Brownian penalisations related to excursion lengths, VII

www.imstat.org/aihp 2009, Vol. 45, No. 2, 421–452

DOI: 10.1214/08-AIHP177

© Association des Publications de l’Institut Henri Poincaré, 2009

Brownian penalisations related to excursion lengths, VII

B. Roynette

, P. Vallois

and M. Yor

aUniversité Henri Poincaré, Institut Elie Cartan, BP239, F-54506 Vandoeuvre-les-Nancy Cedex, France

bLaboratoire de Probabilités et Modèles Aléatoires, Universités Paris VI et VII, 4 Place Jussieu – Case 188, F-75252 Paris Cedex 05, France cInstitut Universitaire de France, France

Received 5 October 2006; revised 6 February 2008; accepted 18 February 2008

Abstract. Limiting laws, ast→ ∞, for Brownian motion penalised by the longest length of excursions up tot, or up to the last zero beforet, or again, up to the first zero aftert, are shown to exist, and are characterized.

Résumé. Il est prouvé que les lois limites, lorsquet→ ∞, du mouvement brownien pénalisé par la plus grande longueur des excursions jusqu’ent, ou bien jusqu’au dernier zéro avantt, ou encore jusqu’au premier zéro aprèst, existent. Ces lois limites sont décrites en détail.

MSC:60F17; 60F99; 60G17; 60G40; 60G44; 60H10; 60H20; 60J25; 60J55; 60J60; 60J65 Keywords:Longest length of excursions; Brownian meander; Penalisation

1. Introduction

(a) Let(Ω, (X_t, t≥0), (Ft, t≥0), (Px, x∈R))denote the canonical realisation of the Wiener process, i.e.Ω= C([0,∞),R);(X_t, t≥0)is the coordinate process onΩ;(Ft, t≥0)denotes its natural filtration, andF∞=

t≥0Ft. (Px, x∈R)is the family of Wiener measures such thatPx(X0=x)=1, for everyx∈R. We write simplyP forP0.

(b) Let(Γ_t, t≥0)denote anR₊-valued,(Ft)adapted process defined onΩ, which satisfies: 0< E_x(Γ_t) <∞for everyt≥0, and everyx∈R. With the help of this processΓ – which we call the penalisation process – we define the family of probabilitiesP_x^{(t )}by:

P_x^{(t )}(Λ_t)=Ex(1Λ_tΓt)

Ex(Γt) (Λ_t∈Ft). (1.1)

In several preceding papers ([11–13,15–17], see also [14] for a survey), we have shown that for many penalisation processes(Γ_t, t≥0), the following holds:

(i) limt→∞P_x^{(t )}(Λ_s)exists, for every fixeds≥0, andΛ_s∈Fs. (1.2) (ii) This limit is of the formE_x(1ΛsM_s^Γ), where(M_s^Γ, s≥0)is a(Fs, P_x)positive martingale. (1.3) Here is our main tool to prove (1.2) and (1.3).

Theorem 1.1. Assume that:

(i) E(Γt|Fs) E(Γt) −→

t→∞M_s a.s.,

(ii) E(M_s)=1, for everys≥0.

Then,

(1) ∀s≥0, ∀Λ_s∈Fs, E(1ΛsΓ_t) E(Γ_t) −→

t→∞E(1ΛsM_s), (2) (Ms, s≥0)is a(Fs, P )positive martingale.

The proof of this Theorem 1.1 is quite elementary. It hinges on Scheffé’s lemma (see [9], p. 37, Theorem 21).

(c) Assume that the hypotheses of Theorem 1.1 are satisfied, and define, fors≥0, andΛs∈Fs:

Q(Λ_s):=E(1ΛsM_s). (1.4)

Then, (1.4) induces a probabilityQon(Ω,F∞). In [11–13,15–17], we have described precisely the main properties of the canonical process(X_t, t≥0)underQ.

(d) The aim of the present paper is to show the existence of the limitP^{(t )}(Λ_s), ast→ ∞,s being fixed, and to study the canonical process(Xt, t≥0)underQ– the Wiener measureP penalized by the process(Γt, t≥0)– when (Γ_t, t≥0)is defined in terms of lengths of excursions. Let us be more precise and fix notation: Fort≥0, we denote byg_t (resp.d_t) the last zero of(X_u, u≥0)before timet, resp.: the first zero after timet:

g_t =sup{s≤t;X_s=0}, (1.5)

d_t =inf{s > t;X_s=0}. (1.6)

Hence,(dt−gt)is the length of the excursion which straddlest. We also introduce:

Σ_t≡Σ_gt =sup{d_s−g_s;d_s≤t}, (1.7)

Σ_t^∗≡Σd_t =sup{ds−gs;gs≤t}. (1.8)

Thus,Σ_t is the length of the longest excursion beforeg_t, whereasΣ_t^∗is the length of the longest excursion befored_t. Consequently:

Σ_t^∗=Σ_t∨(d_t−g_t). (1.9)

We denote by(A_t, t≥0)the so-called age process:

A_t :=t−g_t and A^∗_t :=sup

s≤t

A_s. Hence:

A^∗_t =Σ_t∨(t−g_t) and Σ_t≤A^∗_t ≤Σ_t^∗; Σ_t^∗=A^∗_t ∨(d_t−g_t).

The aim of this article is to study the effects on Brownian motion of penalisations induced by the following processes (Γ_t, t≥0):

(i) Γ_t:=1(Σt≤x)(x >0, fixed). This is studied in Section 2.

(ii) Γt :=h(Σt), whereh is an “integrable function.” This study extends that made in (i), and is developed in Section 3;

(iii) Γ_t:=1_{A∗

t≤x}(x >0, fixed), andΓ_t=1_{Σ∗

t≤x}(x >0, fixed). This is studied in Section 4.

However, in this case, we have not been able to obtain a full proof of the existence of the penalised measure; we present a conjecture (4.5) upon which the existence rests.

(e)Some prerequisites relative to the Brownian meander.

AsΣ_t isFgt-measurable, it turns out that certain features of the part of the trajectory of our Brownian motion (X_s, s≥0)between timesg_t andt play some important role throughout the discussion. We now gather a few useful facts about theBrownian meander:

˜

m^{(t )}_u = 1

√t−gt

X_gt+u(t−gt),0≤u≤1

,

which is a well-defined process, whose law, thanks to the scaling property of Brownian motion does not depend on t >0. (Here, we slightly depart from the classical Brownian terminology, for which it ism^{(t )}≡(| ˜m^{(t )}_u |, u≤1)which is called the Brownian meander.)

The simple fact, obtained by Brownian scaling, that the law ofm˜^{(t )}does not depend ont, can be further extended as follows.

Proposition 1.2. LetT be a finite{Fg_t}stopping time,such that:P (XT =0)=0.Then:

(i) the process(m˜^{(T )}_u , u≤1)is independent fromFgT,and its law does not depend onT; (ii) for any Borel functionf:R→R₊,one has:

E

f (X_T)|FgT

=Kf (A_T), (1.10)

whereKdenotes the Markov kernel defined by:

Kf (y)=1 2

_∞

−∞

|z| y exp

−z²

2y f (z)dz (y∈R₊).

In particular,ifT =T_a^A=inf{t: A_t> a},fora >0,thenX_TA a andFg

T Aa are independent;hence,X_TA

a andT_a^Aare independent,and:

P (X_TA

a ∈dz)=|z| 2aexp

−z²

2a dz. (1.11)

(iii) LetΨ_1,x(z)=

π2x|z|andΨ_2,b(z)=Φ(√^|z|

b)for somex >0,andb >0,where Φ(y)=

2 π

_∞

y

due^−u2/2=P

|G|> y

, (1.12)

forGa standard Gaussian variable.Then, KΨ_1,x(y)=

y

x, KΨ_2,b(y)=1− y

b+y. We note that:

KΨ1,x(y)+KΨ_2,x−y(y)=1 for ally < x. (1.13)

Proof.

• Points (i) and (ii) are very classical; they are proven and applied in [1–3,10].

• Point (iii) follows from elementary computations. Indeed:

KΨ_1,x(y)= 2

πx 1

2

∞

−∞

z²

y e^−z2/(2y)dz= y

x

whilst:

KΨ2,b(y)=1 2

_∞

−∞

|z|

y e^−z2/(2y)Φ |z|

√b

dz

= 2

π _∞

0

dzz

ye^−z2/(2y) _∞

z/√ b

e^−u2/2du

=1 y

2 π

_∞

0

e^−u2/2du _u^√_b

0

ze^−z2/(2y)dz (by Fubini)

= 2

π _∞

0

e^−u2/2

1−e^{−{u2/(2y)}b} du

=1− y

b+y.

2. Penalisation induced byΓ_t=1(Σt≤x)

Here,x >0 is fixed.

Theorem 2.1.

(1)For everys≥0,andΛs∈Fs, Q(Λ_s):= lim

t→∞

E(1Λs1_{Σt≤x})

E(1_{Σt≤x}) exists. (2.1)

(2)This limit induces a probabilityQon(Ω,F_∞)such that:

Q(Λ_s):=E(1ΛsM_s1_{Σs≤x})=E(1ΛsM_s), (2.2)

where:

M_s :=M_s1(Σs≤x), (2.3)

M_s := |X_s| 2

πx +Φ

|X_s|

√x−A_s

1(As≤x) (2.4)

withΦ given by(1.12).Moreover,(M_s, s≥0)is a continuous,positive martingale,such thatM₀=1.

(3)UnderQ,the process(Xt, t≥0)satisfies:

(a) Σ_∞≤xa.s.,and Σ_∞

x is uniformly distributed on[0,1]; (2.5)

(b) A^∗_∞= ∞a.s.; (2.6)

(c)Letg=sup{t: X_t=0}.Then,Q(0< g <∞)=1,and the law ofgis given by:

Q(g≥t )=E

A_t∧TA x

x

, (2.7)

whereT_x^A:=inf{t≥0:At=x}.

(d)Let0≤y≤x,and denoteT_y^A:=inf{t≥0:At=y}.Then:

(i) The process(A_u, u≤T_y^A)is identically distributed underP andQ;

(ii) (A_u, u≤T_y^A)andX_TA

y are independent under eitherP orQ;

(iii) The density of the distribution ofX_TA

y underQequals:

f_X^Q

T Ay

(z)=|z|

2ye^−z2/(2y)

|z| 2

πx +Φ |z|

√x−y

(2.8)

(iv) Q(g > T_y^A)=1−

y x;

(v) The process(Au, u≤T_y^A)and the event(g > T_y^A)are independent underQ.

(4)Moreover,underQ:

(a) The processes(X_u, u≤g)and(X_g+u, u≥0)are independent;

(b) The process(X_g+u, u≥0)is positive,resp.:negative,with probability ¹₂,and (|X_g+u|, u≥0)is aBES(3) process;

(c) Denoting by(Lt, t≥0)the local time process ofXat level0,then:

•

π2xL_∞is exponentially distributed,with mean1;

• Conditionally onL_∞=,the process(Xu, u≤g)is a Brownian motionB stopped atτ:=inf{t: Lt > },and conditioned on{Σ_τ≤x}.

Remark 2.2. Obviously,the probability measureQdefined by(2.1)and the martingale(Ms)depend on the parame- terx.In this section,sincexis fixed,there is no ambiguity.A generalisation of Theorem2.1will be given in Section3 and we shall denoteQ^(x)(resp.M_s^x)the p.m. (resp.the martingale)defined by(2.1) (resp. (2.3)).

Proof of Theorem 2.1.

(1) We begin with the following lemma.

Lemma 2.3.

P (Σ_t≤x)=P

Σ₁≤x t

t→∞∼ x

t (2.9)

(the equality in (2.9) follows from the scaling property).

Proof. We might use the computation in Exercise 4.19 of [10], p. 507, which discusses a result of Knight [7], but we shall proceed from scratch by showing that:

(a) ifSβdenotes an independent exponential time, with parameterβ, then:

β _∞

0

e^−βtP (Σt≤x)dt=P (ΣS_β≤x)=E[sinh(√ 2β|X_TA

x |)] E[cosh(√

2βX_T^A

x )]. (2.10)

Let us admit this result for one moment; then, asβ→0, we obtain:

P (ΣS_β ≤x)∼ 2βE

|X_Tx^A| . Using (1.11), we have

E

|X_TA x |

=1 2

_∞

−∞

z² x exp

−z² 2x

dz=

πx 2 .

Thus, we have obtained:

_∞

0

e^−βtP (Σ_t≤x)dt ∼

β→0

πx

β . (2.11)

(b) We now prove formula (2.10): we first note that:

{ΣS_β≤x} =

gS_β≤T_x^A

= {Sβ≤d_TA

x }, (2.12)

hence:

P (Σ_Sβ≤x)=1−E e^{−βdT Ax}

.

Let(θ_u)denote the usual family of time-translation operators:

X_s◦θ_u=X_s+u (s, u≥0). (2.13)

Since:d_TA

x =T_x^A+T₀◦θ_TA

x , we obtain:

P (Σ_Sβ≤x)=1−E

e^−βTxAE_X

T Ax

e^−βT0 , P (ΣS_β≤x)=1−E

e^−βTxAe⁻

√2β|X_{T A}

x | . Next, we use the independence ofT_x^AandX_TA

x , recalled in Proposition 1.2(ii), which yields:

P (ΣS_β≤x)=1−E e^−βTxA

E e⁻

√2β|X

T Ax| . Formula (2.10) now follows from Wald’s identity:

E e

√2βX

T Ax E

e^−βTxA

=1 (2.14)

(where we have used again the independence of T_x^A andX_TA

x ) together with the symmetry of the distribution of X_TA

x .

(2)We now show relations(2.1)and(2.3).

LetT0=inf{s≥0;X_s=0}andt≥s≥0. First, we observe:

{Σ_t≤x} = {Σ_s≤x} ∩

{d_s> t} ∪

d_s≤t∧(s−A_s+x), Σ_t−ds◦θ_ds≤x

. (2.15)

Hence:

P (Σ_t≤x|Fs)=(1)+(2), where

(1)=1_{Σs≤x}P

{d_s> t}|Fs

, (2)=1_{Σs≤x}P

d_s≤t∧(s−A_s+x), Σ_t−ds ◦θ_ds ≤x

|Fs

.

We now study the asymptotic behavior of (1), then (2), ast→ ∞.

As is well known:

P_y(T0≥t )=P0

|G| ≤|y|

√t

t→∞∼ 2

π|y| 1

√t, (2.16)

whereGdenotes a standardN(0,1)Gaussian variable.

Sinced_s=s+T₀◦θ_s, the equivalence (2.16) implies:

(1)=1_{Σs≤x}P_Xs(T₀> t−s) ∼

t→∞

2

π|X_s|1_{Σs≤x}

1

√t. (2.17)

As for the term (2), we apply both the strong Markov property at timed_s and Lemma 2.3:

(2) ∼

t→∞1_{Σs≤x}

x tP

{d_s≤s−A_s+x}|Fs

.

From the Markov property at times, together with (1.12) and (2.16) we deduce:

(2) ∼

t→∞1_{Σs≤x}

x

t1_{As≤x}Φ

√|X_s| x−As

. Finally

P (Σ_t≤x|Fs) ∼

t→∞

√1 t

√x

|X_s| 2

πx +Φ

√|X_s| x−As

1_{As≤x}

1_{Σs≤x} t→∞∼

√x

√tM_s1_{Σs≤x}=

√x

√tM_s. It is clear that (2.9) implies that:

P (Σ_t≤x|Fs) P (Σ_t≤x) ∼

t→∞

M_s

E[M_s]=Ms, sinceE[Ms] =1 follows from the next point.

(3.a) We begin with the following lemma.

Lemma 2.4.

(1) Let f:(y, a)→f (y, a) be a C_(y,a)^2,1 function, fromR×R+ toR. Then,(f (Xt, At), t ≥0) is a ((Ft), P ) semi-martingale,which decomposes as:

f (X_t, A_t)=f (0,0)+ _t

0

∂f

∂y(X_s, A_s)dXs+ _t

0

1 2

∂²f

∂y² +∂f

∂a

(X_s, A_s)ds +

s≤t

f (0, As)−f (0, As−) . In particular,if:

(i) f (0, a)does not depend ona≥0;

(ii) 1 2

∂²f

∂y² +∂f

∂a =0,

then,(f (X_t, A_t), t≥0)is a((Ft), P )local martingale,with Itô representation:

f (Xt, At)=f (0,0)+ _t

0

∂f

∂y(Xs, As)dXs.

(2) Likewise,letf:(y, a)→f (y, a)be aC^2,1_(y,a)function defined onR+×R+.

Then,(f (|X_t|, A_t), t≥0)is a((Ft), P )semimartingale,which decomposes as:

f

|X_t|, A_t

=f (0,0)+ _t

0

∂f

∂y

|X_s|, A_s

sgn(Xs)dXs

+∂f

∂y(0,0)Lt+ _t

0

1 2

∂²f

∂y² +∂f

∂a

|X_s|, A_s ds +

s≤t

f (0, As)−f (0, As−) ,

where(L_t, t≥0)denotes the local time of(X_t, t≥0)at level0.

In particular,iff satisfies(i)and(ii)above,as well as:

(iii) ∂f

∂y(0,0)=0,

then(f (|X_t|, A_t), t≥0)is a((Ft), P )local martingale,with Itô representation:

f

|X_t|, A_t

=f (0,0)+ _t

0

∂f

∂y

|X_s|, A_s

sgn(Xs)dXs. Proof.

(a) Since the process(A_t, t≥0)has bounded variation, we may apply Itô’s formula tof (X_t, A_t)to obtain:

f (X_t, A_t)=f (0,0)+ _t

0

∂f

∂y(X_s, A_s)dXs+1 2

∂²f

∂y²(X_s, A_s)ds

+γ_t,

where γ_t=

_t

0

∂f

∂a(X_s, A_s)ds+

s≤t

f (0, As)−f (0, As−)

since the continuous part(A^c_t)of(A_t)is equal tot, and moreover ifΔA_s=0, thenX_s=0, andA_s=0.

(b) We use similar arguments, together with the Tanaka decomposition:

|X_t| = _t

0

sgn(Xs)dXs+L_t, t≥0, where(Lt, t≥0)denotes the local time ofXat 0.

We then use the fact that the support of dLs is{s: Xs=0}, and ifsis a zero ofXthen:As=0.

Thus:

_t

0

∂f

∂y

|X_s|, A_s

dLs=∂f

∂y(0,0)Lt.

(3.b)We now show that(M_s, s≥0)is a local martingale.

We define:

T_y^A:=inf{t≥0:At≥y} and T_y^Σ:=inf{t≥0:Σt≥y} (y >0).

Clearly, one has:

T_x^Σ=d_TA

x =T_x^A+T0◦θ_TA x .

We denote:

M_s:=1_{s≤TA x }

|X_s| 2

πx +Φ

√|Xs| x−A_s

(2.18) +1_{s>TA

x }

2

πx(sgnX_TA x )X_s. Then, clearly:

Ms=Ms1_{Σ_s≤x}=Ms1_{s≤TΣ

x }=M_s∧TΣ

x . (2.19)

Thus, it remains to prove that(M_s, s≥0)is a local martingale, and, for this purpose, it suffices to apply point (2) of Lemma 2.4 to the function:

f (y, a)=y 2

πx +Φ y

√x−a

(y≥0, x > a≥0).

We note that (i), (ii) and (iii) of Lemma 2.4 hold:

(α) f (0, a)=Φ(0)=1

hence, (i) is satisfied . (β) ∂²f

∂y²(y, a)= 2

π y

(x−a)^3/2e^{−y2/(2(x−a))};

∂f

∂a(y, a)= −1 2

2 π

y

(x−a)^3/2e^{−y2/(2(x−a))}, so that:1

2

∂²f

∂y² +∂f

∂a =0

hence, (ii) is satisfied .

(γ ) ∂f

∂y(y, a)= 2

πx − 2

π

√ 1

x−ae^{−y2/(2(x−a))}, so that:∂f

∂y(0,0)=0

hence, (iii) is satisfied . Finally, one has:

M_s=1+ s∧T_x^Σ

0

2

πx + 1

(x−A_u)₊Φ

|Xu| (x−A_u)₊

sgn(Xu)dXu, (2.20)

with the natural convention:

Φ y

z

=0 and 1 zΦ

y z

=0, ifz=0 andy >0. (2.21)

(3.c)We now show that(M_s, s≥0)is a true martingale.

SinceT_x^Σis a finite stopping time, the identity (2.19) and Doob’s optional stopping theorem imply that(Ms, s≥0) is a martingale as soon as(M_s, s≥0)is a martingale.

Due to (2.21), we have:

M_s= |X_s| 2

πs +Φ

|Xs| (x−As)₊

. (2.22)

From the preceding (3.b), since(M_s, s≥0)is a positive local martingale, in order to prove that it is a true martin- gale, it suffices to show:E[M_s] =1. One has:E(M_s)=(1)+(2), with:

(1)=E

1_{s≤TA x }

|Xs| 2

πx +Φ

|X_s|

√x−A_s

,

(2)= 2

πxE 1_{s>TA

x}(sgnX_TA x )Xs

.

SinceT_x^Ais a(Fgs)stopping time,{s≤T_x^A}belongs toFgs. Using now Proposition 1.2, we get:

(1)=E

1_{s≤TA x }E

|X_s| 2

πx +Φ

√|X_s| x−As

Fgs

=E 1_{s≤TA

x}

KΨ_1,x(A_s)+KΨ_2,x−As(A_s)

=E

1_{s≤TA x }

A_s x +1−

A_s x

=P s≤T_x^A

.

As for (2), we use again Proposition 1.2:

(2)= 2

πxE 1_{s>TA

x}(sgnX_TA x )(X_TA

x +Xs−X_TA x )

= 2

πxE 1_{s>TA

x}|X_TA x |

= 2

πxP

s > T_x^A E

|X_TA x |

=P s > T_x^A

KΨ1,x(x)=P

s > T_x^A . Finally:

E(M_s)=P s≤T_x^A

+P s > T_x^A

=1. (2.23)

This ends the proof of point (2) in Theorem 2.1.

(4)We now show that, underQ, Σ_∞

x is uniformly distributed on[0, 1].

(4.a) We have:

Q(Σ_s≤x)= lim

t→∞

P (Σs≤x, Σt≤x)

P (Σt≤x) =P (Σt≤x) P (Σt≤x)=1.

Thus:

Q(Σ_∞≤x)=1. (2.24)

(4.b)Proof of point(3.a) (of Theorem 2.1).

Lety∈ [0, x], ands >0. According to (2.2) and Doob’s optional stopping theorem, we have:

Q(Σ_s> y)=Q

T_y^Σ< s

=E[1_{Ty^Σ_<s}M_s] =E[1_{Ty^Σ_<s}M_TΣ y ]. Consequently:

Q(Σ_∞> y)=Q

T_y^Σ<∞

=E[M_TΣ

y ]. (2.25)

However:

M_TΣ

y =

1, if the length of the 1st excursion of length≥yis smaller thanx,

0, otherwise. (2.26)

To computeE(M_TΣ

y ), we use the excursions theory for Brownian motion (see, for instance, [10], Chapter XII). Let e=(e_s, s >0)be the excursion process related to(X_t)underP. We introduce, for any excursionε, its durationζ (ε), and let

U=

ε;ζ (ε)≥y

, Γ =

ε;y≤ζ (ε)≤x

(y < x) (2.27)

andN_t^U the number of excursions starting before timet, and belonging to the setU: N_t^U=

s≤t

1_{e_s∈U}.

We now consider the(Fτ_l)stopping timeS^y: S^y=inf

l >0;ζ (el)≥y

=inf

t >0;N_t^U>0 , whereτ_l=inf{t≥0;L_t> l}.

From [10], Lemma 1.13, Chapter XII and (2.25), one has:

E[M_TΣ y ] =P

ζ (e_S^y)∈U

=n(Γ ) n(U ), withndenoting Itô’s excursion measure.

It is easy to computen(Γ )andn(U ):

n(Γ )= 1

√2π _x

y

√dr

r³= 2

√2π 1

√y − 1

√x

, n(U )= 2

√2π

√1y.

Consequently:

Q(Σ_∞> y)=1− y

x. (2.28)

This ends the proof of point (3.a) in Theorem 2.1.

(4.c)We determine the density function ofL_∞underQ.

Leta >0. We have:

Q(L_∞> a)=Q(τa<∞)= lim

t→∞Q(τa< t ), whereτ_a=inf{s;L_s≥a}.

According to (2.2), (2.3) and the optimal stopping theorem, we get:

Q(τ_a< t )=E[1_{τa<t}M_t] =E[1_{τa<t}M_τa] =P (τ_a< t, Σ_τa≤x).

Consequently, using notation introduced in (4.b) above, we obtain:

Q(L_∞> a)=P (Σ_τa≤x)=exp

−an(ζ≥x) . Since:

n(ζ ≥x)= _∞

x

√dr 2πr³=

2 πx,

we get:

Q(L_∞> a)=exp

−a 2

πx

(a >0). (2.29)

This proves the first part of point (4.c) in Theorem 2.1.

(5)We show that, for anyy < x,E[M_TA y ] =1.

According to identity (2.3), we have:

M_TA y = |X_TA

y | 2

πx +Φ |X_TA

y |

√x−y

. (2.30)

From Proposition 1.2, we deduce:

E[M_TA

y ] =KΨ1,x(y)+KΨ2,x−y(y)= y

x +1− y

x =1. (2.31)

We note that the preceding computation yields another proof of (2.28).

(6)We now show:Q(A^∗_∞= ∞)=1.

Let 0< η <1. We will prove thatQ(A^∗_∞= ∞)≥√

1−η, which will yield the result.

Similarly to the proof of point (4.b) one has:Q(T_y^A<∞)=E[M_TA

y ], for anyy∈ [0, x[. Identity (2.31) implies thatQ(T_y^A<∞)=1.

From point (3.a) of Theorem 2.1:

Q(Σ_∞< x−xη)= 1−η.

Consequently:

Q

{Σ_∞< x−xη} ∩

T_y^A<∞

=

1−η, wherex−xη < y < x.

Let us observe that on the set{Σ_∞< x(1−η} ∩ {T_y^A<∞},X_t does not vanish after timeT_y^A. In particular, one has:

A^∗_∞= ∞. ConsequentlyQ(A^∗_∞= ∞)≥√ 1−η.

We also observe thatΣ_∞<∞andA^∗_∞<∞,Qa.s. imply that:

Q(g <∞)=1, (2.32)

where

g=sup{t: X_t=0}. (2.33)

(7)We then prove simultaneously that the process(A_u, u≤T_y^A)has the same distribution underP andQ, and that it is independent from the r.v.X_TA

y underQ(and underP), and we compute the density ofX_TA

y underQ.

(7.a) From the definition (2.2) ofQ, for every positive functionalF, and every Borel functionh:R→R+, we have:

EQ

F

Au, u≤T_y^A h(X_TA

y )

=E F

Au, u≤T_y^A h(X_TA

y )M_TA y

. (2.34)

Note thatM_TA

y is a function ofX_TA

y , and recall that from Proposition 1.2,X_TA

y and(A_u, u≤T_y^A)areP-independent.

Thus:

EQ

F

Au, u≤T_y^A h(X_TA

y )

=E F

Au, u≤T_y^A E

h(X_TA y )M_TA

y

. (2.35)

(7.b) Now, takingh≡1, and using (2.31), we obtain:

EQ

F

Au, u≤T_y^A

=E F

Au, u≤T_y^A

thus proving the first point announced in (7).

(7.c) We may now write (2.35) as follows:

E_Q F

A_u, u≤T_y^A h(X_TA

y )

=E_Q F

A_u, u≤T_y^A E

h(X_TA y )M_TA

y

=E_Q F

A_u, u≤T_y^A E_Q

h(X_TA y )

, thus proving the independence of(A_u, u≤T_y^A)andX_TA

y underQ.

(7.d)We now compute the density ofX_TA

y underQ.

One has:

E_Q h(X_TA

y )

=E h(X_TA

y )M_TA y

. (2.36)

Hence, from formulae (2.30) and (1.11) we get:

Q(X_TA

y ∈ dz)=|z|

2ye^−z2/(2y)

|z| 2

πx +Φ √|z|

x−y

dz. (2.37)

(8)We show the independence underQof(Au, u≤T_y^A)and of the set{g > T_y^A}, and we computeQ(g > T_y^A).

For every positive functionalF, one has:

EQ

1_{g>TA y }F

Au, u≤T_y^A

=EQ

F

Au, u≤T_y^A Q

g > T_y^A|FT_y^A

.

We now admit for an instant the result of Lemma 2.5, which will be proved below:

Q(g > t|Ft)=Φ

|X_t| (x−A_t)₊

1 M_t1_{A∗

t≤x}. (2.38)

Replacingtby the(Ft)-stopping timeT_y^A, we get:

Q

g > T_y^A|FT_y^A

=Φ |X_TA

y |

√x−y 1

M_TA y

. (2.39)

Since the right-hand side of (2.39) only depends onX_TA

y , we deduce from the above point (7) that:

E_Q 1_(g>T^A

y )F

A_u, u≤T_y^A

=Q

g > T_y^A E_Q

F

A_u, u≤T_y^A . Hence the desired independence property.

Also, from (2.39), (2.34) and Proposition 1.2, we deduce:

Q

g > T_y^A

=E_Q

Φ |X_TA

y |

√x−y 1

M_TA y

=E

Φ |X_TA

y |

√x−y

=1− y

x.

(9) To end the proof of Theorem 2.1, we shall now use the technique of progressive enlargement of filtration (see [6]). We have proven (see point (6) above) thatQ(g <∞)=1, whereg is defined by (2.33). We denote by (Gt, t≥0)the smallest filtration which contains(Ft, t≥0), and which makesga(Gt, t≥0)stopping time. In order to use the technique of enlargement of filtration, we need the following lemma.

Lemma 2.5. (i)For anyt >0,we have:

Z_t:=Q(g > t|Ft)=Φ

|Xt|

√x−A_t 1

Mt

1_{A^∗_t≤x}. (2.40)