Path decompositions for real Levy processes

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PATH DECOMPOSITIONS FOR REAL LEVY PROCESSES

Thomas DUQUESNE

Université Paris-Sud, 91405 Orsay cedex, France Received 7 June 2001, revised 22 January 2002

ABSTRACT. – LetX be a real Lévy process and let X^↑ be the process conditioned to stay positive. We assume that 0 is regular for (−∞,0) and (0,+∞) with respect to X. Using elementary excursion theory arguments, we provide a simple probabilistic description of the reversed paths of X and X^↑ at their first hitting time of (x,+∞)and last passage time of (−∞, x], on a fixed time interval[0, t], for a positive levelx. From these reversion formulas, we derive an extension to general Lévy processes of Williams’ decomposition theorems, Bismut’s decomposition of the excursion above the infimum and also several relations involving the reversed excursion under the maximum.

2003 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – SoitXun processus de Lévy etX^↑le même processus conditionné à rester positif.

On suppose que 0 est régulier pour (−∞,0)et(0,+∞)par rapport àX. Par des arguments simples de théorie des excursions, nous décomposons la loi des trajectoires deXetX^↑retournées aux temps d’entrée de (x,+∞)et de sortie de (−∞, x]. De ces formules de reversion, on déduit une extension au cas des processus de Lévy généraux, des théorèmes de décomposition de Williams, du théorème de décomposition de Bismut de l’excursion au dessus du minimum, ainsi que plusieurs relations faisant intervenir l’excursion sous le maximum retournée.

2003 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

Let(Xt)t0be a real Lévy process, that is a real valued process with homogeneous and independent increments. The supremum (respectively infimum) of X on the time interval[0, t]is denoted bySt (respectivelyIt). We assume that 0 is regular for(0,+∞) and(−∞,0)with respect toX. A classical result says thatX−I (respectivelyX−S) is a strong Markov process for which 0 is regular (see Bingham [5] or Bertoin [4]

Chapter 6 for a proof). Let us denote byL(respectivelyL^∗) the local time at 0 ofX−I (respectivelyX−S): they are uniquely defined up to a multiplicative constant and their normalization is specified in Proposition 2.3.

As Rogers noticed in [17], as soon as the Lévy measure charges the positive numbers, X−Smay hit zero byXjumping across the level of its previous maximum. The classical

E-mail address: [email protected] (T. Duquesne).

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Itô excursion measure ofS−X loses the information about this jump. Let us introduce the relevant definition of the excursion measure under the maximum (respectively above the infimum) denoted byN^∗(respectivelyN). It has the property to record the final jump of the excursion, which represents the amount the excursion overshoots whenXattains a new maximum (respectively infimum). Let(gi, di), i∈I (respectively(gj, dj), j∈I^∗) the excursion intervals ofX−I(respectivelyS−X) above 0. We define the excursions above the infimum and under the supremum by

ωⁱ(s)=X(g_i+s)∧d_i−Xg_i, i∈I, ω^j(s)=X(gj+s)∧dj−Xgj, j ∈I^∗. Then, the point measures

i∈I

δ_(L_gi_,ωi) and

j∈I^∗

δ_(L∗ gj,ω^j)

are distributed respectively as 1_{lη}N(dl dω)and 1_{lη^∗}N^∗(dl dω), whereN and N^∗ are Poisson measures with respective intensitiesdl N (dω)anddl N^∗(dω), and where

η=inft0: N[0, t] × {ζ (ω)= ∞}1, η^∗=inft0: N^∗[0, t] × {ζ (ω)= ∞}1

(ζ (ω)being the lifetime of the pathω). The random variables ηand η^∗ have the same law as respectivelyL_∞andL^∗_∞, that are exponentially distributed or infinite a.s.

In Section 3, Theorem 3.3 provides a decomposition of the law of the excursion under the supremum reversed at its final jump: More precisely, we decompose the law of (ωζ (ω)−ω(ζ (ω)−s)−;0sζ (ω))underN^∗(·{ωζ>0}), in terms of the law ofXand its Lévy measure. Theorems 3.2 and 3.1 give similar results for(Xτx−X(τx−s)−;0s τx)under P(· |Xτ_x > x)and(Xσ_x(t )−X(σ_x(t )−s)−;0sσx(t))under P(· |Xσ_x(t )> x), where we have set for anyx, t >0:

τx=infs0: Xs> x and σx(t)=sups∈ [0, t]: Xsx.

Williams in [19], and many authors after him, explored the connections between the Brownian motion, the three-dimensional Bessel process and the Brownian excursion (see for instance Pitman [16] and Bismut [6]). Many of these identities in the Brownian case hold in a more general setting for totally asymetric Lévy processes: see Bertoin [2] for a generalized Pitman theorem for spectrally negative Lévy processes and Chaumont [7–

9] for Williams’ theorems and Bismut’s decomposition in the spectrally positive case.

Let us mention that Chaumont has also explored the stable case in detail in [10]

and [7], providing several path-constructions and identities concerning the stable meander, the normalised excursion and the stable bridge. In these results the role of the three-dimensional Bessel process is played by the Lévy process conditioned to stay positive. This process, denoted by X^↑, has been introduced by Bertoin in a general setting (see [3]). Bertoin’s construction of X^↑ is recalled in Section 2.2. We use it in combination with Theorems 3.2 and 3.1 to get in Section 4.1 the generalized

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first Williams’ decomposition theorem, then Bismut’s decomposition of the excursion above the infimum in Section 4.2 and the second Williams’ decomposition theorem in Section 4.3.

Let us explain more precisely these results: For anyt >0, we define U_t^∗=X_L∗−1

t if

L^∗_∞> tandU_t^∗= +∞if not. The process(U_t^∗;t0)is a subordinator (see Bertoin [4], Chapter 6) and its drift coefficient is denoted by d^∗. A classical result due to Kesten (see [11]) ensures that P(Xτ_x =x) >0 iffd^∗>0. We assume thatd^∗>0 and that X does not drift to −∞. Then, we can show that σ_x^↑=sup{s 0: X_s^↑x} is finite a.s.

Theorem 4.2 show that

PX^↑

σ_x^↑=x=P(Xτx =x)

and that(x−X(τx−s)−;0sτx)under P(· |Xτx =x)has the same law as(X^↑_s;0 sσ_x^↑)under P(· |X^↑

σ_x^↑=x).

We also prove in Theorem 4.5 a path decomposition of the excursion above the infimum similar to Bismut’s decomposition of the Brownian excursion: we show that for any nonnegative measurable functionalsGandDon the space of càdlàg paths with a finite lifetime and for any nonnegative measurable functionf,

N ζ (ω)

0

dt G(ωs;0st)f (ωt)D(ωt+s;0sζ (ω)−t)

=

+∞

0

dx f (x)u^∗(x)EG(X_s^↑;0sσ_x^↑)|X^↑

σ_x^↑=xED(Xs;0sτ₋x), where τ₋x =inf{s 0: Xs <−x} and where u^∗ is the co-excessive version of the density of the potential measure associated with the subordinatorU^∗.

Section 4.3 is devoted to the proof of Theorem 4.10 that can be seen as an analogue for general Lévy processes of the second Williams’ decomposition theorem that originally concerns the Brownian excursion split at its maximum. Let us describe our result: For anyx >0, we setτ_x^↑=inf{s0: X^↑_s > x}. Proposition 4.7 shows that

PX^↑

τ_x^↑=x>0 iff d^∗>0.

Let us denote by X^↓ the process X conditioned to stay negative (that is defined in Section 2.2); we write g(ω) the instant when the excursion ω attains its maximum.

Theorem 4.10 shows the law ofωg(ω)underNadmits a density with respect to Lebesgue measure that we specify. Under N (· |ωg(ω)=x), the processes (ωs;0sg(ω))and (ωs+g(ω);0sζ (ω)−g(ω))are mutually independent. Furthermore,

– the process(ωs;0sg(ω))is distributed as(X_s^↑;0sτ_x^↑)under P(· |X^↑

τ_x^↑= x);

– the law of(ωs+g(ω);0sζ (ω)−g(ω))is absolutely continuous with respect to the law of(X^↓_s;0sτ₋^↓x)(with an evident notation forτ₋^↓x) and the corresponding density has the formϕ(X^↓

τ_−x^↓ ), where the functionϕis specified.

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Let us mention that we provide two other path decompositions that concern the excursion above the infimum (Theorem 4.6) and the process (X^↑_s;0s σ_x^↑) when X^↑

σ_x^↑> x(Theorem 4.1).

2. Preliminary results 2.1. Notation and basic assumptions

In this section we state our notation and the assumptions made at different stages of the paper. We also recall fondamental results of fluctuation theory that are our starting-point and we give some simple facts concerning excursion theory applied to Lévy processes, that is the main tool we use.

We begin with some notations concerning the canonical space. Let be the space of right-continuous functions with left limits from (0,+∞) toR(the so-called càdlàg functions space) endowed with the Skorokhod’s topology. Let F stand for its Borelσ- algebra. For any pathωin we define its lifetimeζ (ω)by inf{t0:ω(s)=ω(t),∀s t}, with the usual convention inf∅ = ∞. For any timet0, we denote the jump ofωat t by!ω(t)=ω(t)−ω(t−); we also define the path respectively stopped att, stopped just beforet, reversed attand reversed just beforet, by

ω(· ∧t)=ω(s∧t);s0, ω(· ∧t−)=ω(s∧t)−!ω(t)1_[t,+∞)(s);s0, ˆ

ω^t=ω(t)−ω(t−s)−;s0, ωˆ^t⁻= ˆω^t−!ω(t),

with the convention ω(s−)=ω(0)for any non-positive real number s. When ζ (ω)is finite,ωˆ^{ζ (ω)} is well defined and simply denoted byω. We use a non-standard notationˆ for the shifted path at timetdefined by

ω◦θt=ω(s+t)−ω(t);s0.

For anyx >0, we denote byτx(ω)and τ₋x(ω) the first hitting time of respectively (x,+∞)and(−∞,−x):

τx(ω)=infs >0: ω(s) > x, τ₋x(ω)=infs >0: ω(s) <−x

(with the usual convention inf∅ = +∞). For any timet >0, we also denote byσx(t, ω) andσ₋x(t, ω)the last passage time in respectively(−∞, x]and[−x,+∞)on the time interval[0, t]:

σx(t, ω)=sup0st: ω(s)x, σ₋x(t, ω)=sup0st: ω(s)−x (with the convention sup∅ = +∞). We writeσx(ω)=limt→+∞σx(t, ω), the limit being taken in [0,+∞]. Next, we denote respectively by g_t(ω)and g_t(ω), the last infimum time and the last supremum time ofωbeforet:

g_t(ω)=sups∈ [0, t): inf

[0,t]ω=ω(s−)∧ω(s)

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and

g_t(ω)=sups∈ [0, t): sup

[0,t]ω=ω(s−)∨ω(s). We also write g(ω)=limt→+∞g

t(ω) and g(ω)=limt→+∞g_t(ω) (note that these quantities may be infinite).

We denote by X the canonical process on : Xt(ω)=ω(t) and we consider the probability measure P on ( ,F) under which X is a Lévy process started at 0, with characteristic exponentψ:

Ee^iλX^t=e⁻^{t ψ(λ)}, t0, λ∈R. By the Lévy–Khintchine theorem,ψ has the form

ψ(λ)=iaλ+bλ²+ π(dr)1−e^iλr+iλr1_{|r|<1}

, λ∈R,

whereais a real number,bis non-negative and the Lévy measureπis a Radon measure onRnot charging 0, which satisfies

π(dr)1∧ |r|²<+∞.

IfJ = {s0: !Xs=0}, then the point measureN(ds dr)=s∈Jδ(s,!X_s)is a Poisson measure with intensityds π(dr).

Let us recall some path-properties of Lévy processes. For anyt0, we have X^{t (}^law)= (Xs;0st)

(see Bertoin [4]). This identity is refered to as the “duality property”.

In the whole paper (Section 3 excepted), we make the following assumption:

Assumption (A). – The point 0 is regular for(0,+∞)and for(−∞,0) with respect toX.

(In particular X cannot be a subordinator or a compound Poisson process.) As a consequence of (A), we recall the following result (see Millar [15]): For any t 0, the Lévy process X reaches its infimum (respectively supremum) on[0, t] at a unique instant that must beg_t(X)(respectivelyg_t(X)).

For everyt0, we write

St = sup

s∈[0,t]Xs, It= inf

s∈[0,t]Xs.

It is well-known that X−S and X−I are strong Markov processes (see Bertoin [4], Chapter 6). Assumption (A) implies that 0 is regular for itself with respect to both these processes. Rogers has shown in [17] that this implies

P∃t∈(0,+∞): Xt−=It−< Xt

=0 (1)

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and

P∃t∈(0,+∞): Xt−=St−< St

=0. (2) Let us recall briefly the proof: we only need to show for anyε >0

E

s∈J

1_{X_s−=I_s−}1(ε,+∞)(!Xs)

=0.

Apply the compensation formula (see Bertoin [4], p. 7) to get

E

s∈J

1_{X_s−=I_s−}1(ε,+∞)(!Xs)=π(ε,+∞)

+∞

0

dsP(Xs=Is).

But the duality property implies for anys >0, P(Xs=Is)=P(Ss=0)=0, because 0 is regular for(0,+∞). A similar argument proves (2). ✷

We denote the local times ofX−I andX−Sat the level 0 by(Lt)t0and(L^∗_t)t0. They are uniquely determined up to a multiplicative constant specified in a forthcoming lemma. The limit in[0,+∞]ofLt (respectively L^∗_t) whentgoes to infinity is denoted byL_∞(respectivelyL^∗_∞). The quantityL_∞(respectivelyL^∗_∞) is a.s. finite or a.s. infinite according asXdrifts or not to+∞(respectively−∞). IfL_∞(respectivelyL^∗_∞) is finite a.s., then it is exponentially distributed with parameter denoted byp(respectively p^∗).

Eq. (1) and the dual result P(∃t ∈(0,+∞): Xt− =St−> Xt)=0 imply that P- a.s. the sets {s 0: Xs > Is} and {s 0: Xs < Ss} are open sets (we have denoted (gi, di), i∈Iand(gj, dj), j∈I^∗their respective connected components). Let m denote the Lebesgue measure on R. The duality property and Assumption (A) imply that m(s0: Xs=Ss)=m(s0: Xs =Is)=0. Thus,

P-a.s. m

R\

i∈I

(gi, di)

=m

R\

j∈I^∗

(gj, dj)

=0. (3)

Let N and N^∗ be the excursion measures of X above its infimum and under its supremum as defined in the first section. Observe that as soon as the Lévy measure π charges(−∞,0)(respectively(0,+∞)), the set of excursionsωending with a negative jump (respectively positive jump) has a positiveN-measure (respectivelyN^∗-measure).

But thanks to (1) and the dual result, we see that excursions above the infimum and under the supremum leave 0 continuously.

Let(L⁻¹_t )t0and(L^∗−1_t )t0be the right-continuous inverses ofLandL^∗: L⁻_t ¹=inf{s0: Ls> t}, L^∗−_t ¹=inf{s0: L^∗_s > t} (with the convention inf∅ = ∞). Recall that P-a.s.

s0

(L⁻_s₋¹, L⁻_s¹)=

i∈I

(gi, di) and

s0

(L^∗−_s₋¹, L^∗−_s ¹)=

j∈I^∗

(gj, dj). (4)

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For any t0 we define Ut = −X_L−1

t if L_∞> t and Ut = +∞if not. In a similar way, we define U_t^∗=X_L∗−1

t if L^∗_∞> t and U_t^∗= +∞ if not. The processes (L⁻¹, U ) and(L^∗−¹, U^∗)are called the ladder processes. They are two-dimensional subordinators killed at respective ratespandp^∗; their bivariate Laplace exponents are denoted by

κ(α, β)= −log Eexp−αL⁻₁¹−βU1

and κ^∗(α, β)= −log Eexp−αL^∗−₁ ¹−βU₁^∗

(see Bertoin [4], Chapter 6 for a detailed account). Next, we define the two potential measuresU andU^∗associated withUandU^∗:

RU(dx)f (x)=E₀^L^∞dv f (Uv),

RU^∗(dx)f (x)=E₀^L^∗^∞du f (U_u^∗).

Let d^∗ be the drift coefficient of the subordinator U^∗: d^∗ =limβ→+∞κ^∗(0, β)/β.

We recall the following result, due to Kesten [11] (see also Bertoin [4], Chapter 3, Theorem 5): assume thatd^∗is positive, and letu^∗:(−∞,+∞)−→ [0,+∞)be the co- excessive version of the density ofU^∗. Thenu^∗is continuous and positive on(0,+∞), u^∗(0⁺)=1/d^∗, and

P(Xτx =x)=d^∗u^∗(x), x >0,

where for convenience we write τx instead of τx(X). We prove the following simple lemma that will be used in Section 4.

LEMMA 2.1. – Assume (A) and suppose thatd^∗is positive. Then for any nonnegative measurable functionalF on ,

E ^L^∗_∞

0

du F (X_·∧_L∗−1

u )

=

+∞

0

dx u^∗(x)EF (X_·∧τx)|Xτx =x.

Proof. – SetA=₀^+∞dxE[F (X·∧τ_x);Xτ_x =x]and for any positive numberxdefine Hx=L^∗_τ_x. Thanks to (A), we check path by path that

P-a.s. on {τx<∞}, L^∗−_H_x¹=τx. Thus

A=

+∞

0

dxEF (X_·∧_L∗−1

Hx);Xτx =x.

Denote byC the random set{x0: Xτx =x}and define the measuresµandνby µ(dx)=1C(x)m(dx),

ν(du)f (u)=µ(dx)f (Hx).

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Then,

A=E

[0,S_∞)

µ(dx)F (X_·∧_L∗−1 Hx)

=E

[0,L^∗_∞)

ν(du)F (X_·∧_L∗−1

u )

. (5)

For any positive numbera,

ν([0, a])=m{x0: Xτ_x =x;Hxa}

=m{x0: Xτx =x;τxL^∗−_a ¹}=m(C∩ [0, U_a^∗]).

Let us first consider the case U_a^∗ <∞: If there exists some s in [0, a] such that x∈(U_s^∗₋, U_s^∗), then,τx=L^∗−_s ¹andU_s^∗=Xτx > x. Thus,

0sa

(U_s^∗₋, U_s^∗)⊂C^c∩ [0, U_a^∗].

Letx be inC^c∩ [0, U_a^∗]. Then,Xτ_x > x. By (2), it follows thatτxmust be the end-point of some excursion interval ofS−X above 0 that is included in[0, L^∗−_a ¹]. Then, by (4) there exists somesin[0, a]such that

L^∗−_s₋¹< L^∗−_s ¹=τx and Sτ_x−=U_s^∗₋x < U_s^∗=Xτ_x. Hence,

C^c∩ [0, U_a^∗] ⊂

0sa

[U_s^∗₋, U_s^∗).

By combining this with the previous inclusion we get m(C^c∩ [0, U_a^∗])=0sa!U_s^∗. But the Lévy–Itô representation ofU^∗guarantees that P-a.s.

U_a^∗=d^∗a+

0sa

!U_s^∗, 0a < L^∗_∞.

Then, P-a.s. for everyain[0, L^∗_∞),ν([0, a])=d^∗a. Next, observe that for anya > L^∗_∞, ν([0, a])=m(C)=d^∗L^∗_∞. Thus, P-a.s.

ν(dx)=d^∗1_[0,L^∗_∞)(x)m(dx).

The desired result follows from (5) and the identity P(Xτx =x)=d^∗u^∗(x). ✷

Let us introduce some notations: for any positive time t and any path ω, we denote the pre-infimum and the post-infimum path on the interval[0, t]by:



 ω^t

← =ω(· ∧g_t(ω)), ω^t

→ =(ω◦θg

t(ω))(· ∧(t−g

t(ω))).

We also denote the pre-supremum and post-supremum processes on [0, t] by ←ω−^t and−→ω^t:

←ω−^t =ω(· ∧g_t(ω)),

−

→ω^t =(ω◦θg_t(ω))(· ∧(t−g_t(ω))).

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We often use the following lemma in Section 4:

LEMMA 2.2. – Assume (A). LetT be independent ofXand exponentially distributed with parameter α >0. Then,←X^Tand→X ^T are mutually independent and the following identities hold for any nonnegative measurable functional F on :

(i) E[F ( X←^T)] =κ(α,0)E[₀^L^∞dv e⁻^αL⁻¹^v F (X_·∧_L−1 v )].

(ii) E[F ( X→^T)] =_κ(α,0)¹ N (₀^ζds αe⁻^αsF (ω_·∧s)).

Proof. – LetGbe any nonnegative measurable functional on . We have EF←X^T

G→X ^T =

∞

0

dt αe⁻^αtEF←X^t

G→X^t .

By (3) and by the definition of the excursions above the infimum, we have P-a.s.

∞

0

dt αe⁻^αtF←X^t G→X^t

=

i∈I

e⁻^αgⁱF (X_·∧gi)

ζi

0

ds αe⁻^αsGωⁱ(· ∧s).

Apply the compensation formula to get

EF←X^T

G→X ^T =E

L_∞ 0

dv e⁻^αL⁻^v¹F (X_·∧_L−1 v )

N

ζ 0

ds αe⁻^αsG(ω_·∧s)

and the following identities yield (i) and (ii):

N ^ζ

0

ds αe⁻^αs

=N1−e⁻^αζ=κ(α,0)

and

E L_∞

0

dv e⁻^αL⁻¹^v

= 1

κ(α,0). ✷

We now specify the normalization ofLandL^∗thanks to the following proposition.

PROPOSITION 2.3. – Assume (A). Fix the normalization ofL. Then, the normaliza- tion ofL^∗can be chosen in order to have for any nonnegative measurable functionalF on

N ζ

0

dsF (ωˆ^s)

=E ^L^∗_∞

0

du F (X_·∧_L∗−1

u )

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and the dual identity

N^∗ ζ

0

dsFωˆ^s

=E L_∞

0

dvF (X_·∧_L−1 v )

.

Proof. – We denote by←X^T the path←X^T reversed at its lifetimeg_T. Observe that X

←^T =X^T ◦θ_g

T(X^T)

(we use the fact that the minimum of Xover [0, T]is attained P-a.s. at a unique time).

We also denote by→X^T the path→X ^T reversed at its lifetimeT −g

T. Similarly, we see that

X

→^T =X^T_·∧

g_T(X^T). The duality property implies that

←X^T,→X^T (law)

= −→X^T,←X−^T.

LetGbe any nonnegative measurable functional on . Use Lemma 2.2 to get:

EF ←X^T

G →X ^T =E

L_∞ 0

dv e⁻^αL⁻^v¹F X^L⁻^v¹

N ζ

0

ds αe⁻^αsGωˆ^s

.

On the other hand, by replacingXwith−X, we see that Lemma 2.2 also implies

EF−→X^TG←X−^T=E ^L^∗_∞

0

du e⁻^αL^∗−^u ¹G(X_·∧_L∗−1

u )

N^∗

ζ 0

ds αe⁻^αsF (ω_·∧s)

.

Thus, for anyα >0 E

^L^∗_∞

0

du e⁻^αL^∗−^u ¹G(X_·∧_L∗−1

u )

N^∗

ζ 0

ds e⁻^αsF (ω_·∧s)

=E L_∞

0

dv e⁻^αL⁻^v¹F X^L⁻^v¹

N ζ

0

ds e⁻^αsGωˆ^s

. (6)

By lettingα go to 0, we see that the ratio

N^∗(₀^ζds F (ω_·∧s)) E[₀^L^∞dv F (X^L⁻¹^v )]

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does not depend onF, provided it is well-defined (that is the denominator is positive and finite). Furthermore this ratio coincides with

N (₀^ζds G(ω^s)) E[₀^L^∗^∞du G(X_·∧_L∗−1

u )]

for any G such that the denominator is positive and finite. We can choose the normalization ofN^∗, or equivalentlyL^∗, so that both ratios are equal to 1. ✷

In the spectrally positive case, the first identity of Proposition 2.3 has been proved by Le Gall and Le Jan in [12] by a different method.

Immediate applications of Proposition 2.3 are the following identities due to Silverstein (see [18]), also mentioned in Rogers’ paper [17]:

N₀^ζds e⁻^αs⁻^βω^s=1/κ^∗(α, β), N^∗₀^ζds e⁻^αs⁺^βω^s=1/κ(α, β).

We can also derive from Proposition 2.3 the Wiener–Hopf factorisation of the ladder exponents:

κ(α, iβ)κ^∗(α,−iβ)=α+ψ(β).

Indeed, (3) gives the following decomposition:

+∞

0

dt e⁻^αt⁺^iβX^t =

i∈I di

gi

dt e⁻^αt⁺^iβX^t =

i∈I

e⁻^αgⁱ⁺^iβX^gi

ζi

0

ds e⁻^αs⁺^iβωⁱ^(s).

Taking the expectations and using the compensation formula, we get

1/α+ψ(β)=E L_∞

0

du e⁻^αL

−1 u +iβX

L−1 u

N

ζ 0

ds e⁻^αs⁺^iβω^s

,

which yields the Wiener–Hopf factorization thanks to Proposition 2.3.

2.2. The Lévy process conditioned to stay positive or negative

We introduce now the process conditioned to stay positive, respectively negative, denoted byX^↑, respectivelyX^↓. Bertoin in [3] provides a pathwise construction ofX^↑ and X^↓ from concatenation of the excursions of X in(0,+∞), respectively(−∞,0).

Let us recall briefly this construction whose details can be found in [3], Section 3.

Although Bertoin’s construction holds in a general setting, we assume (A). We denote by (Ft)t0 the natural filtration of X completed with the P-null sets of F. Then, X is a semimartingale. Its continuous local martingale part is proportional to a standard Brownian motion and is independent of the non-continuous part. Let us denote by;its

(12)

semimartingale local time at 0. We consider A⁺_t =

t

0

ds1_{Xs>0} and A⁻_t =

t

0

ds1_{Xs0}.

Let us denote byα⁺, respectivelyα⁻, the right-continuous inverse ofA⁺, respectively A⁻:

α_t⁺=inf{s0: A⁺_s > t} and α⁻_t =inf{s0: A⁻_s > t}

(with the usual convention inf∅ = ∞). Letx be a real number. We denote its positive part, respectively negative part, byx₊, respectivelyx₋. We define a new processX^↑by

X_t^↑=X_α+

t +1

2;_α+

t +

0<sα_t⁺

1_{Xs0}(Xs−)₊+1_{Xs>0}(Xs−)₋ ift < A⁺_∞

and by X^↑_t = +∞ if not. When X has no Brownian part, X^↑ can be viewed as the concatenation of the excursions ofXin(0,+∞). Similarly, we defineX^↓by

X^↓_t =X_α−

t −1

2;_α−

t −

0<sα⁻_t

1_{X_s0}(Xs−)₊+1_{X_s>0}(Xs−)₋ ift < A⁻_∞

and by X^↓t = −∞ if not. The laws of X^↑ and X^↓ can be recovered by a harmonic transform: Denote byq_t⁺(x, dy)andq_t⁻(x, dy)the semigroup of the Lévy process killed respectively in (−∞,0] and [0,+∞). One can show (see Silverstein in [18]) that the functions U^∗([0, x]) and U([0, x]) are superharmonic respectively for q⁺ and for q⁻ and that the following kernels

p_t⁺(x, dy)=U([0, y])

U([0, x]q_t⁺(x, dy), x >0, and

p_t⁻(x, dy)=U^∗([0,−y])

U^∗([0,−x])q_t⁻(x, dy), x <0,

define two sub-markovian semigroups. Bertoin has shown in [3], Theorem 3.4 thatX^↑ and X^↓are Markov processes started at 0 with respective semigroupsp⁺and p⁻. IfX does not drift to −∞, respectively +∞, then, p⁺, respectively p⁻, is markovian and X^↑, respectivelyX^↓, has an infinite lifetime. More precisely, ifXdoes not drift to−∞, then, we can show that

X^↑_t <∞, t0 and lim

t→∞X^↑_t = ∞. (7)

Proof. – IfX does not drift to −∞, it is easy to check that limt→∞A⁺_t = ∞, P-a.s.

and then, X^↑t <∞, t >0. IfX drifts to+∞, we have α_t⁺₊_A+

σ0 =σ0+t;t0, where

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σ0=sup{s0: Xs 0}<∞. Thus, X_t^↑₊_A+

σ0 Xσ₀+t−→_t_→∞+∞.

IfX oscillates, we must consider to cases: Suppose first thatπ=0, then,

tlim→∞X_t^↑

s>0

1_{Xs0}(Xs−)₊+1_{Xs>0}(Xs−)₋= ∞.

Ifπ is null, then by assumption (A) there is a Brownian component and limt→∞;t= ∞ that yields the desired result. ✷

In particular cases, we recover “classical” definitions of the process conditioned to stay positive:

– In the Brownian case,U^∗([0, x])=U((−x,0])=xandp⁺is the semigroup of the three-dimensional Bessel process started at 0.

– In the spectrally positive case and the stable case, Chaumont has shown in [7]

and [9] that if the Lévy process does not drift to −∞ and if 0 is regular for (0,+∞), then, for any bounded Ft measurable functional F that is continuous for the Skorokhod topology on :

EF (X^↑)=lim

x→0 lim

T→+∞Ex

F (X)|IT 0.

– In the spectrally negative case, Bertoin (see [3]) gives another construction ofX^↑ that generalizes Pitman’s theorem for Brownian motion (see Pitman [16]).

Let us denote by←X^t the path←X^t reversed at its lifetimeg

t. We denote by(X_s^↓;0 s < A⁻_t )(respectively (X^↑_s;0s < A⁺_t )) the process X^↓ (respectively X^↑) stopped at the random timeA⁻_t (respectivelyA⁺_t ). We need the following theorem due to Bertoin that links the process conditioned to stay positive with excursion theory:

THEOREM 2.4 (Bertoin [3], Theorem 3.1). – For everyt >0, the following identity holds

←X^t, X→^t (law)

=(X^↓_s)₀_s<A−

t , (X^↑_s)₀_s<A+ t

.

Remark. – IfX drifts to+∞, then, the previous identity holds with t= ∞as to say thatX^↑has the same law as the post-infimum process (see Millar in [14]).

In Section 4, we use another identity that is proved in [1] (see also [13]). From now on until the end of this section, we assume thatX does not drift to−∞. For anyt >0 we set

Jt= inf

s∈[t,∞)X^↑_s and

d¯t =inf{s > t: Ss=Xs}.