Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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4.3 Bridges, Excursions, and Meanders 237
Comment 4.3.4.2 The reader will find in Chung [183] another proof of
(4.3.5) based on the following remark, which uses the law of the pair (B
t
,m
B
t
)
established in Subsection 3.1.5:
P(g
t
≤ s, B
s
∈ dx, B
t
∈ dy)=P(B
s
∈ dx, B
u
=0, ∀u ∈ [s, t],B
t
∈ dy)
= P(B
s
∈ dx) P
x
(B
t−s
∈ dy, T
0
>t− s)
= P(B
s
∈ dx) P
0
(B
t−s
+ x ∈ dy, m
B
t−s
> −x)
=
e
−x
2
/(2s)
√
2πs
1
2π(t − s)
exp
−
(x − y)
2
2(t − s)
− exp
−
(x + y)
2
2(t − s)
dx dy .
By integrating with respect to dx, and differentiating with respect to s,the
result is obtained.
Exercise 4.3.4.3 Let t>0befixedandθ
t
=inf{s ≤ t |M
t
= B
s
} where
M
t
=sup
s≤t
B
s
.Provethat
(M
t
,θ
t
)
law
=(|B
t
|,g
t
)
law
=(L
t
,g
t
) .
Hint: Use the equalities (4.1.10)and(4.3.4)andL´evy’s theorem.
4.3.5 Brownian Bridge
The Brownian bridge (b
t
, 0 ≤ t ≤ 1) is defined as the conditioned process
(B
t
,t≤ 1|B
1
= 0). Note that B
t
=(B
t
−tB
1
)+tB
1
where, from the Gaussian
property, the process (B
t
− tB
1
,t ≤ 1) and the random variable B
1
are
independent. Hence (b
t
, 0 ≤ t ≤ 1)
law
=(B
t
− tB
1
, 0 ≤ t ≤ 1). The Brownian
bridge process is a Gaussian process, with zero mean and covariance function
s(1 − t),s ≤ t. Moreover, it satisfies b
0
= b
1
=0.
Each of the Gaussian processes X, Y and Z where
X
t
=(1− t)
t
0
dB
s
1 − s
;0≤ t ≤ 1
Z
t
= tB
(1/t) −1
;0≤ t ≤ 1
Y
t
=(1− t)B
t
1 − t
;0≤ t ≤ 1
has the same properties, and is a Brownian bridge. Note that the apparent
difficulty in defining the above processes at time 0 or 1 may be resolved by
extending it continuously to [0, 1].
Since (W
1−t
− W
1
,t≤ 1)
law
=(W
t
,t≤ 1), the Brownian bridge is invariant
under time reversal.
238 4 Complements on Brownian Motion
We can represent the Brownian bridge between 0 and y during the time
interval [0, 1] as
(B
t
− tB
1
+ ty; t ≤ 1)
and we denote by W
(1)
0→y
its law on the canonical space. More generally, W
(T )
x→y
denotes the law of the Brownian bridge between x and y during the time
interval [0,T], which may be expressed as
x + B
t
−
t
T
B
T
+
t
T
(y − x); t ≤ T
,
where (B
t
; t ≤ T ) is a standard BM starting from 0.
Theorem 4.3.5.1 For every t, W
(t)
x→y
is equivalent to W
x
on F
s
for s<t.
Proof: Let us consider a more general case: suppose ((X
t
; t ≥ 0), (F
t
), P
x
)
is a real valued Markov process with semigroup
P
t
(x, dy)=p
t
(x, y)dy,
and F
s
is a non-negative F
s
-measurable functional. Then, for s ≤ t,andany
function f
E
x
[F
s
f(X
t
)] = E
x
[F
s
P
t−s
f(X
s
)] .
On the one hand
E
x
[F
s
P
t−s
f(X
s
)] = E
x
[F
s
f(y) p
t−s
(X
s
,y) dy]
=
f(y)E
x
[F
s
p
t−s
(X
s
,y)] dy .
On the other hand
E
x
[F
s
f(X
t
)] = E
x
[E
x
[F
s
|X
t
]f(X
t
)] =
dyf(y)p
t
(x, y)E
(t)
x→y
(F
s
) ,
where P
(t)
x→y
is the probability measure associated with the bridge (for a
general definition of Markov bridges, see Fitzsimmons et al. [346]) between x
and y during the time interval [0,t]. Therefore,
E
(t)
x→y
(F
s
)=
E
x
[F
s
p
t−s
(X
s
,y)]
p
t
(x, y)
.
Thus
P
(t)
x→y
|
F
s
=
p
t−s
(X
s
,y)
p
t
(x, y)
P
x
|
F
s
.
(4.3.6)
4.3 Bridges, Excursions, and Meanders 239
Sometimes, we shall denote X under P
(t)
x→y
by (X
(t)
x→y
(s),s≤ t).
If X is an n-dimensional Brownian motion and x = y =0wehave,for
s<t,
W
(t)
0→0
|
F
s
=
t
t − s
n/2
exp
−|X
s
|
2
2(t − s)
W
0
|
F
s
.
(4.3.7)
As a consequence of (4.3.7), identifying the density as the exponential
martingale E(Z), where Z
s
= −
s
0
X
u
t−u
dX
u
, we obtain the canonical
decomposition of the standard Brownian bridge (under W
(t)
0→0
)as:
X
s
= B
s
−
s
0
du
X
u
t − u
, s<t, (4.3.8)
where (B
s
,s ≤ t) is a Brownian motion under W
(t)
0→0
. (This decomposition
may be related to the harness property in Definition 8.5.2.1.)
Therefore, we obtain that the standard Brownian bridge b is a solution of
the following stochastic equation
⎧
⎪
⎨
⎪
⎩
db
t
= −
b
t
1 − t
dt + dB
t
;0≤ t<1
b
0
=0.
Proposition 4.3.5.2 Let X
t
= μt + σB
t
where B isaBM,andforfixedT ,
(X
(T )
0→y
(t),t≤ T ) is the associated bridge. Then, the law of the bridge does not
depend on μ, and in particular
P(X
(T )
0→y
(t) ∈ dx)=
dx
σ
√
2πt
T
T − t
exp
−
1
2σ
2
x
2
t
+
(y − x)
2
T − t
−
y
2
T
(4.3.9)
Proof: The fact that the law does not depend on μ canbeviewedasa
consequence of Girsanov’s theorem. The form of the density is straightforward
from the computation of the joint density of (X
t
,X
T
), or from (4.3.6).
Proposition 4.3.5.3 Let B
(t)
x→z
be a Brownian bridge, starting from x at
time 0 and ending at z at time t,andM
br
t
=sup
0≤s≤t
B
(t)
x→z
(s).Then,for
any m>z∨x,
240 4 Complements on Brownian Motion
P
(t)
x→z
(M
br
t
≤ m)=1− exp
−
(z + x − 2m)
2
2t
+
(z − x)
2
2t
.
In particular, let b be a standard Brownian bridge (x = z =0,t=1).Then,
sup
0≤s≤1
b
s
law
=
1
2
R,
where R is Rayleigh distributed with density x exp
−
1
2
x
2
1
{x≥0}
.If
a
1
(b)
denotes the local time of b at level a at time 1, then for every a
a
1
(b)
law
=(R − 2|a|)
+
. (4.3.10)
Proof: Let B be a standard Brownian motion and M
B
t
=sup
0≤s≤t
B
s
. Then,
for every y>0andx ≤ y, equality (3.1.3)reads
P(B
t
∈ dx , M
B
t
≤ y)=
dx
√
2πt
exp
−
x
2
2t
−
dx
√
2πt
exp
−
(2y − x)
2
2t
,
hence,
P(M
B
t
≤ y|B
t
= x)=
P(B
t
∈ dx , M
B
t
≤ y)
P(B
t
∈ dx)
=1− exp(−
(2y − x)
2
2t
+
x
2
2t
)
=1− exp
−
2y
2
− 2xy
t
.
More generally,
P(sup
s≤t
B
s
+ x ≤ y|B
t
+ x = z)=P(M
B
t
≤ y − x|B
t
= z − x)
hence
P
x
(sup
0≤s≤t
B
(t)
x→z
(s) ≤ y)=1− exp
−
(z + x − 2y)
2
2t
+
(z − x)
2
2t
.
The result on local time follows by conditioning w.r.t. B
1
the equality obtained
in Example 4.1.7.7.
Theorem 4.3.5.4 Let B be a Brownian motion. For every t, the process
B
[0,g
t
]
defined by:
B
[0,g
t
]
=
1
√
g
t
B
ug
t
,u≤ 1
(4.3.11)
is a Brownian bridge B
(1)
0→0
independent of the σ-algebra σ{g
t
,B
g
t
+u
,u≥ 0}.
Proof: By scaling, it suffices to prove the result for t =1.Let
B
t
= tB
1/t
.
As in the proof of Proposition 4.3.4.1,
d
1
=
1
g
1
. Then,
4.3 Bridges, Excursions, and Meanders 241
1
√
g
1
B(ug
1
)=u
√
g
1
B
1
ug
1
=
u
d
1
B
1
g
1
+
1
g
1
(
1
u
− 1)
−
B
1
g
1
=
u
d
1
B
d
1
+
d
1
1
u
− 1
−
B(
d
1
)
.
Knowing that (
B
b
d
1
+s
−
B
b
d
1
; s ≥ 0) is a Brownian motion independent of F
b
d
1
and that
B
b
d
1
= 0, the process
B
u
=
1
√
b
d
1
B
b
d
1
+
b
d
1
u
is also a Brownian motion
independent of F
b
d
1
. Therefore t
B
(
1
t
−1)
is a Brownian bridge independent of
F
b
d
1
and the result is proved.
Example 4.3.5.5 Let B be a real-valued Brownian motion under P and
X
t
= B
t
−
t
0
B
s
s
ds .
This process X is an F
∗
-Brownian motion where F
∗
is the filtration generated
by the bridges, i.e.,
F
∗
t
= σ
B
u
−
u
t
B
t
,u≤ t
.
Let L
t
=exp(λB
t
−
λ
2
t
2
)andQ
|F
t
= L
t
P
|F
t
.ThenQ
|F
∗
t
= P
|F
∗
t
.
Comments 4.3.5.6 (a) It can be proved that |B|
[g
1
,d
1
]
has the same law as
aBES
3
bridge and is independent of
σ(B
u
,u≤ g
1
) ∨ σ(B
u
,u≥ d
1
) ∨ σ(sgn(B
1
)) .
(b) For a study of Bridges in a general Markov setting, see Fitzsimmons
et al. [346].
(c) Application to fast simulation of Brownian bridge in finance can be
found in Pag`es [691], Metwally and Atiya [646]. We shall study Brownian
bridges again when dealing with enlargements of filtrations, in Subsection
5.9.2.
Exercise 4.3.5.7 Let T
a
=inf{t : |X
t
| = a}. Give the law of T
a
under
W
(t)
0→0
.
Hint: : W
(t)
0→0
f(T
a
1
{T
a
<t}
)
= W
f(T
a
)1
{T
a
<t}
t
(t−T
a
)
n/2
e
−
a
2
2(t−T
a
)
.
4.3.6 Slow Brownian Filtrations
If ζ is a random time, i.e., a random variable such that ζ>0 a.s., we define
the σ-field F
−
ζ
of the past up to ζ as the σ-algebra generated by the variables
h
ζ
,whereh is a generic predictable process.
242 4 Complements on Brownian Motion
Likewise, we may define F
+
ζ
as the σ-algebra generated by the variables
h
ζ
,whereh is a generic F-progressively measurable process.
In particular, we consider, as in Dellacherie et al. [241]theσ-algebras F
−
g
t
and F
+
g
t
. The following properties are satisfied:
• Both (F
−
g
t
,t ≥ 0) and (F
+
g
t
,t ≥ 0) are increasing and are called the slow
Brownian filtrations, (F
−
g
t
,t≥ 0) being the strict slow Brownian filtration
and (F
+
g
t
,t≥ 0) the wide slow Brownian filtration.
• For fixed t, there is the double identity
F
+
g
t
= ∩
>0
F
−
g
t
+
= F
−
g
t
∨ σ(sgnB
t
) .
This shows that F
+
g
t
is the σ-algebra of the immediate future after g
t
and
the second identity provides the independent complement σ(sgnB
t
)which
needs to be added to F
−
g
t
to capture F
+
g
t
. See Barlow et al. [50].
4.3.7 Meanders
Definition 4.3.7.1 The Brownian meander of length 1 is the process defined
by:
m
u
:=
1
√
1 − g
1
|B
g
1
+u(1−g
1
)
|;(u ≤ 1) .
We begin with a very useful result:
Proposition 4.3.7.2 The law of m
1
is the Rayleigh law whose density is
x exp(−x
2
/2) 1
{x≥0}
.
Consequently, m
1
law
=
√
2e holds.
Proof: From (4.3.5),
P(B
1
∈ dx, g
1
∈ ds)=1
{s≤1}
|x|dx ds
2π
s(1 − s)
3
exp
−
x
2
2(1 − s)
.
We deduce, for x>0,
P(m
1
∈ dx)=
1
s=0
P(m
1
∈ dx, g
1
∈ ds)=
1
s=0
P(
|B
1
|
√
1 − s
∈ dx, g
1
∈ ds)
= dx 1
{x≥0}
1
0
ds
2x(1 − s)
2π
s(1 − s)
3
exp
−
x
2
(1 − s)
2(1 − s)
=2xdx1
{x≥0}
exp
−x
2
/2
1
0
ds
1
2π
s(1 − s)
= xe
−x
2
/2
1
{x≥0}
dx ,
4.3 Bridges, Excursions, and Meanders 243
where we have used the fact that
1
0
ds
1
π
√
s(1−s)
= 1, from the property of
the arcsin density.
We continue with a more global discussion of meanders in connection with
the slow Brownian filtrations. For any given t, by scaling, the law of the process
m
(t)
u
=
1
√
t − g
t
|B
g
t
+u(t−g
t
)
|,u ≤ 1
does not depend on t. Furthermore, this process is independent of F
+
g
t
and in
particular of g
t
and sgn(B
t
). All these properties extend also to the case when
t is replaced by τ,anyF
−
g
t
-stopping time.
Note that, from |B
1
| =
√
1 − g
1
m
1
where m
1
and
√
1 − g
1
are independent,
we obtain from the particular case of the beta-gamma algebra (see
Appendix A.4.2) G
2
law
=2eg
1
where e is exponentially distributed with
parameter 1, G is a standard Gaussian variable, and g
1
and e are independent.
Comment 4.3.7.3 For more properties of the Brownian meander, see Biane
and Yor [87] and Bertoin and Pitman [82].
4.3.8 The Az´ema Martingale
We now introduce the Az´ema martingale which is an (F
+
g
t
)-martingale and
enjoys many remarkable properties.
Proposition 4.3.8.1 Let B be a Brownian motion. The process
μ
t
=(sgnB
t
)
√
t − g
t
,t≥ 0
is an (F
+
g
t
)-martingale. Let
Ψ(z)=
∞
0
x exp
zx −
x
2
2
dx =1+z
√
2π N(z)e
z
2
/2
. (4.3.12)
The process
exp
−
λ
2
2
t
Ψ(λμ
t
),t≥ 0
is an (F
+
g
t
)-martingale.
Proof: Following Az´ema and Yor [38] closely, we project the F-martingale B
on F
+
g
t
. From the independence property of the meander and F
+
g
t
, we obtain
E(B
t
|F
+
g
t
)=E(m
(t)
1
μ
t
|F
+
g
t
)=μ
t
E(m
(t)
1
)=
π
2
μ
t
. (4.3.13)
Hence, (μ
t
,t ≥ 0) is an (F
+
g
t
)-martingale. In a second step, we project the
F-martingale exp(λB
t
−
1
2
λ
2
t) on the filtration (F
+
g
t
):
244 4 Complements on Brownian Motion
E(exp(λB
t
−
λ
2
2
t)|F
+
g
t
)=E
exp(λm
(t)
1
μ
t
−
λ
2
2
t)|F
+
g
t
and, from the independence property of the meander and F
+
g
t
,weget
E
exp
λB
t
−
λ
2
2
t
|F
+
g
t
=exp
−
λ
2
2
t
Ψ(λμ
t
) , (4.3.14)
where Ψ is defined in (4.3.12)as
Ψ(z)=E(exp(zm
1
)) =
∞
0
x exp
zx −
x
2
2
dx .
Obviously, the process in (4.3.14)isa(F
+
g
t
)-martingale.
Comment 4.3.8.2 Some authors (e.g. Protter [726]) define the Az´ema
martingale as
π
2
μ
t
, which is precisely the projection of the BM on the wide
slow filtration, hence in further computations as in the next exercise, different
multiplicative factors appear.
Note that the Az´ema martingale is not continuous.
Exercise 4.3.8.3 Prove that the projection on the σ-algebra F
+
g
t
of the F-
martingale (B
2
t
− t, t ≥ 0) is 2(t − g
t
) − t, hence the process
μ
2
t
− (t/2) = (t/2) − g
t
is an (F
+
g
t
)-martingale.
4.3.9 Drifted Brownian Motion
We now study how our previous results are modified when working with a BM
with drift. More precisely, we consider X
t
= x+μt+σB
t
with σ>0. In order
to simplify the proofs, we write g
a
(X)forg
a
1
(X)=sup{t ≤ 1:X
t
= a}.The
law of g
a
(X) may be obtained as follows
g
a
(X)=sup{t ≤ 1:μt + σB
t
= a − x}
=sup{t ≤ 1:νt + B
t
= α},
where ν = μ/σ and α =(a − x)/σ. From Girsanov’s theorem, we deduce
P(g
a
(X) ≤ t)=E
1
{g
α
≤t}
exp
νB
1
−
ν
2
2
, (4.3.15)
where
g
α
= g
α
1
(B)=sup{t ≤ 1:B
t
= α}.
4.3 Bridges, Excursions, and Meanders 245
Then, using that |B
1
| = m
1
√
1 − g
1
where m
1
is the value at time 1 of the
Brownian meander,
P(g
a
(X) ≤ t) = exp
−
ν
2
2
E
1
{g
α
<t}
exp
νm
1
1 − g
α
(4.3.16)
where is a Bernoulli random variable; furthermore, the random variables
g
α
,, and m
1
are mutually independent. Therefore, since m
1
follows the
Rayleigh law
P(m
1
∈ dy)=y exp
−
y
2
2
1
{y≥0}
dy ,
we obtain
P(g
a
(X) ≤ t) = exp
−
ν
2
2
t
0
1
π
u(1 − u)
exp
−
(a − x)
2
2uσ
2
Υ (ν, u) du
:= Ψ (x, a, t), (4.3.17)
where
Υ (ν, u)=E(exp(νm
1
√
1 − u))
=
1
2
∞
0
e
νy
√
1−u
ye
−y
2
/2
dy +
∞
0
e
−νy
√
1−u
ye
−y
2
/2
dy
,
that is,
Υ (ν, u)=
∞
0
cosh(νy
√
1 − u) ye
−y
2
/2
dy.
Lemma 4.3.9.1 Let X
t
= νt + B
t
. We have, for t<1
P(g
a
(X) >t|F
t
)=1
{T
a
(X)≤t}
e
ν(α−X
t
)
H(ν, |α − X
t
|, 1 − t) ,
where, for y>0
H(ν, y, s)=e
−νy
N
νs − y
√
s
+ e
νy
N
−νs − y
√
s
.
Proof: From the absolute continuity relationship, we obtain, for t<1
W
(ν)
(g
a
(X) ≤ t|F
t
)=ζ
−1
t
W
(0)
(ζ
1
1
{g
a
(X)≤t}
|F
t
),
where
ζ
t
=exp
νX
t
−
tν
2
2
. (4.3.18)
Therefore, from the equality
{g
a
(X) ≤ t} = {T
a
(X) ≤ t}∩{d
a
t
(X) > 1}
we obtain
246 4 Complements on Brownian Motion
W
(0)
(ζ
1
1
{g
a
≤t}
|F
t
)
=exp
νX
t
− ν
2
/2
1
{T
a
(X)≤t}
W
(0)
exp[ν(X
1
− X
t
)]1
{d
a
t
(X)>1}
|F
t
).
Using the independence properties of Brownian motion and equality (4.3.3),
we get
W
(0)
exp[ν(X
1
− X
t
)]1
{d
a
t
(X)>1}
|F
t
= W
(0)
exp[νZ
1−t
]1
{T
a−X
t
(Z)>1−t}
|F
t
= Θ(a − X
t
, 1 − t)
where Z
t
= X
1
− X
t
law
= X
1−t
is independent of F
t
under W
(0)
and
Θ(x, s):=W
(0)
e
νX
s
1
{T
x
≥s}
= e
sν
2
/2
− W
(0)
e
νX
s
1
{T
x
<s}
.
By conditioning with respect to F
T
x
, we obtain (see Subsection 3.2.4 for the
computation of H)
W
(0)
e
νX
s
1
{T
x
<s}
= e
νx
W
(0)
1
{T
x
<s}
e
ν
2
2
(s−T
x
)
W
(0)
e
ν(X
s
−X
T
x
)−
ν
2
2
(s−T
x
)
|F
T
x
= e
νx
W
(0)
1
{T
x
<s}
e
ν
2
2
(s−T
x
)
= e
νx+sν
2
/2
H(ν, |x|,s) .
Therefore,
Θ(a − X
t
, 1 − t)=e
(1−t)ν
2
/2
(1 − e
ν(a−X
t
)
H(ν, |a − X
t
|, 1 − t))
and
W
(ν)
(g
a
(X) ≤ t|F
t
)=1
{T
a
(X)≤t}
exp
(t − 1)ν
2
2
Θ(a − X
t
, 1 − t)
= 1
{T
a
(X)≤t}
1 − e
ν(a−X
t
)
H(ν, |a − X
t
|, 1 − t)
.
4.4 Parisian Options
In this section, our aim is to price an exotic option which we describe below,
in a Black and Scholes framework: the underlying asset satisfies the stochastic
differential equation
dS
t
= S
t
((r − δ) dt + σdW
t
) (4.4.1)
where W is a Brownian motion under the risk-neutral probability Q,and
w.l.g. σ>0. In a closed form,