine the asymptotic behavior of the numerator and the denominator as q → 0. Let us begin with the denominator :

ϕ(q) ∼ qϕ ′ (0) = q c - R 1l {|y|≥1} yν(dy) as q → 0. (3.96)
Next, we consider the numerator. We have :

q 2 + 1 q ∞ 0 (e -qy -1 + qy)ν(dy) ∼ q 2 1 + ∞ 0 y 2 ν(dy) as q → 0 . (3.97)
Since the support of ν is included in [-k, -1] ∪ [0, ∞[, then RF (q) can be simplified :

RF (q) = -1 -k ν(dy) -y 0 e -q(b+y) -1 F (b)db. (3.98)
But RF (0) = 0 and the derivative of RF (q) is bounded, then |RF (q)| ≤ Cq, for any 0 ≤ q ≤ q 0 . (3.94) follows immediately.

It is now easy to check that F is in L 1 (R + ). The function F is non-negative, then the monotone convergence theorem implies :

∞ 0 F (x)dx = lim q→0 ∞ 0 e -qx F (x)dx ≤ sup 0<q≤q 0 F (q) < ∞ . (3.99)
As a result F is integrable.

Step 4 Definition of F .

The function F can be extended to the whole line, setting F (x) = 0, for any x ≤ 0. However F may have a jump at 0. Let F be the following continuous extension of F :

F (x) := F (x)1l [0;∞[ (x) + (1 + x)1l [-1; 0] (x), ∀x ∈ R . (3.100)
Let q → F (iq) be the Fourier transform of F :

F (iq) = ∞ -∞ e -iqx F (x)dx = ∞ -1 e -iqx F (x)dx ∀q ∈ R . (3.101)
Step 5 q → F (iq) is in L 1 (R).

Since F ∈ L 1 (R + ), then F ∈ L 1 (R) and q → F (iq) is continuous. So, if we establish :

F (iq) ≤ C 1 + q 2 ,
(3.102) then F will be an element of L 1 (R).

It is proved in Proposition 2.7, that ϕ(iq) = 0, q ∈ R iff q = 0. Therefore we are allowed to replace q by iq in (3.95). By using the identity :

0 -1 (1 + x)e -iqx dx = - e iq -1 -iq q 2 , (3.103)
and (3.94) we may deduce :

1. (3.102) holds for any |q| ≤ q 0 (for some q 0 > 0), 2. F (iq) may be written as follows :

F (iq) = 1 -e iq q 2 + 1 ϕ(iq) -c - ∞ 1 yν(dy) + i q -1
-k e -iqy -1 ν(dy) + RF (iq) .

(3.104)

The inequality (3.102) will be a direct consequence of following estimates : Step 6

ϕ(iq) ∼ - q 2 2 R y 2 1l {|y|<1} ν(dy) (|q| → ∞) (3
The n first derivatives of F belong to L 1 (R).

Obviously any k derivative of q → 1e iq q 2 is continuous, bounded by C q 2 , |q| ≥ 1, and therefore belongs to L 1 (R).

The second term in the right hand-side of (3.104) may be written this way : N (q) ϕ(iq) .

By proceeding likewise Step 5 and using (2.31) it can be proved that the n first derivatives of N are bounded. As a result

N (q) ϕ(iq) ≤ C q 2 , |q| ≥ 1.
As for the asymptotic behavior of N (q) ϕ(iq) in a neighborhood of 0, it can be proved through a similar reasoning that this ratio is bounded, for any |q| ≤ 1.

Step 7 Proof of (2.32).

Since the n derivatives of q → F (iq) belong to L 1 (R), then We would like to point out that the asymptotic development of F (θ, µ, ρ, x) which is given by (2.25) provides the rate of convergence of 1

x n F (x) = i n 2π R e iqx d
√ x T x + x ϕ ′ (-γ 0 (0)) to the Gaussian distribution. Let us suppose that E[X 1 ] > 0. Let A x be the distribution function of 1 √ x T x + x ϕ ′ (-γ 0 (0))
and let A x be its characteristic function :

A x (t) = P 1 √ x T x + x ϕ ′ (-γ 0 (0)) ≤ t , t ∈ R A x (θ) = E e iθ √ x (Tx+ x ϕ ′ (-γ 0 (0)) ) , θ ∈ R.
Thanks to (2.25), it is not difficult to check that, if x is large enough we obtain :

sup θ>0 | A x (θ) -A(θ)| ≤ kθ √ x . (4.1)
Here k is a positive constant and A is the characteristic function of N 0; -ϕ ′′ (-γ 0 (0)) ϕ ′3 (-γ 0 (0)) .

Berry-Essen's inequalities (see for instance [START_REF] Loève | Probability theory[END_REF], p 285) imply the existence of two positives constants k 1 and k 2 such that :

sup t∈R |A x (t) -A(t)| ≤ k 1 a 0 kθ √ x dθ θ + k 2 a , for any a > 0, (4.2)
where A is the distribution function of the previous Gaussian distribution. Choosing a = x 1/4 , we get :

sup t∈R |A x (t) -A(t)| ≤ k 3 x 1/4 , x ≥ 1, (4.3)
for some k 3 > 0.

Theorem 4.2

Let us suppose that ν(R) < ∞.

(i) For any θ ≥ 0, the operator Λ θ defined by (2.14) is a linear and non-negative operator, whose norm is equal to

λ λ + θ in L ∞ (R + ).
(ii) Let γ ∈ [0, r ν [ and θ > 0 or θ = 0 and E(X 1 ) < 0 (recall that r ν has been defined by (1.13)). Then : a) Λ θ is a bounded operator from B γ to B γ . More precisely :

Λ θ f γ ≤ c θ,γ f γ ∀f ∈ B γ , (4.8)
with

c θ,γ = ν(-γ) ν(-γ) -ϕ(-γ) + θ , (4.9)
and ν the Laplace transform of ν (cf (3.47)).

b) There exists A straightforward calculation shows that

γ ∈]0, r ν [ such that ϕ(-γ) < θ. Therefore Λ θ is a B γ - contraction since : 0 < c θ,γ < 1 . ( 4
|Λ θ h(x)| ≤ λ λ + θ h ∞ , for any x ≥ 0. If we take h : x → 1, we have Λ θ h ∞ = λ λ + θ . Therefore, |Λ θ | L ∞ (R + ) = λ λ + θ .
(ii) Let f be an element of B γ , then |f (x)| ≤ f γ e -γx , ∀x ≥ 0. It follows that :

|Λ θ f (x)| ≤ 1 α θ f γ e -γx ∞ -∞ ν(dy) (x-y)∧x -∞
e -c 0 a e -α θ |a|e -(2x-a)α θ e γ(a+y) da , (4.13) for any γ ∈ [0, r ν [. By using (4.12), we get :

|Λ θ f (x)| ≤ 1 α θ f γ e -γx ∞ -∞ ν(dy)e γy 0 -∞ e -(c 0 -α θ -γ)a da + ∞ 0 e -(c 0 +α θ -γ)a da
Computing the integral with respect to da, leads directly to (4.8). ⊓ ⊔ Proposition 4.3

Let us assume ν(R) < ∞, r ν > 0, µ ≥ 0, θ > 0, or θ = 0 if E(X 1 ) < 0. Let γ be in [0, r ν [, such that ϕ(-γ) < θ. Then the function F (θ, µ, ρ, .) belongs to B γ and the equation (2.10) has a unique solution in B γ .

To prove Proposition 4.3, we need the following preliminary.

Lemma 4.4

Suppose either θ > 0, or θ = 0 if E(X 1 ) < 0, then for any x > 0, 2) Let us now turn to the case θ = 0 and E(X 1 ) < 0. By iterating the functional equation (2.10), we come to :

lim n→∞ Λ n θ F (θ, µ, ρ, .)(x) = 0 . ( 4
F (θ, µ, ρ, x) = n-1 p=0 Λ p θ [(F 0 + F 1 )(θ, µ, ρ, .)] (x) + Λ n θ F (θ, µ, ρ, .)(x) . (4.15)
The B γ -norm of Λ θ is strictly less than 1, then the series in

G(0, µ, ρ, x) = Λ 0 G(0, µ, ρ, .)(x) ≤ G(0, µ, ρ, .) ∞ , x ≥ 0 . (4.16)
As (4.12) is a strict inequality then (4.16) is a strict one too if G(0, µ, ρ, .) ∞ = 0. According to b) and c), there exists x 0 ≥ 0 such that : G(0, µ, ρ, x 0 ) = G(0, µ, ρ, .) ∞ . This implies G(0, µ, ρ, .) ∞ = 0. ⊓ ⊔

Proof of Proposition 4.3 Using the explicit expression of F 0 and F 1 (cf. (2.12) and (2.13)), by a straightforward calculation, enables us to infer that both F 0 (θ, µ, ρ, .) and F 1 (θ, µ, ρ, .) belong to B γ (for a detailed proof, cf. [START_REF] Volpi | Processus associés à l'équation de diffusion rapide; Etude asymptotique du temps de ruine et de l'overshoot[END_REF]). Due to Lemma 4.4 and (4.15)we may obtain :

F (θ, µ, ρ, x) = ∞ n=0 Λ n θ (F 0 + F 1 )(θ, µ, ρ, .)) (x) , (4.17)
Because

F 0 + F 1 ∈ B γ and
F (θ, µ, ρ, x) - p n=0 Λ n θ [(F 0 + F 1 )(θ, µ, ρ, .)] (x) < c p+1 θ,γ K e -γx , (4.18)
where

K = ∞ n=0 Λ n θ [(F 0 + F 1 )(θ, µ, ρ, .)]
γ < ∞ and c θ,γ is defined by (4.9).

2. Let us consider the case when the support of ν is included in ] -∞, 0]. ϕ is well defined on ] -∞, 0] and r ν = ∞. Moreover K x = L x = 0 and F 1 (θ, µ, ρ, x) = 0 for any x ≥ 0. As a result, (2.10) is reduced to :

F (θ, µ, ρ, x) = e -(c + α θ )x + Λ θ F (θ, µ, ρ, .)(x) . (4.19)
If either θ > 0, or θ = 0 and E(X 1 ) < 0, from both (1.25) and (1.26) we can infer the existence of a unique real number γ 0 (θ) such that :

-γ 0 (θ) < 0 and ϕ(-γ 0 (θ)) = θ . (4.20)

A direct (but fastidious !) calculation shows that x → e -γ 0 (θ)x is a solution of (2.10).

For more details see [START_REF] Volpi | Processus associés à l'équation de diffusion rapide; Etude asymptotique du temps de ruine et de l'overshoot[END_REF]. Hence F (θ, µ, ρ, x) = F (θ, 0, 0, x) = e -γ 0 (θ)x . where ρ i > 0, β i > 0 and m i ∈ N, for any i ∈ {1, 2, • • • , n},.

Since the Laplace transform of ν is explicit, we obtain immediately its meromorphic extension to the whole plane (B ν = ∞) and (2.23). Moreover :

ν(q) = ∞ 0 e -qx n i=1 ρ i e -β i x x m i dx = n i=1 ρ i ∞ 0
x m i e -(q+βi)x dx . (4.24)

If we set y = (q + β i )x, we have :

ν(q) = n i=1 ρ i (q + β i ) m i +1

Calculation of the C i coefficients

Let us suppose in this section that (H) holds. a) We claim that C i (θ, µ, ρ, .) is a polynomial function. We may assume that -γ i (θ) is a zero of ϕ θ with multiplicity n i , and F (θ, µ, ρ, z) has the following asymptotic expansion in a neighborhood of -γ i (θ) :

F (θ, µ, ρ, z) = K i,n i (θ, µ, ρ) (z + γ i (θ)) n i + K i,n i -1 (θ, µ, ρ) (z + γ i (θ)) n i -1 + • • • + K i,
C i (θ, µ, ρ, x) = K i,n i (θ, µ, ρ) (n i -1)! x n i -1 + K i,n i -1 (θ, µ, ρ) (n i -2)! x n i -2 +• • •+K i,1 (θ, µ, ρ) . (4.30) b)
We suppose in this item that -γ i (θ) is a single zero of ϕ θ = ϕθ. Then C i (θ, µ, ρ, x) does not depend on x. Furthermore C i (θ, µ, ρ, x) is given by the following :

C i (θ, µ, ρ) = Res F (θ, µ, ρ, z); -γ i (θ) . (4.31)

Note, that in this case, according to (2.17), -γ i (θ) is a single pole of F (θ, µ, ρ, •). i) When the real part of -γ i (θ) is bigger than -r ν , C i (θ, µ, ρ) can be determined as follows :

C i (θ, µ, ρ) = 1 ϕ ′ (-γ i (θ))

-γ i (θ)γ * 0 (θ) 2 + ∞ 0 e (γ i (θ)-ρ)ye -µy -γ i (θ) + ρµ -e -(γ * 0 (θ)+ρ)ye -µy γ * 0 (θ) + ρµ ν(dy) +RF (θ, µ, ρ, .)(-γ i (θ)) -RF (θ, µ, ρ, .)(γ * 0 (θ)) ,

where it is supposed that e ay -1 a = y if a = 0.

It has to be observed that if the support of ν is included in ]0, ∞[, then the formula above reduces to the two first lines, since RF (θ, µ, ρ, •) = 0. What is more, if θ = µ = ρ = 0 then

C i (0, 0, 0) =        ϕ ′ (0) ϕ ′ (-γ i (0))
if E(X 1 ) < 0 1 if i = 0 and E(X 1 ) ≥ 0 0 if i ≥ 1 and E(X 1 ) ≥ 0 (4.33)

Wiener-Hopf factorization

Let us recall the Wiener-Hopf decomposition (cf. [START_REF] Bertoin | Lévy processes[END_REF], page 165) : for any θ > 0, we have : θ θ + ϕ(-q) = ψ + θ (q)ψ - θ (q) , (4.37) where ψ + θ (q) := E e iqSτ θ , ψ - θ (q) := E e iq(Sτ θ -Xτ θ ) , (4.38) and τ θ is an exponential r. v. with parameter θ, independent from process (X t , t ≥ 0) and S t := sup s≤t X s . Since : P(S τ θ > a) = P(T a < τ θ ) = E(e -θT a ) = F (θ, 0, 0, a) , (4.39)

it is easy to deduce the following identity :

ψ + θ (q) = 1 + iq F (θ, 0, 0, iq) . (4.40) Equation (2.17) implies that the Wiener-Hopf factor ψ + θ verifies a functional equation. In particular, if ν(] -∞, 0]) = 0, combining equations (2.18) and (4.40) an explicit form of ψ + θ (q) may be obtained. Due to (4.37), ψ - θ (q) is also explicit.

4.8 Proof of Proposition 2.7

1. It is clear that ϕ θ is holomorphic in {q ∈ C ; Re q > -r ν } and the only real zeros of ϕ θ in this domain are -γ 0 (θ) and γ * 0 (θ).

2. We claim that the complex zeros of ϕ θ in {q ∈ C ; -γ 0 (θ) ≤ Re q ≤ γ * 0 (θ)} are -γ 0 (θ) and γ * 0 (θ).

a) Suppose first that -γ 0 (θ) < Re q < γ * 0 (θ). We have :

|e ϕ θ (q) | = |E(e -qX 1 -θ )| ≤ E e -Re q X 1 -θ = e ϕ θ (Re q) < 1 , since ϕ θ < 0 on ]γ 0 (θ), γ * 0 (θ)[. Then ϕ θ (q) = 0.

b) Let q = -γ 0 (θ) + ib, b ∈ R. We compute ϕ θ (q) :

ϕ θ (q) =ϕ θ (-γ 0 (θ)) - b 2 2 + ∞ -∞
e γ 0 (θ)y (cos(by) -1)ν(dy)

+ i -bγ 0 (θ) + cb - ∞ -∞
e γ 0 (θ)y sin(by) -by1l {|y|<1} ν(dy) .

(4.41)

-ϕ

  2θ ′′ (0) are Laplace transforms of probability measures on [0, ∞[×[0, ∞[ and [0, ∞[ respectively.

- 1 ϕ

 1 ′′ (0) √ 2θ , it is well known (cf. [12] page 96 formula (8.6)) that it is the Laplace transform of the first hitting time to level 1 ϕ ′′ (0) for a standard Brownian motion started at 0. Let us prove that C 0 (0, µ, ρ) is a Laplace transform. We modify the identity (4.34) in Section 4.5 via the relation : e -ay -1 + ay a 2 = -y 0 (zy)e -az dz.

y 0 -

 0 e -(-γ 0 (0)+ρ-µ)z dz ν(dy) ρ-µ)z dz ν(dy) + RF (0, µ, ρ, .)(-γ 0 (0)) .

-a) 2 2t

 2 1l {a<b; b>0} dadb . (3.23) Combining (3.23) and (3.21) leads to : ∆ = e (µ-ρ)x

  .26) obtained by derivation and the changing of variable t → β γ t. This allows to first compute explicitly the integral with respect to dt in (3.24) :∆ = 2e (µ-ρ)x ∞ 0 ν(dy) e -µy x x-y da e -(c 0 +µ-ρ)a x a∨0 e -α θ (2b-a) db (3.27)In a second step we evaluate the integral with respect to db : da e -(c 0 +µ-ρ)a e -α θ |a|e -α θ (2x-a) . (3.28) Let us introduce two cases xy ≥ 0 and xy < 0 : c 0 +µ-ρ+α θ )a da+ e (µ-ρ)x α θ ]x,∞[ ν(dy) e -µy 0 x-y e -(c 0 +µ-ρ-α θ )a da + x 0 e -(c 0 +µ-ρ+α θ )a da e (-2α θ +µ-ρ)x α θ ∞ 0 ν(dy) e -µy x x-y e -(c 0 +µ-ρ-α θ )a da . (3.29) If the integral is computed according to da, we may easily obtain (3.18). ⊓ ⊔ Lemma 3.3 In (3.14), the third expectation is equal to :

dy) ∞ 0 eeee

 0 cb F (θ, µ, ρ, b)db ∞ 0 e -(q+c)x e -α θ |x-y-b| dx cb F (θ, µ, ρ, b)db ∞ 0 e -(q+c)x e -α θ (x-y-b) dx cb F (θ, µ, ρ, b)db b+y 0 e -(q+c)x e α θ (x-y-b) dx cb F (θ, µ, ρ, b)db ∞ b+y e -(q+c)x e -α θ (x-y-b) dx .

1 - 1 e 1 e 1 0e

 1111 -qy -1 + qy1l {|y|<1} ν(dy) ≤ k|q| , -qy -1 + qy1l {|y|<1} ν(dy) ≤ k|q|, (3.63) -qy -1 + qy1l {|y|<1} ν(dy) ≤ k|q| , (3.64) (ii) ∀d > 0, ∃k 0 > 0 such that sup Re q≤d -1-∞ e -qy ν(dy) ≤ k 0 (3.65)

  iqy -1)ν(dy) ≤ |q| k ν [-k, -1] , (3.106) |RF (iq)| ≤ 2 k ν [-k, -1] .(3.107)

. 10 )

 10 Proof of Theorem 4.2 (i) Relation (3.31) implies that Λ θ is a non-negative operator. It is easy to check that the function ℓ : ℓ(x) := 1l {a+y≤x ; a≤x} e -c 0 a e -α θ |a|e -(2x-a)α θ (4.11) is increasing, then : ℓ(x) < ℓ(∞) = e -c 0 a e -α θ |a| ∀x ∈ R . (4.12)

  (4.15) converges. The remaining term Λ n θ F (θ, µ, ρ, .)(x) converges in B γ to some function G(θ, µ, ρ, x). It is easy to check what follows : a) G(0, µ, ρ, .) is a bounded and non negative function , b) G(0, µ, ρ, .) is a continuous function on [0, ∞[ , c) lim x→∞ G(0, µ, ρ, x) = 0 , d) Λ 0 G(0, µ, ρ, .) = G(0, µ, ρ, .) . Using (i) in Theorem 4.2 leads to :

4. 4 - 1 + 1 e 1 e

 4111 Few examples of ν satisfying (2.23) We give three classes of measures ν which satisfy (H) and (2.23) : a) Let us suppose that ν has finite exponential moments : qy1l {|y|<1} ν(dy) < ∞ . (4.21) In that case, ν | [1,∞[ and ϕ are holomorphic functions in the whole plane C, then B ν = ∞. Moreover, for any B > 0 : sup Re q≥-B ∞ -qy ν(dy) ≤ ∞ By ν(dy) < ∞ . (4.22) Then (2.23) holds. The condition (4.21) is fulfilled if, for instance, ν has a compact support. b) Let ν be a linear combination of gamma distributions : ν(dy) := n i=1 ρ i e -β i y y m i 1l {y≥0} dy . (4.23)

∞ 0 y

 0 m i e -y dy = n i=1 ρ i m i ! (q + β i ) m i +1 .(4.25)This implies that ν resp. ϕ is holomorphic in C -{-β 1 , • • • , -β n } resp. meromorphic in C.c) The example above may be generalized as follows :ν(dy) := φ(y)1l {y≥0} dy ,(4.26)where φ ≥ 0, bounded on [0, y 0 ], and for every y ≥ y 0 :φ(y) := ρ 0 e -β 0 y y m 0 -1 + n i=1 (ρ i e -β i y + ρ i e -β i y )y m i -1 + O(e -β n+1 y ) , (4.27) with y 0 ≥ 0, ρ 0 ≥ 0, β 0 > 0, Re β i > 0, ρ i ∈ C * , m i ∈ N * et β n+1 ≥ sup 1≤i≤n Re β i .Then ν and ϕ are meromorphic functions in {q ; Re q ≥ β n+1 }, B ν = β n+1 and (2.23) holds.d) If ν 1 and ν 2 satisfy (2.23), then ν = ν 1 + ν 2 satisfies (2.23) too.

  .14)

	Proof of Lemma 4.4 1) Suppose θ > 0. Since F is bounded by 1, and the norm of Λ θ is rem 4.2) : Λ n λ θ F (θ, µ, ρ, .) ∞ ≤ λ + θ	λ λ + θ	(cf. Theo-

n

. This proves (4.14).

  Λ θ is a contraction in B γ , the serie converges in B γ , which directly implies the result.

		⊓ ⊔
	Remark 4.5
	1. Under the conditions stated in Proposition 4.3, we have actually proved that p
	F (θ, µ, ρ, x) can be approximated by	Λ n θ [(F
	n=0

0 + F 1 )(θ, µ, ρ, .)] (x). More precisely :

  1 (θ, µ, ρ) z + γ i (θ) + • • • . (4.28) Since e zx = e -γ i (θ)x 1 + (z + γ i (θ))x + (z + γ i (θ)) 2

	2!	x 2 + • • •	(4.29)
	then relations (3.80) and (3.81) imply that :

Acknowledgments: We wish to thank the referee for carefully reading the manuscript and making useful suggestions.

Stochastic interpretation of w -and w +

We would like to give a stochastic interpretation of the probability measure w - defined by (2.2). Let (K, L) be a two-dimensional r.v. with probability distribution w -. Obviously the event {K = 0} (= {L = 0} a.s.) occurs with probability -γ 0 (0) 2E(X 1 ) . Conditionally on {L > 0}, the distribution of (K, L) is of type α e γ 0 (0)l -1 1l {k>0;l>0} ν l (dk)dl where ν l is the positive measure defined in Theorem 2.1. This leads us to consider the positive measure :

in which γ > 0, ν l is the image of ν by y → yl. In addition, ν is a positive measure on ]0; ∞[ which satisfies :

here α is the normalization factor :

. Proposition 4.1 Let (K * , L * ) be a two dimensional r.v. with distribution w (γ) defined by (4.4) . Then L * has a density function given by α(e γl -1)ν([l, ∞[)1l {l>0} .

Conditionally on L * = l, the distribution of

It is obvious that choosing γ = γ 0 (0) (resp. γ = γ * 0 (0)) allows to recover the probability measure w -(resp. w + ) introduced in (2.2) (resp. (2.5)). Consequently, Proposition 4.1 gives a stochastic interpretation of the limit law of (K x , L x ) as x → ∞, in the case where ν (] -∞; 0[) = 0.

Study of Λ θ

To investigate uniqueness in (2.10), we will prove that Λ θ is a contraction on the Banach space :

B γ is equipped with the norm :

Considering i = 0 in (4.33) allows to recover the result given in [START_REF] Doney | Hitting probabilities for spectrally positive Lévy processes[END_REF].

ii) As for the case Re (-γ i (θ)) < -r ν , the previous ν-integrals and ϕ, have to be replaced by their meromorphic extensions.

c) In this item we can focus on C 0 (θ, µ, ρ), which is the dominent term in (2.25).

i) When θ > 0 or θ = 0 and E(X 1 ) = 0, then -γ 0 (θ) is a single zero of ϕ θ and -γ 0 (θ) > -r ν . Therefore C 0 (θ, µ, ρ) is given by (4.32) with i = 0. Let us recall that when θ = 0 and E(X 1 ) < 0 (resp. E(X 1 ) > 0), then γ * 0 (0) = 0 (resp. γ 0 (0) = 0).

ii) In the case E(X 1 ) = 0, then γ 0 (0) = γ * 0 (0) = 0 is a double zero of ϕ, but a simple pole of F (θ, µ, ρ, .). Thus, a direct calculation shows :

In particular : iii) The constant C 0 (θ, µ, ρ) is positive because µ → F (θ, µ, q -γ 0 (θ)) is decreasing and

> 0 otherwise.

Girsanov transformation

Let (X t , t ≥ 0) be a Lévy process. It is well known that there is a family of probability measures P (λ) , 0 ≤ λ ≤ γ so that, under P (λ) , (X t , t ≥ 0) is still a Lévy process and :

Consequently ϕ (λ) (q) = ϕ(qλ)ϕ(-λ), where ϕ (λ) is associated with (X t , t ≥ 0) under P (λ) . Under (H), there exists λ such that ϕ(-λ) = θ and ϕ ′ (0)ϕ ′ (-λ) < 0. Since E(X 1 ) = -ϕ ′ (0), and E (λ) (X 1 ) = -ϕ ′(λ) (0) = -ϕ ′ (-λ), then E(X 1 )E (λ) (X 1 ) < 0. This trick allows to only consider the case E(X 1 ) > 0 (or E(X 1 ) < 0), and then simplify the proofs of Theorems 2.1 and 2.3 and the result given in subsection 4.1.

But ϕ θ (-γ 0 (θ)) = 0, then Re (ϕ θ (q)) ≤ -b 2 2 . Consequently ϕ θ (q) = 0 iff b = 0.

c) The same reasoning applies to the case q = -γ * 0 (θ) + ib.

3. Let us prove that ϕ θ has a finite number of zeros in the strip {q ∈ C; -B ≤ Re q ≤ β θ }, for some β θ in ]0, γ * 0 (θ)[. Proposition 2.7 is a consequence of both items 1. and 2. as described above and of the following properties : (i) According to (2.23) there is R > R 0 > 0, k > 0 so that : |ϕ θ (q)| ≥ k|q 2 | for any q such that -B ≤ Re q ≤ 0, |Im q| > R.

(4.42)

As a result, ϕ θ (q) = 0.

(ii) The function ϕ θ is meromorphic in {q ∈ C ; Re q > -B} and therefore admits at most a finite number of poles in the compact domain {q ∈ C; -B ≤ Re q ≤ 0, |Im q| ≤ R}.


Pierre Gallian
email: pierre.gallian@efs.sante.fr

André Cabié

Pascale Richard

Laure Paturel

Rémi N Charrel

Boris Pastorino

Isabelle Leparc-Goffart

Pierre Tiberghien

Xavier De Lamballerie

Zika virus in asymptomatic blood donors, Martinique: 2016

Text:

To the editor: Zika fever is an Aedes-borne disease caused by a flavivirus (Zika virus, ZIKV, genus Flavivirus). During the past years ZIKV spread in Polynesia, South-America and the Caribbean. Many aspects of Zika fever natural history remain unknown, e.g. the proportion of asymptomatic cases and the duration of viremia. Estimating the prevalence of Zika infections is difficult because a large proportion of infected individuals do not seek medical attention and seroprevalence studies are hampered by antigenic cross-reactivity with other flaviviruses, e.g. dengue virus. [START_REF] Charrel | Background review for diagnostic test development for Zika virus infection[END_REF] Here, we present a study of ZIKV infection in blood donors from Martinique island (French West Indies, Caribbean region), with novel epidemiological, biological and clinical information that refine the picture of Zika fever in adults.

Following the ZIKV outbreak in 2013-2014 in French Polynesia and the evidence that a significant proportion (3%) of donors were viremic,

12

ZIKV has been recognized in Martinique since the end of 2015, leading to implementation by the French Blood Bank of a systematic individual Nucleic Acid Testing (NAT) of blood donations. Virus detection was performed on plasma samples using a semi-automated platform consisting in a Microlab-STARlet® (Hamilton, Switzerland) and the NucleoSpin® 96 Virus Extraction Kit (Macherey-Nagel, Germany) for nucleic acids extraction, and a CFX96 thermocycler (Bio-Rad Laboratories, California) and the RealStar® Zika Virus RT-PCR Kit_1.1 (Altona Diagnostics, Germany) for RT-PCR testing.

Between January-19 th and June-10 th 2016, 4,129 consecutive blood donations were tested (mean age: 41.9 years; sex ratio (M/F): 0.88). Positive NAT detection occurred in 76 blood donations (1.84%) with the most intense detection rate (3%) during weeks 17-20 (mean age: 41.8 years; sex ratio: 1.2). Post-donation enquiry consisted in a phone call at day 7 post-donation to identify symptoms compatible with ZIKV infection. When the donor declared no sign, a new call was organized 14 days post-donation. This information was obtained from 75 viremic donors: 34 (45.3%) remained asymptomatic; 41 (54.7%) reported symptoms (1-6 days post-donation, figure 1A) such as fever, conjunctivitis, myalgia, arthralgia and rash (figure 1B). There was a trend for higher values of molecular viral load in symptomatic vs asymptomatic donors: mean values 5.36 vs 4.93 log 10 RNA genomic equivalents/mL, Independent Wilcoxon test p-value=0.0013 (figure 1C). This difference does not imply that symptomatic donors reached higher viremia than asymptomatic donors: sampling could occur in the former only during the early steps of viremia, while in the latter, it could also occur during the decreasing phase of viremia. The range of viral loads (2.09-6.50 log 10 RNA copies/mL) was comparable to that previously described in French Polynesian asymptomatic blood donors. The seroprevalence after seroneutralisation was 13.5% in early March and 42.2% in early June. An ELISA ratio >4 was associated with a positive seroneutralisation in more than 95% of cases. Sera with an ELISA ratio >5 were associated with seroneutralisation in 100% of cases.

During the same period, the cumulative curve of incident cases detected by NAT in blood donors was almost linear (figure 2A). This allowed performing a simulation analysis and determining, based on the incident cases detected amongst blood donors, the number of donors infected by ZIKV at different time points. Starting from the hypothesis of a random distribution of (i) the first day of detectable viremia in infected donors during all days of the period studied, and (ii) the day of donation during the days open to collection during the same period, we estimated the number of cases for which donation would occur during the phase of asymptomatic viremia detectable by NAT. We deduced the expected prevalence of ZIKV infection in donors for theoretical durations of asymptomatic viremia in plasma of 4 to 6 days (figure 2B), by reference to dengue 15 and chikungunya fever.

16

A faithful correlation with actual seroprevalence results was observed for a duration of asymptomatic viremia of 6 days (prevalence estimated at 12.5% vs 13.5% by seroneutralisation in early March, and at 43.5% vs 42.2% in early June). This is consistent with the observation that the longest period reported between blood donation and symptoms (pre-symptomatic viremia) was 6 days (figure 1a). Since detection methods using whole blood samples have allowed detecting virus RNA up to 68 days following the onset of symptoms, 17 it will be important to determine if such samples contain infectious virus, and re-assess ZIKV transfusion risk accordingly.

The estimated number of clinically suspect cases in Martinique (French Public Health institute) was ~7,600, i.e. ~2% of the population, in early March (week 8) and ~28,900, i.e. ~7.6% of the population, in early June (week 22).

18

With reference to seroprevalence results, this suggests that the proportion of cases which did not seek medical attention was 80-85% and did not change significantly along the study period. Our analyses suggest therefore that ZIKV has spread in the general population silently, with a minor impact of the acute disease on medical structures (e.g., 10-35 weekly consultations for Zika syndromes in emergency departments, 10-20 in obstetric departments). The most significant medical impact consisted in 19 Zika-related GBS confirmed cases and Zika infection in 342 pregnant women at the end of our study (June-2016). We conclude that ZIKV individual NAT screening in Martinique during 5 months of circulation of the virus allowed the detection of approximately 2% of contaminated blood donations. The proportion of truly asymptomatic cases of Zika disease among Martiniquan blood donors infected by ZIKV was ca. 45% and the proportion of cases that did not require medical attention was even higher (80-85%). The duration of plasma asymptomatic and pre-symptomatic viremia was estimated to be close to 6 days. We suggest that, once the duration of asymptomatic viremia has been estimated (e.g., from early analysis of seroprevalence and PCR detection datasets), NAT incidence studies in blood donors could improve prevalence estimates in the general population in the case of a disease with frequent mild or asymptomatic clinical presentations such as Zika fever. The French Blood Bank experience of individual ZIKV NAT screening and epidemiological studies performed from blood donors may be of interest at the time the U.S. Food and Drug Administration is recommending that blood banks screen all blood donations for ZIKV or utilize a FDA approved pathogen reduction technology. 2B.

1 Most

 1 ZIKV infections are asymptomatic or result in mild febrile disease with rash and conjunctivitis. Attention was recently drawn to non-vectored to-fetus transmission, and to severe forms such as Guillain-Barré syndrome,

13

  Men declared symptoms less frequently than women (45.2% vs 66.7%, Chi² test p-value=0.06) but viral load and time to declaration of symptoms were not significantly different in men and women. No significant association of viremia with ABO, Rhesus and Kell blood groups was detected.We performed seroprevalence analyses at two time points: in 418 donors sampled in early March (were tested using an anti-Zika virus NS1 IgG ELISA kit (anti-Zika Virus ELISA IgG, Euroimmun, Germany) and positives were further processed by microneutralisation in a 96-well format, using Vero cells, 50 pfu of the MRS_OPY_Martinique_PaRi_2015 strain 14 and a threshold titer of 40 as recommended by the French National Reference Centre for Arboviruses.

19 Therefore

 19 the situation is significantly different from that previously observed in Martinique during the 2014 Chikungunya outbreak, characterized by a high proportion of cases seeking medical attention 20 .

Figure 1 .

 1 Figure 1. Clinical symptoms and Zika viral loads in blood donors.The distribution of symptomatic donors according to time of inaugural symptoms (1A) and the frequency of reported symptoms (1B) are Zika viral loads (log10 copies/mL) are provided in both asymptomatic and symptomatic blood donors (1C). Zika viral load was determined by realtime RT-PCR in 41 donors reporting clinical signs and 34 who did not. The diagram reports for each series the quantitative distribution of viral load together with the median value ± SD.

Figure 2 :

 2 Figure 2: Epidemiological dynamics of Zika outbreak in MartiniqueFigure2Areports the cumulative incidence of Zika virus NAT detection in blood donors (normalized for 100 weekly blood donations). Figure2Bshows the simulated prevalence of Zika virus infections in blood donors at different time points, based on incident cases and on different hypotheses for the duration of asymptomatic viremia (4-6 days). It also presents the actual seroprevalence data obtained by seroneutralisation at weeks 8 and 22.

Conflict-of-interest disclosure: The authors declare no competing financial interests.