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.

Full text

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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.
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4Multidimensional extensions of Bougerol’s identity
4.1The law of the couple (sinh(βt),sinh(Lt)) . . . . . . . . . . . . . . . . . .
4.2 Another two-dimensional extension . . . . . . . . . . . . . . . . . . . . .
4.3 A three-dimensional extension . . . . . . . . . . . . . . . . . . . . . . . .
10
10
12
14
5 The diffusion version of Bougerol’s identity
5.1 Bougerol’s diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Relations involving Jacobi processes . . . . . . . . . . . . . . . . . . . . .
16
16
18
6Bougerol’s identity and peacocks 19
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
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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
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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}.
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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