SOME EXPLICIT FORMULAS FOR THE BROWNIAN BRIDGE, BROWNIAN MEANDER AND BESSEL PROCESS UNDER UNIFORM SAMPLING

DOI:10.1051/ps/2015009 www.esaim-ps.org

SOME EXPLICIT FORMULAS FOR THE BROWNIAN BRIDGE, BROWNIAN MEANDER AND BESSEL PROCESS UNDER UNIFORM SAMPLING

Mathieu Rosenbaum

and Marc Yor

Abstract.We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge given in [M. Rosenbaum and M. Yor, S´eminaire de Probabilit´es XLVI (2014) 359–375], together with various relationships between the laws of these four processes. Finally, we consider the variable BUT1/√

T1, where B is a Brownian motion,T1 its first hitting time of level one and U a uniform random variable independent of B. This variable is shown to be centered in [R. Elie, M. Rosenbaum and M. Yor,Electron. J. Probab.37(2014) 1–23; M. Rosenbaum and M. Yor, S´eminaire de Probabilit´es XLVI (2014) 359–375]. The results obtained here enable us to revisit this intriguing property through an enlargement of filtration formula.

Mathematics Subject Classification. 60G40, 60J55, 60J65.

Received November 24, 2014.

1. Introduction

Let (B_t, t≥0) be a standard Brownian motion,T₁its first hitting time of level one andm≥0. The goal of the paper [3] was the study of some very natural Brownian functionals, namely the normalized integrals of the Brownian motion up to its first hitting time, that is the variables

A^(m)= 1 T₁^1+m/2

_T1

0 |B_s|^msgn(B_s)ds, A˜^(m)= 1 T₁^1+m/2

_T1

0 |B_s|^mds.

This led us to investigate the properties of the random variableαdefined by α=B_UT1

√T₁, (1.1)

withU a uniform random variable on [0,1], independent ofB. Indeed, it is closely connected toA^(m)and ˜A^(m). For example, for man odd integer, we see that them-th moment of αis the expectation ofA^(m). In [3], it is

Keywords and phrases.Brownian motion, Brownian bridge, Brownian meander, pseudo-Brownian bridge, Bessel process, uniform sampling, local times, hitting times, enlargement of filtration.

1 LPMA, University Pierre et Marie Curie - Paris 6, Paris, France.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2015

EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING

first shown that the random variableαis centered. Intrigued by this property, we determined the distribution of this variable, which is expressed in [8] under the following form, where =

L denotes equality in law:

α=

L ΛL₁−1

2|B1|, (1.2)

with L_t the local time at point 0 of B at time t and Λ a uniform random variable on [0,1], independent of (|B1|, L1). The centering property is easily recovered from (1.2) since

E

ΛL₁−1 2|B1|

= 1

2E[L1− |B1|] = 0.

In fact, in [8], a preliminary to the proof of (1.2) is to obtain the law of a triplet of random variables defined in terms of the pseudo-Brownian bridge introduced in [1], see Section 2 below. In this paper, we show that the law of this triplet enables us to derive several unexpected simple formulas for various quantities related to some very classical Brownian type processes, namely the Brownian bridge, the Brownian meander and the three dimensional Bessel process. More precisely, we focus on distributional properties of these processes when sampled with an independent uniform random variable. Thus, this work can be viewed as a modest complement to the seminal paper by Pitman, see [6], where the laws of these processes when sampled with (several) independent uniform random variables are already studied.

The paper is organized as follows. In Section 2, we give some preliminary results related to the law of (|B1|, L1). Indeed, they play an important role in the proofs. Distributional properties for the Brownian bridge are established in Section 3 whereas the Brownian meander and the three dimensional Bessel process are investigated in Section4. Finally, in Section5, we reinterpret the fact thatαis centered through the lenses of an enlargement formula for the Brownian motion with the timeT₁due to Jeulin, see [5]. In particular we show that this centering property can be translated in terms of the expectation of the random variableU/(R_UR²₁), whereR is a three dimensional Bessel process andU a uniform random variable on [0,1] independent ofR.

2. Some preliminary results about the law of (|B

|, L

)

Before dealing with the Brownian bridge, the Brownian meander and the three dimensional Bessel process, we give here some preliminary results related to the distribution of the couple (|B₁|, L₁). These results will play an important role in the proofs of our main theorems.

It is well-known that the law of (B₁, L₁) admits a density onR×R⁺. Its value at point (x, l) is given by

√1

2π(|x|+l) exp

−(l+|x|)² 2

· (2.1)

Forl≥0, we set

H(l) = e^l2/2 _+∞

l dxe^−x2/2. (2.2)

The following consequences of (2.1) shall be useful in the sequel.

Proposition 2.1. Let l≥0. We have the double identity

E[L₁||B₁|=l] =E[|B₁||L₁=l] =H(l). (2.3) Furthermore, one has

H(l) =lE 1

N²+l²

, (2.4)

whereN denotes a standard Gaussian random variable.

Proof. We start with the proof of (2.3). Of course, it can be deduced from (2.1) at the cost of some integrations.

We prefer the following arguments. First, the equality on the left hand side of (2.3) stems from the symmetry of the law of (|B₁|, L₁), which is obvious from (2.1). Thus, we now have to show the second equality in (2.3).

This easily follows from the identity

E[φ(L₁)|B₁|] =E _L1

0 dxφ(x)

, (2.5)

which is valid for any bounded measurable functionφ. Indeed, assuming (2.5) for a moment, using the fact that L₁=

L |B₁|, we may write (2.5) as

_+∞

0 dle^−l2/2φ(l)E[|B₁||L₁=l] = _+∞

0 dlφ(l) _+∞

l dxe^−x2/2. Hence, since this is true for every bounded measurable functionφ, we get

e^−l2/2E[|B₁||L₁=l] = _+∞

l dxe^−x2/2, which is the desired result for (2.3).

It remains to prove (2.5) for a generic bounded measurable functionφ. Remark that the formula φ(L_t)|B_t|=

_t

0 dB_sφ(L_s)sign(B_s) + _t

0 dL_sφ(L_s)

is a very particular case of the balayage formula (see [7], p. 261). It now suffices to take expectation on both sides of this last equality to obtain (2.5).

We now give the proof of the second part of Proposition2.1. First, note that E

1 N²+l²

= _+∞

0 dve^−vl2E e^−vN2

.

Using the Laplace transform ofN², we obtain E

1 N²+l²

= _+∞

0 dv e^−vl2/2 2√

1 +v· (2.6)

Then remark that thanks to the change of variablex²= (1 +v)l², we get H(l) = e^l2/2

_+∞

l dxe^−x2/2= _+∞

0 dv l

2√

1 +ve^−vl2/2.

This together with (2.6) gives the second part of Proposition2.1.

3. The Brownian bridge under uniform sampling

3.1. A key result on the pseudo-Brownian bridge

Before giving our theorem on the uniformly sampled Brownian bridge, we recall a result related to the pseudo- Brownian bridge established in [8], and which is the key to most of the proofs in this paper. The pseudo-Brownian bridge was introduced in [1] and is defined by

B_uτ1

√τ₁, u≤1

,

EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING

with (τ_l, l >0) the inverse local time process:

τ_l= inf{t, Lt> l}.

This pseudo-Brownian bridge is equal to 0 at time 0 and time 1 and has the same quadratic variation as the Brownian motion. Thus, it shares some similarities with the Brownian bridge, which explains its name. LetU be a uniform random variable on [0,1] independent ofB. The following theorem is proved in [8].

Theorem 3.1. There is the identity in law B_Uτ1

√τ₁ , 1

√τ₁, L_Uτ1

=L

1

2B₁, L₁, Λ

,

with Λa uniform random variable on[0,1], independent of(B₁, L₁).

In other words,L_Uτ1 is a uniform random variable on [0,1], independent of the pair B_Uτ1

√τ₁ , 1

√τ₁

, which is distributed as (¹₂B₁, L₁).

To illustrate and partially check on Theorem3.1, we give the following corollary.

Corollary 3.2. For anyy∈R andλ >0, one has E

L^y_τ1exp

−λ² 2 τ₁

= exp (−λ(2|y|+ 1)). As a consequence, we get

E

L^y_τ1|τ₁=t

= (2|y|+ 1) exp

−2(y²+|y|) t

·

Note that some similar formula, where τ₁ is replaced by T₁, is presented in [3] (Prop. 3.1). A proof of Corollary3.2may be obtained from Ray–Knight’s theorem which asserts that (L^y_τ1, y ≥0) and (L^−y_τ1 , y≥0) are two independent BESQ(0) processes, starting from 1. Here we give a proof based on Theorem3.1.

Proof. Consider the measure onRdefined forf non negative measurable by I(f) =E

τ₁f(B_Uτ1) exp

−λ² 2 τ₁

,

withU a uniform random variable on [0,1], independent ofB. It is easily seen that it admits the density y→E

L^y_τ1exp

−λ² 2 τ₁

with respect to the Lebesgue’s measure (dy) onR. On the other hand, we may write from Theorem3.1:

I(f) =E 1

L²₁f B₁

2L₁

exp

−λ² 2L²₁

· Using the well-known identity in law:

(B₁, L₁) =

L R₁(ε(1−U), U),

whereU is a uniform random variable on [0,1],εa symmetric Bernoulli variable, independent ofU, andR₁the value at time 1 of a BES(3) process, independent ofU andε, we obtain

I(f) =E 1

R₁²U²f ε

2(1 U −1)

exp

− λ² 2R²₁U²

· It is easily shown that forμ≥0,

E 1

R²₁exp

− μ² 2R²₁

= exp(−μ).

Therefore,

I(f) = ₁

0 du 1 u²E

f

ε 2

1 u−1

exp

−λ u

= _∞

−∞dyf(y) exp (−λ(2|y|+ 1)),

which ends the proof of Corollary3.2.

We note that, conversely, we may recover the joint law of (B_Uτ1, τ₁) from Corollary3.2. For example:

• The density ofB_Uτ1 is given by

y→E 1

τ₁L^y_τ1

= 1

(2|y|+ 1)²·

• The density of ^B√Uτ_τ1¹ is given by

y→E 1

√τ₁L^yτ_τ1¹

= 2

πexp(−2y²), that is ^B√^Uττ1¹ is distributed as ¹₂B₁.

• We have

E

φ(B_Uτ1) 1 (√

τ₁)^p

= _∞

−∞dyφ(y) 1

(2|y|+ 1)^p+2E[(R1)^p].

Consequently, conditional onB_Uτ1 =y, √¹τ1 is distributed as _(2|y|+1)¹ R₁.

3.2. Some consequences for the Brownian bridge

To deduce some properties of the Brownian bridge from Theorem3.1, the idea is to use an absolute continuity relationship between the law of the pseudo-Brownian bridge and that of the Brownian bridge shown by Biane, Le Gall and Yor in [1]. More precisely, forF a non negative measurable function onC([0,1],R), we have

E

F B_uτ1

√τ₁, u≤1

= 2

πE

F(b(u), u≤1) 1 λ⁰₁

, (3.1)

where

b(u), u ≤ 1

denotes the Brownian bridge and (λ^x_u, u ≤ 1, x ∈ R) its family of local times. Let U be again a uniform random variable on [0,1] independent of b. The following theorem is easily deduced from Theorem3.1together with equation (3.1).

Theorem 3.3. For any non negative measurable functionsf andg, we have

E

f

b(U), λ⁰₁ g

λ⁰_U λ⁰₁

= π

2E

f 1

2B₁, L₁

L₁

E[g(Λ)], (3.2) with Λa uniform random variable on[0,1], independent of(B₁, L₁).

EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING

Thus,λ⁰_U/λ⁰₁ is a uniform random variable on [0,1], independent of the pair

b(U), λ⁰₁

which is distributed according to (3.2) withg= 1.

The following corollary of Theorem3.3provides some surprisingly simple expressions for some densities and (conditional) expectations of quantities related to the Brownian bridge.

Corollary 3.4. The following properties hold:

• The variableλ⁰₁ admits a density on R⁺. Its value at pointl≥0 is given by lexp(−l²/2).

Hence,λ⁰₁has the same law as√

2E, withEan exponential random variable. Therefore,λ⁰₁is Rayleigh distributed.

• The density ofb(U)at point y given λ⁰₁=l is given by E[λ^y₁|λ⁰₁=l] = (2|y|+l) exp

−(2y²+ 2|y|l) . Consequently, there is the formula

E λ^y₁

λ⁰₁

= exp(−2y²).

• The density ofb(U)at point y is given by E[λ^y₁] =

_+∞

2|y|

dzexp(−z²/2).

Thus, we haveb(U) =

L

√2E(V /2), withEan exponential variable independent ofV which is uniformly distributed on[−1,1].

The first part of Corollary 3.4 is obviously deduced from Theorem 3.3 and is in fact a very classical result, see [1,2,4,7]. We now prove the second part.

Proof. Letf be a non negative measurable function. First note that E

f(b(U))|λ⁰₁=l

=E ₁

0 duf(b(u))|λ⁰₁=l

=

Rdyf(y)E

λ^y₁|λ⁰₁=l . Hence the density ofb(U) at pointy givenλ⁰₁=l is equal to

E

λ^y₁|λ⁰₁=l .

Now, let hdenote the density of the couple (B₁, L₁) given in equation (2.1). From Theorem3.3, we easily get that the density ofb(U) at pointy givenλ⁰₁=l is equal to

2 π

2

lh(2y, l) lexp(−l²/2) =√

2πh(2y, l) exp(l²/2).

The first statement in the second part of Corollary 3.4 readily follows from equation (2.1). For the second statement, we use the fact that

E λ^y₁

λ⁰₁

=E _+∞

0 dl1

lE[λ^y₁|λ⁰₁=l]

lexp(−l²/2) =√ 2π

_+∞

0 dlh(2|y|, l).

Using the definition ofh, this last expression is equal to exp

−2y²

.

The last identity in Corollary3.4is easily deduced from Theorem3.3together with Proposition2.1. Note that this formula can also be found in ([9], p. 400).

4. The Brownian meander and the three dimensional Bessel process under uniform sampling

In this section, we reinterpret Theorem 3.1 in terms of the Brownian meander and the three dimensional Bessel process.

4.1. The Brownian meander

We first turn to the translation of Theorem3.1in terms of the Brownian meander, denoted by

m(u), u≤1 . To do so, we use an equality in law shown by Biane and Yor in [2]. More precisely we have

((m(u), i_u), u≤1) =

L

(|b(u)|+λ⁰_u, λ⁰_u), u≤1 , where

i_u= inf

u≤t≤1m_t. Thus, we can reinterpret Theorem3.3as follows.

Theorem 4.1. For any non negative measurable functionsf andg, we have

E

f(m(U), m(1))g i_U

m(1)

= π

2E

f 1

2|B₁|+ΛL₁, L₁

L₁g(Λ)

, with Λa uniform random variable on[0,1], independent of(B₁, L₁).

Let (˜λ^y₁, y ≥0) denotes the family of local times of m at time 1. Similarly to what we have done for the Brownian bridge, we are able to retrieve from Theorem4.1simple expressions for the laws ofm(1) andm(U).

We state these results in the following corollary.

Corollary 4.2. The following properties hold:

• The variable m(1)is Rayleigh distributed.

• The density of m(U)at pointy≥0 is given by E[˜λ^y₁] = 2

_2y

y exp(−z²/2)dz.

Thus, we havem(U) =

L

√2EW, withE an exponential variable independent ofW which is uniformly distributed on[1/2,1].

Proof. The proof of the first part of Corollary 4.2is obvious from Theorem 4.1. We now consider the second part. Letf be a non negative measurable function. Using Theorem4.1together with equation (2.1), we get that

E f

m(U)

= _+∞

0 dx _+∞

0 dll(x+l)e^−(x+l)2/2E f x

2 +Λl ·

Now remark that

E f x

2 +Λl

=1 l

_x/2+l

x/2 dνf(ν).

Therefore, by Fubini’s theorem, we get E

f m(U)

= _+∞

0 dx _+∞

0 dl(x+l)e^−(x+l)2/2 _x/2+l

x/2 dνf(ν)

=

dνf(ν) _2ν

0 dx _+∞

ν−x/2dl(x+l)e^−(x+l)2/2.

EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING

Thus, the density ofm(U) at pointν is given by _2ν

0 dx _+∞

ν−x/2dl(x+l)e^−(x+l)2/2= _2ν

0 dxexp

−(x/2 +ν)² 2

= 2 _2ν

ν dze^−z2/2.

This ends the proof.

In fact, as it is the case for the Brownian bridge, we can give explicit formulas for several other quantities related to the Brownian meander, for example the law ofm(U) givenm(1). However, these expressions are not so simple and therefore probably less interesting than those obtained for the Brownian bridge.

4.2. The three dimensional Bessel process

Finally, let (R_t, t≥0) be a three dimensional Bessel process starting from 0 and J_u= inf

u≤t≤1R_t.

Using Imhof’s absolute continuity relationship between the law of the meander and that of the three dimensional Bessel process, see [1,4], we may rewrite Theorem4.1as follows.

Theorem 4.3. For any non negative measurable functionsf andg, we have E

f(R(U), R(1))g J_U

R(1)

=E

f 1

2|B₁|+ΛL₁, L₁

L²₁g(Λ)

, with Λa uniform random variable on[0,1], independent of(B₁, L₁).

We finally give the following corollary.

Corollary 4.4. The following properties hold:

• The density ofR(1)at point y≥0 is given by 2

πy²exp(−y²/2).

• R(U) =

L

√U R(1)and its density at point y≥0 is given by

2 2

πy _+∞

y exp(−z²/2)dz.

• The law ofR(U)given R(1)is the same as the law ofm(U)given m(1).

The first two parts of Corollary4.4 are in fact easily deduced from basic properties of the three dimensional Bessel process. The last part is a consequence of Imhof’s relation.

5. The centering property of α revisited through an enlargement of filtration formula

In this last section, we revisit the centering property of the variable α=B√_UT1

T₁,

which is proved in [3] and leads to various developments in [8]. Our goal here is to show that this result can be recovered from simple properties of the three dimensional Bessel process sampled at uniform time, together with an enlargement of filtration formula for the Brownian motion with the timeT₁due to Jeulin, see [5].

5.1. Some preliminary remarks on the uniformly sampled Bessel’s process

Let U be a uniform random variable on [0,1], independent of the considered Bessel’s process R. We start with the two following lemmas on the conditional expectation of the uniformly sampled Bessel’s process.

Lemma 5.1. We have

E[R_U|R₁=r] = 1 2

r+E

U

R_U|R₁=r

· (5.1)

Consequently,

E R_U

R²₁

=1 2

2 π+E

U R_UR²₁

·

Lemma 5.2. We have

E U

R_U|R1=r

=H(r), with H defined by (2.2). Consequently,

E R_U

R²₁

=E U

R_UR₁²

= 2

π· Remark that we already proved the equality

E R_U

R₁²

= 2

π

in [3]. This was in fact the cornerstone of our first proof of the centering property of α. In the enlargement of filtration approach used here, we will see that instead ofR_U/R²₁, the random variable U/(R_UR²₁) appears naturally.

5.2. Proofs of Lemmas 5.1 and 5.2

5.2.1. Proof of Lemma5.1

The first part of Lemma5.1 follows from the identity E

R_t t |R1

=R₁+E ₁

t

dv vR_v|R1

, t≤1, (5.2)

after multiplying both sides by t and integrating in t from 0 to 1. To show (5.2), we use time inversion with t= 1/wand

R_w=wR_1/w,

another three dimensional Bessel process. With this notation, using the Ito’s representation of the Bessel’s process, we get

E[R_w−R₁|R₁] =E _w

1

dt R_t|R₁

,

from which (5.2) is easily obtained. The second statement in Lemma 5.1readily follows.

EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING

5.2.2. Proof of Lemma5.2

We start with the proof of the first part of Lemma5.2. Let ρ=E

₁

0 dv v

R_v|R1=r

.

Using the same time inversion trick as for the proof of Lemma 5.1together with the Markov property, we get ρ=

_+∞

1

dw w²E

1

R_w|R₁=r

= _+∞

0

dt (1 +t)²E_r

1 R_t

,

whereP_rdenotes the law of a Bessel’s processR starting fromr. We then use the Doob’s absolute continuity relationship, that is

Pr

Ft = X_t∧T0 r W_r

Ft,

where W_r is the Wiener’s measure associated to a Brownian motion starting at pointr, X is the canonical process andT₀ is the first hitting time of 0 byX (see for example [7], Chap. XI). This together with the fact

that X_t∧T0

X_t = 1_T0>t gives

E_r 1

R_t

= 1

rW_r[T₀> t] = 1

rW₀[T_r> t].

Therefore,

ρ= 1 rE^W0

_Tr

0

dt (1 +t)²

= 1 rE^W0

T_r 1 +T_r

=rE 1

N²+r²

·

Using equation (2.4), this is equal to H(r). This ends the proof of the first part of Lemma 5.2. Using the expression for the density ofR₁ given in Corollary4.4, the proof of the second part readily follows remarking that

E U

R_UR₁²

= 2

π _+∞

0 dr _+∞

r dxe^−x2/2= 2

π _+∞

0 dxxe^−x2/2= 2

π·

5.3. An enlargement of filtration approach to the centering property of α

We now revisit the centering property ofαthrough an enlargement of filtration formula. Let (Ft) denote the filtration of the Brownian motion (B_t) and (F_t) the filtration obtained by initially enlarging (Ft) withT₁. It is shown in [5] that (B_t) is a (F_t) semi-martingale. More precisely,

B_t=β_t− _t∧T1

0

ds 1−B_s+

_t∧T1

0

ds1−B_s

T₁−s, (5.3)

where (β_t) is a (F_t) Brownian motion (in particular it is independent ofT₁). Taking expectation on both sides of (5.3) at timet=U T₁, we get

E[α] =−E

√1 T₁

_UT1

0

ds 1−B_s

+E

√1 T₁

_UT1

0

ds1−B_s T₁−s

·

Using the change of variables=uT₁in both integrals, we get E[α] =−E

T₁

_U

0

du 1−B_uT1

+E

√1 T₁

_U

0 du1−B_uT1 1−u

·

SinceU is independent ofB and uniformly distributed on [0,1], we get E[α] =−E

T_{1 1}

0 du (1−u) 1−B_uT1

+E

1

√T_{1 1}

0 du(1−B_uT1)

.

Thus,

2E[α] =−E T₁

₁

0 dv v

1−B_T1(1−v)

+E 1

√T₁

· We now use Williams time reversal result:

T₁,

1−B_T1(1−v), v≤1

=L (γ₁,(R_vγ1, v≤1)), where

γ₁= sup{s >0, R_s= 1}·

Hence we obtain

2E[α] =−E V

R_{V γ1}/√ γ₁

+E

√1 T₁

,

withV a uniform random variable on [0,1], independent ofR. From the absolute continuity relationship between the laws of

R_vγ1/√

γ₁, v≤1 and (R_v, v≤1),see [1], we get

2E[α] = 2

π−E V

R_VR²₁

· Hence E[α] = 0 if and only if

E V

R_VR²₁

= 2

π·

From Lemma 5.2, the last equality holds. Moreover, it has been shown without the help of our previous re- sults [3,8]. Thus, the use of the enlargement formula of [5] provides an alternative proof of the centering property ofα.

6. A few words of conclusion

Together with [3,8], this paper is our third work where various aspects of the law of α=B_UT1

√T₁

are investigated. For example, we have considered its centering property, the explicit form of its density, which may be directly deduced from equations (1.2) and (2.1), and its Mellin transform. In the present paper, starting from the pseudo-Brownian bridge, we obtain some results relative to the Brownian bridge, the Brownian meander and the three dimensional Bessel process. Imhof type relations between these processes allow to go from one to another.

EXPLICIT FORMULAS FOR BROWNIAN-TYPE PROCESSES UNDER UNIFORM SAMPLING

References

[1] P. Biane, J.-F. Le Gall and M. Yor, Un processus qui ressemble au pont brownien, S´eminaire de Probabilit´es XXI (1987) 270–275.

[2] P. Biane and M. Yor, Quelques pr´ecisions sur le m´eandre brownien.Bull. Sci. Math.112(1988) 101–109.

[3] R. Elie, M. Rosenbaum and M. Yor, On the expectation of normalized Brownian functionals up to first hitting times.Electron.

J. Probab.37(2014) 1–23.

[4] J.P. Imhof, Density factorizations for Brownian motion, meander and the three-dimensional bessel process, and applications.

J. Appl. Probab.(1984) 500–510.

[5] T. Jeulin, Semimartingales et grossissement d’une filtration. In vol. 833 ofLect. Notes Math.Springer (1980).

[6] J.W. Pitman, Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times.

Electron. J. Probab.4(1999) 1–33.

[7] D. Revuz and M. Yor, Continuous martingales and Brownian motion. In vol. 293. Springer (1999).

[8] M. Rosenbaum and M. Yor, On the law of a triplet associated with the pseudo-brownian bridge. S´eminaire de Probabilit´es XLVI(2014) 359–375.

[9] G.R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics. In vol. 59. SIAM (2009).