Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

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Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi

To cite this version:

Bernard Roynette, Pierre Vallois, Agnès Volpi. Asymptotic behavior of the hitting time, overshoot

and undershoot for some Lévy processes. ESAIM: Probability and Statistics, EDP Sciences, 2008, 12,

pp.58-93. �10.1051/ps:2007034�. �hal-00141550�

Asymptotic Behavior of the Hitting Time, Overshoot and Undershoot for some L´ evy

Processes

Bernard ROYNETTE ⁽¹⁾ , Pierre VALLOIS ⁽¹⁾ and Agn` es VOLPI ^(1),(2)

26th March 2007

(1) D´epartement de math´ematique, Institut ´ Elie Cartan,Universit´e Henri Poincar´e, BP 239, 54506 Vandœuvre-l`es-Nancy cedex, France.

(2) ESSTIN, 2 rue Jean Lamour, Parc Robert Bentz, 54500 Vandœuvre-l`es-Nancy, France.

e-mail:roynette@iecn.u-nancy.fr vallois@iecn.u-nancy.fr volpi@esstin.uhp-nancy.fr Abstract

Let (X t , t ≥ 0) be a L´evy process started at 0, with L´evy measure ν . We consider the first passage time T x of (X t , t ≥ 0) to level x > 0, and K x := X T

− x the overshoot and L x := x − X T

the undershoot. We first prove that the Laplace transform of the random triple (T x , K x , L x ) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that (f T x , K x , L x ) converges in distribution as x → ∞ , where T f x denotes a suitable renormalization of T x .

Keywords: L´evy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation.

AMS 2000 Subject classification: 60E10, 60F05, 60G17, 60G40, 60G51, 60J65, 60J75, 60J80, 60K05.

Introduction

1. Let (X _t , t ≥ 0) be a L´evy process, which is right continuous with left limits and starts at 0. Let

X t = σB t − c 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 one-

dimensional Brownian motion started at 0. (J _t , t ≥ 0) is a pure jump L´evy

process which is independent from (B _t , t ≥ 0). In addition J ₀ = 0. Recall (see for instance Theorem 2.1, Chap. 2, [13]) 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 _T

− x , (0.3)

L _x := x − X _T

. (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 com- pany 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 clas- sical or the Cram´er-Lundberg one) was initiated by F. Lundberg [16] and H.

Cram´er [4], [3]. 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 [8], 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 [17], 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 [2] 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 L´evy 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 [11] has considered the limit distribution of normalized

passage times.

(c) In both [10] and [7] analytical conditions are given to ensure that the ratio X T

/x almost surely converges to 1, as x → ∞ , in the discrete time model as well as in the case of L´evy processes.

(d) R. Doney and A. Kyprianou [6] 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 normal- ization 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 L´evy 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 ( f T _x , K _x , L _x ) with x → ∞ . In addition T f _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 ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_{< ∞}} ´

, 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 b (θ, µ, ρ, · ) of F (θ, µ, ρ, · ) :

F b (θ, µ, ρ, q) :=

Z + ∞

0

e ^{− qy} F (θ, µ, ρ, y)dy , (0.6) we proved in Theorem 2.5 that, under (H), the function F b (θ, µ, ρ, · ) solves an equation which looks like an integral equation. Then an asymptotic develop- ment 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 be- havior 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 L´evy process (X _t , t ≥ 0) with canonical decomposition (0.1), i.e. E (e ^iqX

) = e ^tψ(q) (q ∈ R ). With the L´evy-Khintchine formula, we get :

ψ(q) = − q ²

2 − icq + Z

R

(e ^iqy − 1 − iqy1l _{{| y | <1 }} )ν(dy), q ∈ R . (1.7) where ν is the L´evy measure which satisfies

Z

R

¡ y ² ∧ 1 ¢

ν(dy) < ∞ . It is well known (cf [18], example 25.12) that as soon as

Z

R | y | 1l _{{| y |≥ 1 }} ν(dy) < ∞ then

E [ | X ₁ | ] < ∞ , (1.8)

and

E (X ₁ ) = − iψ ^′ (0) = − c + Z

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 − Z

R

y1l _{{| y | <1 }} ν(dy). (1.10)

1.2 The following assumptions will be needed throughout the paper : Z ₋ 1

−∞

e ^sy ν(dy) < ∞ , ∀ s ∈ ] − ∞ , 0[ (1.11)

and Z _∞

1

e ^sy ν(dy) < ∞ , for some s > 0. (1.12) It is convenient to introduce :

r ν := sup © s ≥ 0;

Z _∞

1

e ^sy ν(dy) < ∞ ª

. (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}

) = e ^tϕ(q) , (1.14)

and

ϕ(q) = q ²

2 + cq + Z

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 +

Z

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( ∞ ) :=

x lim →∞ 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

Z _∞

1

e ^r

^y ν(dy) = ∞ then ϕ( − r _ν ) := lim

q →− r

ϕ(q) = ∞ . 3. In the case :

r _ν < ∞ and Z _∞

1

e ^r

^y ν(dy) < ∞ , (1.17) then ϕ( − r ν ) is a real number and :

ϕ( − r _ν ) = r ² _ν

2 − cr _ν + Z

R

(e ^r

^y − 1 − r _ν y1l _{{| y | <1 }} )ν (dy). (1.18) Let c ν be the real number defined as :

r _ν ²

2 − c _ν r _ν + Z

R

(e ^r

^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 and 2 for illustration).

− γ

(0) u

− r

ϕ(u)

0 = γ

(0)

a) E (X ₁ ) = − ϕ ^′ (0) < 0

− r

− γ

(0) = 0 γ

(0) u ϕ(u)

b) E (X ₁ ) = − ϕ ^′ (0) > 0

− r

u

ϕ(u)

− γ

(0) = 0 = γ

(0)

c) E (X ₁ ) = − ϕ ^′ (0) = 0 Figure 1: Graph of ϕ

0 γ ₀ ^∗ (θ)

− γ 0 (θ)

− θ

− r ν u

ϕ θ (u)

Figure 2: Graph of ϕ _θ

From the Figures 1 and 2, 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)

½ > 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)

½ < 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 sat- isfy (1.11), (1.12) and (1.21). On principle, let us consider these three conditions as as- sumption (H).

Note that, under (H), there is κ > 0 so that (1.23)-(1.26) holds.

2 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

µ 1

√ x

³

T _x + x

ϕ ^′ ( − γ ₀ (0))

´

, K _x , L _x

¶

converges in distribution when x → ∞ to the 3-dimensional law N ³

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)l − 1)1l _{{ k ≥ 0;l ≥ 0 }} ν _l (dk) dl +

Z

R

µZ ₋ y 0

(1 − e ^γ

^(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 → y − l, 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)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 .

2. a) We may observe that time and positions become asymptotically independent.

However the two components of the positions are not independent. In subsec- tion 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 _T

− X _T

is the jump size of (X _t , t ≥ 0) at T _x . It is easy to infer from Theorem 2.1 that

µ 1

√ x µ

T x + x

ϕ ^′ ( − γ ₀ (0))

¶

, X T

− X _T

¶

converges in distribution, as x → ∞ , to N µ

0; − ϕ ^′′ ( − γ ₀ (0)) ϕ ^{′ 3} ( − γ 0 (0))

¶

⊗ w e where

e

w is the probability measure on R ₊ : e

w(ds) = − 1 E (X 1 )

"

γ ₀ (0)

2 δ ₀ (ds) + e ^γ

^(0)s − 1 − γ ₀ (0)s

γ 0 (0) 1l _{{ s ≥ 0 }} ν(ds) +

Z

R

µZ ₋ y 0

(1 − e ^γ

^(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

− X _T

b−

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

e

w(ds) = − 1 E (X ₁ )

"

γ 0 (0)

2 δ ₀ (ds) + e ^γ

^(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 µ 1

√ x

³

T _x + x ϕ ^′ (0)

´

, K _x , L _x

¶

converges in distribution, as x → ∞ , to N ³

0; − ϕ ^′′ (0) ϕ ^{′ 3} (0)

´ ⊗ w ⁺ , where w ⁺ is de- fined 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) + (1 − e ^{− γ}

^(0)l )1l _{{ k ≥ 0, l ≥ 0 }} ν _l (dk) dl

¸ . (2.5) 2. When E (X 1 ) = 0, then ³ 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 s 1

ϕ ^′′ (0) by a standard Brownian motion started at 0. Let us recall that

Z _∞

0

e ^{− θx} ̺(dx) = e ⁻

q

. Moreover ϕ ^′′ (0) = 1 +

Z

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 Z

R

µZ ₋ 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 e x =

 

 

 



 

 

 

√ 1 x

³

T x + x

ϕ ^′ ( − γ ₀ (0))

´

when E (X 1 ) < 0

√ 1 x

³ 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 e _x to the associated Gaussian dis- tribution, 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).

2.2 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 ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_{< ∞}} ´

, (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 = q

c ² ₀ + 2(λ + θ) , (2.11)

F ₀ (θ, µ, ρ, x) = e ^{− (c}

^+α

^)x , (2.12)

F ₁ (θ, µ, ρ, x) = e ^{− (c}

^+α

^)x α _θ (µ − ρ + c ₀ + α _θ )

Z

[0,x]

³

e ^{( − ρ+c}

^+α

^)y − e ^{− µy} ´ ν(dy)

+ e ^{− ρx}

α _θ (µ − ρ + c ₀ − α _θ ) Z

]x, ∞ [

³ e ^{− (ρ+α}

^{− c}

^{)(y − x)} − e ^{− µ(y − x)} ´ ν(dy)

+ e ^{(µ − ρ)x} − e ^{− (c}

^+α

^)x α _θ (µ − ρ + c ₀ + α _θ )

Z

]x, ∞ [

e ^{− µy} ν(dy)

− e ^{− (c}

^+α

^)x α _θ (µ − ρ + c ₀ − α _θ )

Z _∞

0

³

e ^{− (ρ+α}

^{− c}

^)y − e ^{− µy} ´

ν (dy) , (2.13) and Λ _θ is the operator :

Λ _θ G(x) = 1 α _θ

Z _∞

−∞

ν(dy)

Z (x − y) ∧ x

−∞

e ^{− c}

^a ³

e ^{− α}

^{| a |} − e ^{− (2x − a)α}

´

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 L´evy 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 b (θ, µ, ρ, · ) of F (θ, µ, ρ, · ) :

F(θ, µ, ρ, q) = b Z _∞

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 b (θ, µ, ρ, .) verifies some kind of integral equation which remains valid when ν is a L´evy measure with no necessary finite mass.

Before stating this result, let us denote by R the operator : Rh(q) :=

Z 0

−∞

ν(dy) Z ₋ y

0

³

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 b (θ, µ, ρ, q) = 1 ϕ(q) − θ

à q − γ ^∗ ₀ (θ)

2 +

Z _∞

0

"

e ^{− (q+ρ)y} − e ^{− µy} q + ρ − µ

− e ^{− (γ}

^(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 b (θ, µ, ρ, q) is given by the following explicit formula :

F b (θ, µ, ρ, q) = 1 ϕ(q) − θ

à q − γ ₀ ^∗ (θ)

2 +

Z _∞

0

"

e ^{− (q+ρ)y} − e ^{− µy} q + ρ − µ

− e ^{− (γ}

^(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) = b 1 ϕ(q) − θ

µ 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 b (0, 0, 0, q) = Z _∞

0

e ^{− qy} P (T _y < ∞ )dy = 1

q + E (X ₁ )

ϕ(q) . (2.20) Therefore (2.20) generalizes identity (3.3) in [8].

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)x .

c) Let us briefly indicate how Theorem 2.5 enables to obtain the asymptotic behavior of ¡ e 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 b (θ, µ, ρ, · ). According to (2.17), the complex zeros of ϕ _θ are the poles of F(θ, µ, ρ, b · ). 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 7→

Z _∞

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 _{− B}

. For simplicity, this extension will be denoted ϕ .

Our last assumption on ν is :

∀ B ∈ ]0, B _ν [, ∃ K, R ₀ > 0, so that :

¯ ¯

¯ ¯ Z _∞

1

e ^{− qy} ν(dy)

¯ ¯

¯ ¯ ≤ K | q | , for any q so that

½ − 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}

:= { q ∈ C ; − B ≤ Re q ≤ β _θ } .

This set of zeros of ϕ _θ in D _{− B,β}

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 ) 6 = 0

(resp. otherwise, i.e. θ = 0 and E (X ₁ ) = 0).

The zeros of ϕ _θ which are the poles of F b (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}

^(θ)x +

X p

i=1

a _i ³

C _i (θ, µ, ρ, x)e ^{− γ}

^(θ)x + C _i (θ, µ, ρ, x)e ^{− γ}

^(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 :

x lim →∞ F (θ, µ, ρ, x)e ^γ

^(θ)x = C ₀ (θ, µ, ρ). (2.26) This property has already been reached in [2] in the particular case : θ = µ = ρ = 0 :

F (0, 0, 0, x) = P (T _x < ∞ ) ∼ C ₀ (0, 0, 0)e ^{− γ}

^(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}

^(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.

2.3 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 Z

R

| y | 1l _{{| y | >1 }} ν(dy) < ∞ and E (X ₁ ) = − c + Z

R

y1l _{{| y |≥ 1 }} ν(dy) < 0 . (2.30)

and Z _∞

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

3.1 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}

^(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 expan- sion 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 = s 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 µ θ

x ² , µ, ρ, x

¶

= E ³

e ^{− θ}

^{− µK}

^{− ρL}

1l _{{ T}

_{< ∞}} ´

∼ C ₀ µ θ

x ² , µ, ρ

¶ e ⁻

q

ϕ′′(0)

as x → ∞ ,

∼ C ₀ (0, µ, ρ) e ⁻

q

ϕ′′(0)

as x → ∞ . (3.4) (ii) We would like to demonstrate in this item that C 0 (0, µ, ρ) and e ⁻

q

are Laplace transforms of probability measures on [0, ∞ [ × [0, ∞ [ and [0, ∞ [ respectively.

As for e ⁻

q

√ 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

s 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 = −

Z y

0

(z − y)e ^{− az} dz.

Then, we obtain : C ₀ (0, µ, ρ) = 1

ϕ ^′′ (0)

· 1 − 2

Z _∞

0

e ^{− ρy} µZ _y

0

(z − y)e ^{(ρ − µ)z} dz

¶ ν (dy)

− 2 Z 0

−∞

µZ ₋ y 0

(y + b) E ¡

e ^{− µK}

^{− ρL}

1l _{{ T}

_{< ∞}} ¢ 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 Z _∞

0

e ^{− µz} ÃZ

[z, ∞ [

(y − z)e ^{− ρ(y − z)} ν (dy)

! dz

− 2 Z _∞

0

Z _∞

0

e ^{− µu} e ^{− ρz} Z 0

−∞

µZ ₋ y 0

(y + b)n(b, du, dz)db

¶ ν(dy)

¸

= Z _∞

0

Z _∞

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 ³

e ^{− θT}

^{− µK}

^{− ρL}

| 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}

^{(θ) − γ}

^(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 µ

e ⁻

T

+

(−γ0(0))

− µK

− ρL

| T _x < ∞

¶

∼ C ₀ (0, µ, ρ) C 0 (0, 0, 0) e ⁻

1 2

ϕ′′(−γ0(0)) ϕ′3(−γ0(0))

θ

as x → ∞ . (3.9) Let us observe that e ⁻

1 2

ϕ′′(−γ0(0)) ϕ′3(−γ0(0))

θ

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 +

Z _∞

0

e ^{− ( − γ}

^(0)+ρ)y − e ^{− µy}

− γ ₀ (0) + ρ − µ ν(dy)

− Z _∞

0

e ^{− ρy} − e ^{− µy}

ρ − µ ν(dy) + RF (0, µ, ρ, .)( − γ ₀ (0))

¸

. (3.10)

It follows : C 0 (θ, µ, ρ)

C ₀ (0, 0, 0) = 1

C ₀ (0, 0, 0)ϕ ^′ ( − γ ₀ (0))

·

− γ 0 (0)

2 +

Z _∞

0

e ^{− µy} µZ y

0 − e ^{− ( − γ}

^{(0)+ρ − µ)z} dz

¶ ν(dy)

− Z _∞

0

e ^{− µy} µZ y

0

e ^{− (ρ − µ)z} dz

¶

ν(dy) + RF (0, µ, ρ, .)( − γ ₀ (0))

¸ .

(3.11)

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 −

Z _∞

0

(e ^γ

^(0)z − 1)e ^{− ρz} ÃZ

[z; ∞ [

e ^{− µ(y − z)} ν(dy)

! dz +

Z 0

−∞

µZ ₋ y 0

(e ^γ

^(0)z − 1) E ¡

e ^{− µK}

^{− ρL}

1l _{{ T}

_{< ∞}} ¢ 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 −

Z _∞

0

(e ^γ

^(0)z − 1)e ^{− ρz} ÃZ

[z; ∞ [

e ^{− µ(y − z)} ν (dy)

! dz +

Z _∞

0

Z _∞

0

e ^{− µu} e ^{− ρz} Z ₀

−∞

µZ _{− y}

0

(e ^γ

^(0)z − 1) P (T _b < ∞ )n(b, du, dz)db

¶ ν (dy)

¸

= Z _∞

0

Z _∞

0

e ^{− µu} e ^{− ρz} w ⁻ (du, dz) . (3.13)

with w ⁻ defined in (2.1).

⊓

⊔

3.2 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 , expo- nentially distributed with parameter ν( R ) (cf [18], theorem 21.1). The process (X t+τ

− X τ

; t ≥ 0) is again a L´evy process distributed as (X t , t ≥ 0). This prop- erty 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

(B _t − c ₀ t) > x,

• T _x = τ ₁ if sup

0 ≤ t ≤ τ

(B _t − c ₀ t) < x and J _τ

+ B _τ

− c ₀ τ ₁ > x,

• T _x > τ ₁ otherwise. Then, conditionally on { T _x > τ ₁ } , T _x − τ ₁ is distributed as T b _{x − X}

where ( T b _x ; x > 0) is an independent copy of (T _x ; x > 0). ( T b _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 ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_<τ

_} ´ + E ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_=τ

_} ´ + E ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ τ}

_<T

_{< ∞}} ´

. (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 ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_<τ

_} ´

= e ^{− (c}

^+α

^)x . (3.15) Proof of Lemma 3.1

Let ( B e _t , t ≥ 0) be the Brownian motion with drift − c ₀ :

B e _t := B _t − c ₀ t ∀ t ≥ 0 . (3.16) We set T e _x := inf { t ≥ 0; B e _t > x } , x ≥ 0.

Then, { T _x < τ ₁ } = { T e _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 e x , we have :

E ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_<τ

_} ´

= E ³

e ^{− (λ+θ) T e}

´

. (3.17)

According to ([12], exercise 5.10 page 197) we can conclude that (3.15) holds. ⊓ ⊔ Lemma 3.2 We have :

E ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_=τ

_} ´

= e ^{− (c}

^+α

^)x α _θ (µ − ρ + c ₀ + α _θ )

Z

[0,x]

³ e ^{( − ρ+c}

^+α

^)y − e ^{− µy} ´ ν (dy)

+ e ^{− ρx}

α _θ (µ − ρ + c 0 − α _θ ) Z

]x, ∞ [

³ e ^{− (ρ+α}

^{− c}

^{)(y − x)} − e ^{− µ(y − x)} ´ ν(dy)

+ e ^{(µ − ρ)x} − e ^{− (c}

^+α

^)x α _θ (µ − ρ + c ₀ + α _θ )

Z

]x, ∞ [

e ^{− µy} ν(dy)

− e ^{− (c}

^+α

^)x α _θ (µ − ρ + c ₀ − α _θ )

Z _∞

0

³

e ^{− (ρ+α}

^{− c}

^)y − e ^{− µy} ´

ν (dy) .

(3.18) Proof of Lemma 3.2

Let us write : Y ₁ := J _τ

. We observe that on { T _x = τ ₁ } , Y ₁ > 0. Morever : { T _x = τ ₁ } = { sup

t ≤ τ

B e _t < x, B e _τ

+ Y ₁ > x } , (3.19) and

K _x = B e _τ

+ Y ₁ − x , L _x = x − B e _τ

, (3.20)

where ( B e _t , t ≥ 0) is defined by the relation (3.16). Conditioning by (τ ₁ , Y ₁ ) implies that :

∆ := E ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ T}

_=τ

_} ´

=e ^{− ρx} Z _∞

0

dt e ^{− (λ+θ)t} Z _∞

0

ν (dy) E ³

e ^{− (µ − ρ) B e}

^{− µ(y − x)} 1l

{ sup

B e

<x ; x − y ≤ B e

}

´

(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 ([12] page 95), i.e. :

P µ

B _t ∈ da; sup

u ≤ t

B _u ∈ db

¶

= 2(2b − a)

√ 2πt ³ e ⁻

1l _{{ a<b; b>0 }} dadb . (3.22) Let us apply Girsanov’s formula :

P µ

B e _t ∈ da; sup

u ≤ t

B e _u ∈ db

¶

= 2(2b − a)

√ 2πt ³ e ^{− c}

^{a −}

20

2

t e ⁻

1l _{{ a<b; b>0 }} dadb . (3.23) Combining (3.23) and (3.21) leads to :

∆ = e ^{(µ − ρ)x} Z _∞

0

ν(dy) e ^{− µy} Z x

x − y

da e ^{− (c}

^{+µ − ρ)a} Z x

a ∨ 0

db (2b − a) Z _∞

0

√ 2

2πt ³ e ⁻

1 2

(2(λ+θ)+c

)t+

dt . (3.24) Let us recall the classical identities (cf. [14] p 118, or [9] sections 8.432 6 page 959, and 8.469 3 page 967) :

K

2

(δ) := 1 2

Z _∞

0

√ 1

t e ⁻

^(t+

⁾ dt = r π

2δ e ^{− δ} ∀ δ > 0 , (3.25)

and Z _∞

0

√ 1

t ³ e ⁻

^(βt+

⁾ dt = r 2π

γ e ^{− √ βγ} ∀ β > 0, ∀ γ > 0 . (3.26) obtained by derivation and the changing of variable t → q

β γ t.

This allows to first compute explicitly the integral with respect to dt in (3.24) :

∆ = 2e ^{(µ − ρ)x} Z _∞

0

ν(dy) e ^{− µy} Z x

x − y

da e ^{− (c}

^{+µ − ρ)a} Z x

a ∨ 0

e ^{− α}

^{(2b − a)} db

(3.27)

In a second step we evaluate the integral with respect to db :

∆ = e ^{(µ − ρ)x} α _θ

Z _∞

0

ν(dy) e ^{− µy} Z _x

x − y

da e ^{− (c}

^{+µ − ρ)a} ³

e ^{− α}

^{| a |} − e ^{− α}

^{(2x − a)} ´

. (3.28) Let us introduce two cases x − y ≥ 0 and x − y < 0 :

∆ = e ^{(µ − ρ)x} α _θ

Z

[0,x]

ν(dy) e ^{− µy} Z x

x − y

e ^{− (c}

^{+µ − ρ+α}

^)a da

+ e ^{(µ − ρ)x} α θ

Z

]x, ∞ [

ν(dy) e ^{− µy}

·Z 0 x − y

e ^{− (c}

^{+µ − ρ − α}

^)a da + Z x

0

e ^{− (c}

^{+µ − ρ+α}

^)a da

¸

− e ^{( − 2α}

^{+µ − ρ)x} α _θ

Z _∞

0

ν(dy) e ^{− µy} Z _x

x − y

e ^{− (c}

^{+µ − ρ − α}

^)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 :

E ³

e ^{− θT}

^{− µK}

^{− ρL}

1l _{{ τ}

_<T

_{< ∞}} ´

= 1

α _θ Z _∞

−∞

ν(dy)

Z _{(x − y) ∧ x}

−∞

e ^{− c}

^a ³

e ^{− α}

^{| a |} − e ^{(2x − a)α}

´

F (θ, µ, ρ, x − a − y)da . (3.30) Morever :

e ^{− α}

^{| a |} − e ^{− (2x − a)α}

≥ 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.

3.3 Proof of Theorem 2.5

Assume that ν satisfies the assumptions given in Theorem 2.5. The proof of Theo- rem 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 b (θ, µ, ρ, q) = F c ₀ (θ, µ, ρ, q) + F c ₁ (θ, µ, ρ, q) + Λ d _θ F (θ, µ, ρ, .)(q) q ∈ D ₀ , (3.33) with D ₀ := { z ∈ C , Re(z) > 0 } .

The identity (2.12) implies :

c F ₀ (θ, µ, ρ, q) = 1

c + α _θ + q , (3.34)

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

Z _∞

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) :=

Z _∞

0

e ^{− (q+c+α}

^)x α _θ (µ + c + α _θ − ρ)

ÃZ

[0,x]

³

e ^(c+α

^{− ρ)y} − e ^{− µy} ´ ν(dy)

! dx

= 1

α _θ (q + c + α _θ )(µ + c + α _θ − ρ) Z _∞

0

³

e ^{− (q+ρ)y} − e ^{− (q+µ+c+α}

^)y ´ ν (dy) , (3.36)

I ₂ (θ, µ, ρ, q) :=

Z _∞

0

e ^{− (q+ρ)x} α _θ (µ + c − α _θ − ρ)

ÃZ

]x, ∞ [

³

e ^{− (ρ+α}

^{− c)(y − x)} − e ^{− µ(y − x)} ´ ν(dy)

! dx

= 1

α _θ (µ + c − α _θ − ρ) Z _∞

0

à e ^{− (ρ+α}

^{− c)y} − e ^{− (q+ρ)y}

q + c − α _θ + e ^{− (q+ρ)y} − e ^{− µy} q + ρ − µ

!

ν(dy) ,

(3.37)

I ₃ (θ, µ, ρ, q) :=

Z _∞

0

e ^{− (q+ρ − µ)x} − e ^{− (q+c+α}

^)x α _θ (µ − ρ + c + α _θ )

ÃZ

]x, ∞ [

e ^{− µy} ν (dy)

! dx

= − 1

α _θ (µ − ρ + c + α _θ ) Z _∞

0

ν(dy)

à e ^{− (q+ρ)y} − e ^{− µy}

q + ρ − µ − e ^{− (q+µ+c+α}

^)y − e ^{− µy} q + c + α _θ

! , (3.38)

I ₄ (θ, µ, ρ, q) := − Z _∞

0

e ^{− (q+c+α}

^)x

α _θ ( − ρ + µ + c − α _θ ) dx ³

ν c ⁺ (ρ + α _θ − c) − ν c ⁺ (µ) ´

= ν c ⁺ (µ) − ν c ⁺ (ρ + α _θ − c) α _θ (q + c + α _θ )( − ρ + µ + c − α _θ ) ,

(3.39) and ν c ⁺ denotes the Laplace transform of ν _|

∞]

: ν c ⁺ (q) := ν b _|

∞]

(q) = Z _∞

0

e ^{− qy} ν(dy), q ∈ ] − r _ν , ∞ [ . (3.40) Therefore :

F c ₁ (θ, µ, ρ, q) = − ν c ⁺ (q + ρ)

α _θ (q + c + α _θ )(q + ρ − µ) + ν c ⁺ (q + ρ)

α _θ (q + c − α _θ )(q + ρ − µ) + 2 ν c ⁺ (ρ + α _θ − c)

(µ + c − α _θ − ρ)(q + c − α _θ )(q + c + α _θ )

− 2 ν c ⁺ (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 c ₁ (θ, µ, ρ, q) = 2 C _θ (q)

à ν c ⁺ (q + ρ) − ν c ⁺ (µ)

q + ρ − µ − ν c ⁺ (µ) − ν c ⁺ (ρ + α _θ − c) µ − ρ + c − α _θ

!

. (3.43)