# Bougerol's identity in law and extensions

**ABSTRACT** We present a list of equivalent expressions and extensions of Bougerol's

celebrated identity in law, obtained by several authors. We recall well-known

results and the latest progress of the research associated with this celebrated

identity in many directions, we give some new results and possible extensions

and we try to point out open questions.

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**ABSTRACT:**We deal with complex-valued Ornstein-Uhlenbeck (OU) process with parameter $\lambda\in\mathbb{R}$ starting from a point different from 0 and the way that it winds around the origin. The fact that the (well defined) continuous winding process of an OU process is the same as that of its driving planar Brownian motion under a new deterministic time scale (a result already obtained by Vakeroudis in \cite{Vak11}) is the starting point of this paper. We present the Stochastic Differential Equations (SDEs) for the radial and for the winding process. Moreover, we obtain the large time (analogue of Spitzer's Theorem for Brownian motion in the complex plane) and the small time asymptotics for the winding and for the process, and we deal with the exit time from a cone for a 2-dimensional OU process. Some Limit Theorems concerning the angle of the cone (when our process winds in a cone) and the parameter $\lambda$ are also presented. Furthermore, we discuss the decomposition of the winding process of complex-valued OU process in "small" and "big" windings, where, for the "big" windings, we use some results already obtained by Bertoin and Werner in \cite{BeW94}, and we show that only the "small" windings contribute in the large time limit. Finally, we study the windings of complex-valued OU process driven by a Stable process and we obtain the SDE satisfied by its (well defined) winding and radial process.09/2012; - SourceAvailable from: Peter Kern[Show abstract] [Hide abstract]

**ABSTRACT:**We point out an easy link between two striking identities on exponential functionals of the Wiener process and the Wiener bridge originated by Bougerol, and Donati-Martin, Matsumoto and Yor, respectively. The link is established using a continuous one-parameter family of Gaussian processes known as $\alpha$-Wiener bridges or scaled Wiener bridges, which in case $\alpha=0$ coincides with a Wiener process and for $\alpha=1$ is a version of the Wiener bridge.03/2014;

Page 1

arXiv:1201.5823v1 [math.PR] 27 Jan 2012

Bougerol’s identity in law and extensions

S. Vakeroudis∗†

January 30, 2012

Abstract

We present a list of equivalent expressions and extensions of Bougerol’s cele-

brated identity in law, obtained by several authors. We recall well-known results

and the latest progress of the research associated with this celebrated identity in

many directions, we give some new results and possible extensions and we try to

point out open questions.

AMS 2010 subject classification: Primary: 60J65, 60J60, 60-02, 60G07;

secondary: 60G15, 60J25, 60G46, 60E10, 60J55, 30C80, 44A10.

Key words: Bougerol’s identity, time-change, hyperbolic Brownian motion, subordina-

tion, Gauss-Laplace transform, planar Brownian motion, Ornstein-Uhlenbeck processes,

two-dimensional Bougerol’s identity, local time, multi-dimensional Bougerol’s identity,

Bougerol’s diffusion, peacock, convex order, Bougerol’s process.

Contents

1Introduction2

2Extensions of Bougerol’s identity to other processes

2.1Brownian motions with drifts . . . . . . . . . . . . . . . . . . . . . . . .

2.2Hyperbolic Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . .

4

4

5

3Bougerol’s identity and subordination

3.1General results

3.2Bougerol’s identity in terms of planar Brownian motion . . . . . . . . . .

3.3The Ornstein-Uhlenbeck case. . . . . . . . . . . . . . . . . . . . . . . .

7

7

7

9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗Laboratoire de Probabilités et Modèles Aléatoires (LPMA) CNRS : UMR7599, Université Pierre et

Marie Curie - Paris VI, Université Paris-Diderot - Paris VII, 4 Place Jussieu, 75252 Paris Cedex 05,

France. E-mail: stavros.vakeroudis@upmc.fr

†Probability and Statistics Group, School of Mathematics, University of Manchester, Alan Turing

Building, Oxford Road, Manchester M13 9PL, United Kingdom.

1

Page 2

4Multidimensional extensions of Bougerol’s identity

4.1The law of the couple (sinh(βt),sinh(Lt)) . . . . . . . . . . . . . . . . . .

4.2Another two-dimensional extension . . . . . . . . . . . . . . . . . . . . .

4.3 A three-dimensional extension . . . . . . . . . . . . . . . . . . . . . . . .

10

10

12

14

5The diffusion version of Bougerol’s identity

5.1Bougerol’s diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2Relations involving Jacobi processes . . . . . . . . . . . . . . . . . . . . .

16

16

18

6Bougerol’s identity and peacocks19

7Further extensions and open questions 20

A Appendix: Tables of Bougerol’s Identity and other equivalent expres-

sions

A.1 Table: Bougerol’s Identity in law and equivalent expressions (u > 0 fixed)

A.2 Table: Bougerol’s Identity for other 1-dimensional processes (u > 0 fixed)

A.3 Table: Bougerol’s Identity in terms of planar Brownian motion (u > 0 fixed) 22

A.4 Table: Multi-dimensional extensions of Bougerol’s Identity . . . . . . . .

A.5 Table: Diffusion version of Bougerol’s Identity (relations involving the Ja-

cobi process) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

20

21

22

23

Bibliography24

1Introduction

Bougerol’s celebrated identity in law has been the subject of research for several authors

since first formulated in 1983 [Bou83]. A reason for this study is on the one hand its

interest from the mathematical point of view and on the other hand its numerous appli-

cations, namely in Finance (pricing of Asian options etc.)-see e.g. [Yor92, Duf00, Yor01].

However, one still feels that some better understanding remains to be discovered.

This paper is essentially an attempt to collect all the known results (up to now) and

to give a (full) survey of the several different equivalent expressions and extensions (to

other processes, multidimensional versions, etc.) in a concise way. We also provide a

bibliography, as complete as possible. For the extended proofs, we address the reader to

the original articles.

Bougerol’s remarkable identity states that [Bou83, ADY97] and [Yor01] (p. 200), with

(Bu,u ≥ 0) and (βu,u ≥ 0) denoting two independent linear Brownian motions§, we have:

for fixed t, sinh(Bt)

(law)

= βAt(B),

(1)

where Au(B) =

(1), see e.g. the corresponding Chapters in [ReY99] and in [ChY12]. In what follows,

sometimes, for simplicity, we will use the notation Auinstead of Au(·).

§When we simply write: Brownian motion, we always mean real-valued Brownian motion, starting

from 0. For 2-dimensional Brownian motion, we indicate planar or complex BM.

?u

0dsexp(2Bs) is independent of (βu,u ≥ 0). For a first approach of

2

Page 3

Alili, Dufresne and Yor [ADY97] obtained the following simple proof of Bougerol’s identity

(1):

Proof. On the one hand, we define St≡ sinh(Bt); then, applying Itô’s formula, we have:

?t

On the other hand, a time-reversal argument for Brownian motion yields: for fixed t ≥ 0,

?t

where (γs,s ≥ 0) denotes another 1-dimensional Brownian motion, independent from

(Bs,s ≥ 0).

Applying once more Itô’s formula to Qt, we have:

St=

0

?1 + S2

sdBs+1

2

?t

0

Ssds .

(2)

βAt(B)=

0

eBsdγs

(law)

= eBt

?t

0

e−Bsdγs≡ Qt,

(3)

dQt=1

2Qtdt + (QtdBt+ dγt) =1

2Qtdt +

?

Q2

t+ 1 dδt,

(4)

where δ is another 1-dimensional Brownian motion, depending on B and on γ. From (2)

and (4) ,we deduce that S and Q satisfy the same Stochastic Differential Equation with

Lipschitz coefficients, hence, we obtain (1).

With some elementary computations, from (1) (e.g. identifying the densities of both

sides, for further details see [Vakth11, BDY12a]), we may obtain the Gauss-Laplace trans-

form of the clock At: for every x ∈ R, with a(x) ≡ argsinh(x) ≡ log?x +√1 + x2?

E

√At

2At

?

1

exp

?

−x2

??

=a′(x)

√t

exp

?

−a2(x)

2t

?

.

(5)

where a′(x) = (1 + x2)−1/2.

For further use, we note that Bougerol’s identity may be equivalently stated as:

sinh(|Bu|)

(law)

= |β|Au(B).

(6)

Using now the symmetry principle (see [And87] for the original Note and [Gal08] for a

detailed discussion):

sinh(¯Bu)

(law)

=

¯βAu(B),

(7)

where, e.g.¯Bu≡ sup0≤s≤uBs.

In the remainder of this article, we give several versions and generalizations of Bougerol’s

identity (1). In particular, in Section 2 we give extensions of this identity to other pro-

cesses (i.e. Brownian motion with drift, hyperbolic Brownian motion, etc.). Section 3 is

devoted to some results that we obtain from subordination and some applications to the

study of Bougerol’s identity in terms of planar Brownian motion and of complex-valued

Ornstein-Uhlenbeck processes. In Section 4, we give some 2 and 3 dimensional extensions

of Bougerol’s identity, first involving the local time at 0 of the Brownian motion B, and

second by studying the joint law of 2 and 3 specific processes. In particular, in Subsection

3

Page 4

4.2 we give a new 2-dimensional extension. In Section 5, we generalize Bougerol’s iden-

tity for the case of diffusions, named "Bougerol’s diffusions", followed by some studies

in terms of Jacobi processes. Section 6 deals with Bougerol’s identity from the point

of view of "peacocks" (see this Section for the precise definition, as introduced in e.g.

[HPRY11]). In Section 7 we propose some possible directions for further investigation of

this "mysterious" identity in law with its versions and extensions and we give an as full

as possible list of references (to the best of author’s knowledge) up to now. Finally, in the

Appendix, we present several tables of Bougerol’s identity and all the equivalent forms

and extensions that we present in this survey. These tables can be read independently

from the rest of the text.

We also note that (sometimes) the notation used from Section to Section may be

independent.

2 Extensions of Bougerol’s identity to other processes

2.1Brownian motions with drifts

Alili, Dufresne and Yor, in [ADY97], showed the following result:

Proposition 2.1. With µ,ν two real numbers, for every x fixed, the Markov process:

X(µ,ν)

t

≡ (exp(Bt+ µt))

?

x +

?t

0

exp(−(Bs+ µs))d(βs+ νs)

?

,

(8)

for every t ≥ 0, has the same law as (sinh(Y(µ,ν)

diffusion with infinitesimal generator:

t

), t ≥ 0), where (Y(µ,ν)

t

,t ≥ 0) is a

1

2

d2

dy2+

?

µtanh(y) +

ν

cosh(y)

?

d

dy,

(9)

starting from y = argsinh(x).

Proof. It suffices to apply Itô’s formula to both processes X(µ,ν)and sinh(Y(µ,ν)).

It follows now:

Corollary 2.2. For every t fixed,

sinh(Y(µ,ν)

t

)

(law)

=

?t

0

exp(Bs+ µs)d(βs+ νs).

(10)

In particular, in the case µ = 1 and ν = 0:

sinh(Bt+ εt)

(law)

=

?t

0

exp(Bs+ s)dβs,

(11)

with ε denoting a symmetric Bernoulli variable taking values in {−1,1}.

4

Page 5

Remark 2.3. With µ = −1/2 and ν = 0, we have that, sinh

Indeed, with Yt≡ Y(−1/2,0)

?t

=

0

?t

Hence:

?

Y(−1/2,0)

t

?

is a martingale.

t

, Itô’s formula yields:

sinh(Yt) =

0

cosh(Ys) dYs+1

2

?t

2tanh(Ys) ds

0

sinh(Ys) ds

?t

cosh(Ys)

?

dBs−1

?

+1

2

?t

0

sinh(Ys) ds

=

0

cosh(Ys)dBs.

Mt≡ sinh(Yt) = β?t

0ds(cosh2(Ys))≡ β?t

0ds(1+sinh2(Ys)),

(12)

and for this Markovian martingale, we have:

Mt=

?t

0

?1 + M2

?

sdγs,

(13)

where γ is another Brownian motion, independent from B and from β.

It can also be seen directly from (8) that

martingales. This property is true because:

X(−1/2,0)

t

,t ≥ 0

?

is the product of two orthogonal

X(−1/2,0)

t

=Bu

Ru

???

u=A(1/2)

t

,

(14)

with A(ν)

(Rt,t ≥ 0) a 2-dimensional Bessel process started at 0. Further details about this ratio

are discussed in Sections 5 and 7. We also remark that, with the notation of Section 1,

A(0)

t

≡ At.

t

=?t

0ds exp(2B(ν)

s ), (B(ν)

t ,t ≥ 0) denoting a Brownian motion with drift, and

2.2Hyperbolic Brownian motion

Alili and Gruet in [AlG97] generalized Bougerol’s identity in terms of hyperbolic Brownian

motion:

Proposition 2.4. We use the notation introduced in the previous Subsection and we

denote by Ξ an arcsine variable such that B(ν), R and Ξ are independent. Let φ be a

function defined by:

?

Then, for fixed t, we have:

?

In particular, with ν = 0, we recover Bougerol’s identity:

φ(x,z) =

2excosh(z) − e2x− 1, for z ≥ |x|.

(15)

βA(ν)

t

(law)

= (2Ξ − 1)φB(ν)

t ,

?

R2

t+ (B(ν)

t )2

?

.

(16)

βAt

(law)

= (2Ξ − 1)φ

?

Bt,

?

R2

t+ B2

t

?(law)

= sinh(Bt).

(17)

5

Page 6

This is an immediate consequence of the following:

Lemma 2.5.

(i) The law of the functional A(ν)

t

is characterized by: for all u ≥ 0,

?+∞

E

?

exp

?

−u2

2

A(ν)

t

??

= e−ν2t/2

?

R

dx eνx

|x|

dz

z

√2πt3e−z2t/2J0(φ(x,z)), (18)

where J0stands for the Bessel function of the first kind with parameter 0 [Leb72].

(ii) In particular, taking ν = 0, for u ≥ 0 and x ∈ R, we have:

?

Proposition 2.4 follows now immediately from Lemma 2.5 by using the classical rep-

resentation of the Bessel function of the first kind with parameter 0 (see e.g. [Leb72]):

exp

−x2

2t

?

E

?

exp

?

−u2

2

At

???Bt= x

?

=

?+∞

|x|

dzz

te−z2t/2J0(φ(x,z)).

(19)

J0(z) =1

π

?+1

−1

dr

√1 − r2cos(zr),

(20)

and remarking that (with Ξ denoting again an arcsine variable), for all real ξ:

J0(ξ) = E [exp(iξ(2Ξ − 1))].

(21)

Proof. (Lemma 2.5)

With Iµand Kµdenoting the modified Bessel functions of the first and the second kind

respectively with parameter µ =

?ρ2+ ν2(for ρ and ν reals), we define the function

?2Iµ(u)Kµ(v) u ≤ v;

First, using the skew product representation of planar Brownian motion, the following

formula holds (for further details, we address the interested reader to [AlG97]):

Gµ: R2→ R+by:

Gµ(u,v) =

2Iµ(v)Kµ(u) u ≥ v.

(22)

?∞

0

dt exp

?

−ρ2

2

t

?

E

?

exp

?

−u2

2

A(ν)

t

??

=

?+∞

−∞

dy eνyGµ(u,uey).

(23)

Using the integral representation (see e.g. [Leb72], problem 8, p. 140):

Iµ(x)Kµ(y) =1

2

?∞

log(y/x)

dr e−µrJ0

??

2cosh(r)xy − x2− y2?

, y ≥ x.

(24)

we can invert (23) in order to obtain part i) of Lemma 2.5.

Part ii) follows with the help of Cameron-Martin relation.

6

Page 7

3Bougerol’s identity and subordination

In this Section, we consider (Zt= Xt+ iYt,t ≥ 0) a standard planar Brownian motion

(BM), starting from x0+ i0,x0 > 0 (for simplicity and without loss of generality, we

suppose that x0 = 1). Then, a.s., (Zt,t ≥ 0) does not visit 0 but it winds around it

infinitely often, hence θt = Im(?t

?t

where (Bu+ iγu,u ≥ 0) is another planar Brownian motion, starting from log1 + i0 .

Thus:

?u

For further study of the Bessel clock H, see e.g. [Yor80]. We also define the first hitting

times Tθ

c

≡ inf{t : |θt| = c}.

0

dZs

Zs),t ≥ 0 is well defined [ItMK65]. There is the

well-known skew-product representation:

log|Zt| + iθt≡

0

dZs

Zs

= (Bu+ iγu)

???

u=Ht=?t

0

ds

|Zs|2

,

(25)

H−1

u

≡ inf{t : Ht> u} =

0

dsexp(2Bs) := Au(B).

c≡ inf{t : θt= c} and T|θ|

3.1 General results

Bougerol’s identity in law combined with the symmetry principle of André [And87, Gal08]

yields the following identity in law (see e.g. [BeY12, BDY12a]): for every fixed l > 0,

Hτl

(law)

= τa(l)

(26)

where (τl,l ≥ 0) stands for a stable (1/2)-subordinator. An example of this kind of

identities in law is given for the planar Brownian motion case in the next Subsection.

The main point in [BeY12] is that (26) is not extended in the level of processes indexed

by l ≥ 0.

3.2Bougerol’s identity in terms of planar Brownian motion

Vakeroudis [Vak11] investigated Bougerol’s identity in terms of planar Brownian motion

and obtained some striking identities in law:

Proposition 3.1. Let (βu,u ≥ 0) be a 1-dimensional Brownian motion independent of

the planar Brownian motion (Zu,u ≥ 0), starting from 1. Then, for any b ≥ 0 fixed, the

following identities in law hold:

i) HTβ

b

(law)

= TB

a(b)

ii) θTβ

b

(law)

= Ca(b)

iii)¯θTβ

b

(law)

= |Ca(b)|,

where CAis a Cauchy variable with parameter A and¯θu= sup0≤s≤uθs.

Proof. i) We identify the laws of the first hitting times of a fixed level b by the processes

on each side of (7) and we obtain:

TB

a(b)

(law)

= HTβ

b,

7

Page 8

which is i).

ii) It follows from i) since:

θu

(law)

= γHu,

with (γs,s ≥ 0) a Brownian motion independent of (Hu,u ≥ 0) and (Cu,u ≥ 0) may be

represented as (γTB

iii) follows from ii), again with the help of the symmetry principle.

u,u ≥ 0).

Using now these identities in law, we can apply William’s "pinching" method [Wil74,

MeY82] and recover Spitzer’s celebrated asymptotic law which states that [Spi58]:

2

logtθt

(law)

−→

t→∞C1,

(27)

with C1 denoting a standard Cauchy variable (for other proofs, see also e.g. [Wil74,

Dur82, MeY82, BeW94, Yor97, VaY11a]). One can also find a characterization of the

distribution of Tθ

c

in [Vak11]. First, applying Bougerol’s identity (1) in terms

of planar Brownian motion, we have:

cand of T|θ|

Proposition 3.2. For fixed c > 0,

sinh(Cc)

(law)

= β(Tθ

c)

(law)

=

?

Tθ

cN ,

(28)

where (Cc,c ≥ 0) stands for a standard Cauchy process and N ∼ N(0,1).

Furthermore, we can obtain the following Gauss-Laplace transforms which are equiv-

alent to Bougerol’s identity exploited for planar Brownian motion:

Proposition 3.3. For x ≥ 0 and m =

??π

??

πT|θ|

c

π

2c,

c E

2Tθ

c

exp

?

−

x

2Tθ

c

??

??

=

1

√1 + x

c2

(c2+ log2(√x +√1 + x));

(29)

c E

2

exp

?

−

x

2T|θ|

c

=

1

√1 + x

2

(√1 + x +√x)m+ (√1 + x −√x)m.

(30)

Proof. For the proof of (29), it suffices to identify the densities of the two parts of (28)

and to recall that the density of a Cauchy variable with parameter c, equals:

c

π(c2+ y2).

For (30), we apply Bougerol’s identity with u = T|γ|

c

≡ inf{t : |γt| = c} and we obtain:

?

sinh(BT|γ|

c)

(law)

= β(T|θ|

c )

(law)

=T|θ|

c

N .

(31)

8

Page 9

Once again, we identify the densities of the two parts. For the left hand side, we use the

following Laplace transform: for λ ≥ 0, E

p. 71 in Revuz and Yor [ReY99]). Using now the well-known result [Lev80, BiY87]:

?

e−λ2

2T|γ|

b

?

=

1

cosh(λb)(see e.g. Proposition 3.7,

E

?

exp(iλBT|γ|

c)

?

=

1

cosh(λc)=

?∞

1

cosh(πλc

π

c

cosh(xπ

π)=

?∞

?∞

−∞

ei(λc

π)y1

2π

1

cosh(y

1

cosh(xπ

2)dy

x=cy

=

π

−∞

eiλx1

2π

2c)dx =

−∞

eiλx1

2c

2c)dx ,

(32)

we obtain the density of BT|γ|

c

which equals:

?

2ccosh(yπ

2c)

?−1

=

?

c(e

yπ

2c+ e−yπ

2c)

?−1

,

and finishes the proof.

Vakeroudis and Yor in [VaY11a, VaY11b] investigated further the law of these random

times.

3.3 The Ornstein-Uhlenbeck case

Vakeroudis in [Vak11] investigated also the case of Ornstein-Uhlenbeck processes. In

particular, we consider now a complex valued Ornstein-Uhlenbeck (OU) process:

Zt= z0+˜Zt− λ

?t

0

Zsds,

(33)

where˜Ztis a complex valued Brownian motion, z0∈ C (for simplicity and without loss of

generality, we suppose again z0= 1) and λ ≥ 0 and T(λ)

(θZ

the symmetric conic boundary of angle c for Z. Then, we have the following:

c

≡ T|θZ|

c

≡ inf?t ≥ 0 :??θZ

t

??= c?

tis the continuous winding process associated to Z) denoting the first hitting time of

Proposition 3.4. Consider (Zλ

Uhlenbeck processes, the first one complex valued and the second one real valued, both

starting from a point different from 0, and define T(λ)

any b ≥ 0. Then, an Ornstein-Uhlenbeck extension of identity in law ii) in Proposition

3.1 is the following:

t,t ≥ 0) and (Uλ

t,t ≥ 0) two independent Ornstein-

b (Uλ) = inf?t ≥ 0 : eλtUλ

t= b?, for

θZλ

T(λ)

b

(Uλ)

(law)

= Ca(b),

(34)

where a(x) = argsinh(x) and CAis a Cauchy variable with parameter A.

Proof. First, for Ornstein-Uhlenbeck processes, is well known that [ReY99], with (Bt,t ≥ 0)

denoting a complex valued Brownian motion starting from 1, using Dambis-Dubins-

Schwarz Theorem,

Zt = e−λt

?

1 +

?t

0

eλsd˜Zs

?

= e−λt(Bαt),

(35)

9

Page 10

Let us consider a second Ornstein-Uhlenbeck process (Uλ

one. Taking now equation (35) for Uλ

t,t ≥ 0) independent of the first

t(1-dimensional case), we have:

eλtUλ

t= δ(e2λt−1

2λ

),

(36)

where (δt,t ≥ 0) is a real valued Brownian motion starting from 1.

Second, applying Itô’s formula to (35), and dividing by Zs, we obtain (αt=?t

?dZs

hence:

0e2λsds =

e2λt−1

2λ):

Im

Zs

?

= Im

?dBαs

Bαs

?

,

θZ

t= θB

αt.

(37)

By inverting αt, it follows now that:

T(λ)

c

=

1

2λln

?

1 + 2λT|θ|B

c

?

.

(38)

Similarly, for the 1-dimensional case, we have:

T(λ)

b(Uλ) =

1

2λln?1 + 2λTδ

?, equivalently: α(t) = Tδ

b)= θB

b

?.

(39)

Equation (37) for t =

1

2λln?1 + 2λTδ

θZλ

T(λ)

b

bbbecomes:

(Uλ)= θZλ

1

2λln(1+2λTδ

u=Tδ

b

(law)

= Ca(b),

where the last equation in law follows precisely from statement ii) in Proposition 3.1.

4 Multidimensional extensions of Bougerol’s identity

4.1The law of the couple (sinh(βt),sinh(Lt))

A first 2-dimensional extension of Bougerol’s identity was obtained by Bertoin, Dufresne

and Yor in [BDY12a] (for a first draft, see also [DuY11]). With (Lt,t ≥ 0) denoting the

local time at 0 of B, we have:

Theorem 4.1. For fixed t, the 3 following 2-dimensional random variables are equal in

law:

(sinh(Bt),sinh(Lt))

(law)

= (βAt,exp(−Bt) λAt)

(law)

= (exp(−Bt) βAt,λAt),

(40)

where (λu,u ≥ 0) is the local time of β at 0.

10

Page 11

Remark 4.2. Theorem 4.1 can be equivalently stated as: for fixed t, the 3 following

2-dimensional random variables are equal in law:

(sinh(|Bt|),sinh(Lt))

Using now Paul Lévy’s celebrated identity in law (see e.g. [ReY99]):

(law)

= (|β|At,exp(−Bt) λAt)

(law)

= (exp(−Bt) |β|At,λAt).

(41)

?(¯Bt− Bt,¯Bt),t ≥ 0?(law)

= ((|Bt|,Lt),t ≥ 0),

(42)

we can reformulate (40) or (41), and we obtain:

(sinh(¯Bt− Bt),sinh(¯Bt))

(law)

=

?(¯β − β)At,exp(−Bt)¯βAt

?exp(−Bt) (¯β − β)At,¯βAt

?

?.

(law)

=

(43)

Remark 4.3. Considering only the second processes of the first and the third part of (40)

(or equivalently of (41)), we obtain a "local time" version of Bougerol’s identity:

sinh(Lt)

(law)

= λAt,

(44)

which (as was shown in [BeY12]), similar to the Brownian motion case, is true only for

fixed t and not in the level of processes.

Proof. (Theorem 4.1)

From Remark 4.2 it suffices to prove (41).

First, we denote Sp, p ≥ 0, an exponential variable with parameter p independent from B

and gt= sup{u < t : Bu= 0}. We know that?Bu,u ≤ gSp

|Bt| have the same law. Hence, using the following computation: for every l ≥ 0, with

(τl,l ≥ 0) denoting the time L reaches l,

P?LSp≥ l?= P (Sp≥ τl) = E [exp(−pτl)] = exp(−l?2p),

we deduce that the common density of LSpand |BSp| is:

?2pexp(−u?2p), u ≥ 0.

Equivalently, we have:

√2e(|β(1)|,λ(1))

where on the left hand side e and e′are two independent copies of S1independent from

β.

For the second identity in law in Theorem 4.1, it suffices to remark that

?and

?

BgSp+u,u ≤ Sp− gSp

?

are independent, hence LSpand BSpare also independent. We also know that Ltand

(law)

= (e,e′),

(βAt,exp(−Bt) λAt)

(law)

= (

?

Atβ1,exp(−Bt)

?

Atλ1),

and use a time reversal argument.

For the first identity in law, we use an exponential time Spand we compute the joint

Mellin transforms in both sides in order to show that:

√2e(sinh(|B|Sp),sinh(LSp))

For further details, we address the reader to [BDY12a].

(law)

=

√2e(exp(−BSp)

?

ASp|β1|,

?

ASpλAt).

11

Page 12

Using now Tanaka’s formula, we can also obtain the following identity in law for

2-dimensional processes:

Corollary 4.4.

(sinh(Bt),Lt)t≥0

(law)

=

?

exp(−Bt) βAt,

?t

0

exp(−Bs)dλAs

?

t≥0

,

(45)

where in each part, the second process is the local time at level 0 and time t of the first

one.

4.2 Another two-dimensional extension

In this Subsection, we will study the joint distribution of:

?

where (ξ1

Hence, we obtain a new 2-dimensional extension, which states the following:

?X(1)

u,X(2)

u

?=exp(−Bu)

?u

0

dξ(1)

v exp(Bv), exp(−2Bu)

?u

0

dξ(2)

v exp(2Bv)

?

,

(46)

v,v ≥ 0), (ξ2

v,v ≥ 0) and (Bu,u ≥ 0) are three independent Brownian motions.

Proposition 4.5. For u fixed,

?X(1)

u,X(2)

u

?

(law)

=

?

?

sinh(B(1)

u),1

2sinh(2B(2)

u)

?

(47)

(law)

=β(1)

(

?u

0dv exp(2Bv)),β(2)

(

?u

0dv exp(4Bv))

?

.

(48)

With pu(x,y) now denoting the density function of (sinh(B(1)

u ),1

2sinh(2B(2)

u )), we have:

pu(x,y) = E

1

2π

exp?−x2/2?u

0dv exp(2Bv)?

??u

0dv exp(2Bv)

exp?−y2/2?u

0dv exp(4Bv)?

??u

0dv exp(4Bv)

.

(49)

Proof. Let us define:

X(α)

u

= exp(−αBu)

?u

0

dξvexp(αBv),

(50)

where α = 1,2. By Itô’s formula, we have:

?

= ξu+

0

Hence:

X(α)

u

= ξu+

?u

?u

0

exp(−αBv)(−αdBv) +α2

?

2exp(−αBv)dv

?

? ??v

0

dξhexp(αBh)

?

−αdBvX(α)

v

+α2

2

X(α)

v

dv.

X(1)

u

= ξ(1)

u −

?u

?u

0

dBvX(1)

v

+1

2

?2?

?u

0

X(1)

v

dv

=

0

dη(1)

v

??

1 +

?

X(1)

v

+1

2

?u

0

X(1)

v

dv ,

(51)

12

Page 13

and

X(2)

u

= ξ(2)

u − 2

?u

?u

??

0

dBvX(2)

v

+ 2

?u

?2?

0

X(2)

v

dv

=

0

dη(2)

v

1 + 4

?

X(2)

v

+ 2

?u

0

X(2)

v

dv,

(52)

where (η(1)

variation:

v ,v ≥ 0) and (η(2)

v ,v ≥ 0) are two dependent Brownian motions, with quadratic

d < η(1),η(2)>v=

2X(1)

?2???

v X(2)

v dv

??

1 +

?

X(1)

v

1 + 4

?

X(2)

v

?2?.

(53)

Thus, we deduce that the infinitesimal generator of

?

X(1)

u ,X(2)

u

?

∂

is:

1

2

??1 + x2

1

? ∂2

∂x2

1

+?1 + 4x2

2

? ∂2

∂x2

2

+ 4x1x2

∂2

∂x1∂x2

?

+x1

2∂x1

+ 2x2

∂

∂x2

.

(54)

Let us now study the couple:

(x(1)

t ,x(2)

t ) =

?

sinh(B(1)

t ),1

2sinh(2B(2)

t )

?

,

(55)

where (B(1)

formula we have:

t ,t ≥ 0) and (B(2)

t ,t ≥ 0) are two dependent Brownian motions. By Itô’s

x(1)

t

= sinh(B(1)

?t

=

0

t )

=

0

cosh(B(1)

v) dB(1)

v

+1

2

?t

+1

0

sinh(B(1)

v) dv

?t

??

1 + (x(1)

v )2?

dB(1)

v

2

?t

0

x(1)

v

dv,

(56)

and:

x(2)

t

=

1

2sinh(2B(2)

?t

?t

t )

=

0

cosh(2B(2)

v) dB(2)

v

+

?t

0

sinh(2B(2)

v) dv

=

0

??

1 + 4(x(2)

v )2?

dB(2)

v

+ 2

?t

0

x(2)

v

dv.

(57)

Moreover:

d < sinh(B(1)),1

2sinh(2B(2)) >v

= cosh(B(1)

v) cosh(2B(2)

v) d < B(1),B(2)>v

= 2sinh(B(1)

v)1

2sinh(2B(2)

v) dv.

(58)

13

Page 14

Hence, the dependence of (B(1)

t ,t ≥ 0) and (B(2)

d < B(1),B(2)>v= tanh(B(1)

t ,t ≥ 0) is characterized by:

v) tanh(2B(2)

v) dv.

(59)

Finally, we have that (x(1)

Hence:

u ,x(2)

u ) has the same infinitesimal generator with

?

X(1)

u ,X(2)

u

?

.

?

?

exp(−Bu)

?u

2sinh(2B(2)

0

dξ(1)

v exp(Bv), exp(−2Bu)

?u

0

dξ(2)

v exp(2Bv), u ≥ 0

?

(law)

= sinh(B(1)

u),1

u), u ≥ 0

?

.

(60)

Thus, if we fix u:

?

v ,v ≥ 0) and (β(2)

is another Brownian motion independent from them.

Now, from (61), we obtain (49).

sinh(B(1)

u),1

2sinh(2B(2)

u)

?

(law)

=

?

β(1)

(

?u

0dv exp(2Bv)),β(2)

(

?u

0dv exp(4Bv))

?

,

(61)

where (β(1)

v ,v ≥ 0) are two dependent Brownian motions and (Bv,v ≥ 0)

4.3A three-dimensional extension

Alili, Dufresne and Yor, in [ADY97], obtained a 3-dimensional extension of Bougerol’s

identity:

Proposition 4.6. The two following processes have the same law:

?

eBt

?t

0

eBudβu,Bt,βt;t ≥ 0

?

(law)

= {sinh(Bt),B′

t,G′

t;t ≥ 0},

(62)

where:

?

B′

G′

t=?t

0tanh(Bs)dBs+?t

0

0

dcs

cosh(Bs);

t=?t

dBs

cosh(Bs)−?t

0tanh(Bs)dGs,

(63)

with (Gt,t ≥ 0) denoting another Brownian motion, independent from B.

Remark 4.7. We remark that with:

?

α(x) =

tanh(x) −

1

cosh(x)

1

cosh(x)

tanh(x)

?

,

(64)

we have:

?dB′

t

dG′

t

?

= α(Bt)

?dBt

dβt

?

,

(65)

and

??B′

t

G′

t

?

,t ≥ 0

?

is a 2-dimensional Brownian motion.

14

Page 15

Proof. (Proposition 4.6)

First proof: Using Itô’s formula, we deduce easily that each of these triplets is a Markov

process with infinitesimal generator (in C2(R3)):

1

2(1 + x2)d2

dx2+1

2

d2

dy2+1

2

d2

dz2+ x

d2

dxdy+

d2

dxdz+ xd

dx.

(66)

The proof finishes by the unicity (in law) of the solutions of the corresponding martingale

problem.

Second proof: First, we admit that the identity in law is true. Then, if we replace

on the left hand side (Bs) by (B′

s) and (βs) by (G′

s), we have necessarily:

sinh(Bt)

(law)

= eB′

t

?t

0

eB′

udG′

t,

(67)

which is essentially a (partial) inversion formula of the transformation (65).

Equation (67) can be proved by using Itô’s formula on the right hand side.

Gruet in [ADY97] also remarked that:

Proposition 4.8. There exist two independent linear Brownian motions V and W and

a diffusion J starting from 0 satisfying the following equation

dJt= dWt+1

2

tanh(Jt)dt,

(68)

such that,

?dβt

dBt

?

= α(−Jt)

?dVt

dWt

?

.

(69)

Hence, the two following 3-dimensional processes:

?

exp

?

Bt+t

2

??t

0

exp

?

−Bs−s

2

?

dβs,Bt,βt;t ≥ 0

?

and

(sinh(Jt),Bt,βt;t ≥ 0),

are equal.

Proof. This result follows from a geometric proof and it is essentially an explanation of

the second proof of Proposition 4.6, at least for ν = 0. For this purpose, we can compare

the writing of a hyperbolic Brownian motion in the half-plane of Poincaré, decomposed

in rectangular coordinates with the equidistant coordinates [Vin93]. For further details,

see the Appendix in [ADY97] due to Gruet.

15

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