Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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5.9 Enlargements of Filtrations 327
In particular, an honest time is F
∞
-measurable. If X is a transient diffusion,
the last passage time Λ
a
(see Proposition 5.6.2.1) is honest. Jeulin [493]
established that an F
∞
-measurable random time is honest if and only if it
is equal, on {τ<t},toanF
t
-measurable random variable.
A key point in the proof of the next Proposition 5.9.4.10 is the following
description of F
τ
-predictable processes: if τ,anF
∞
-measurable random time,
is honest, and if H is an F
τ
-predictable process, then there exist two F-
predictable processes h and
h such that
H
t
= h
t
1
{τ>t}
+
h
t
1
{τ≤t}
.
(See Jeulin [493] for a proof.)
Proposition 5.9.4.10 Let τ be honest. Then, if X is an F-local martingale,
there exists an F
τ
-local martingale
X such that
X
t
=
X
t
+
t∧τ
0
dX, μ
τ
s
Z
τ
s−
−
τ∨t
τ
dX, μ
τ
s
1 − Z
τ
s−
.
Proof: Let M be an F-martingale which belongs to H
1
and G
s
∈F
τ
s
.We
define a G
τ
predictable process H as H
u
= 1
G
s
1
]s,t]
(u). For s<t, one has,
using the decomposition of G
τ
predictable processes:
E(1
G
s
(M
t
− M
s
)) = E
∞
0
H
u
dM
u
= E
τ
0
h
u
dM
u
+ E
∞
τ
h
u
dM
u
.
Noting that
t
0
h
u
dM
u
is a martingale yields E
∞
0
h
u
dM
u
=0,
E(1
G
s
(M
t
− M
s
)) = E
τ
0
(h
u
−
h
u
)dM
u
= E
∞
0
dA
τ
v
v
0
(h
u
−
h
u
)dM
u
.
By integration by parts, with N
t
=
t
0
(h
u
−
h
u
)dM
u
,weget
E(1
G
s
(M
t
− M
s
)) = E(N
∞
A
τ
∞
)=E
∞
0
(h
u
−
h
u
)dM,μ
τ
u
.
Now, it remains to note that
328 5 Complements on Continuous Path Processes
E
∞
0
H
u
dM,μ
τ
u
Z
u−
1
{u≤τ}
−
dM,μ
τ
u
1 − Z
u−
1
{u>τ}
= E
∞
0
h
u
dM,μ
τ
u
Z
u−
1
{u≤τ}
−
h
u
dM,μ
τ
u
1 − Z
u−
1
{u>τ}
= E
∞
0
h
u
dM,μ
τ
u
−
h
u
dM,μ
τ
u
= E
∞
0
h
u
−
h
u
dM,μ
τ
u
to conclude the result in the case M ∈ H
1
. The general result follows by
localization.
Example 5.9.4.11 Let W be a Brownian motion, and τ = g
1
, the last time
when the BM reaches 0 before time 1, i.e., τ =sup{t ≤ 1:W
t
=0}.Usingthe
computation of Z
g
1
in Subsection 5.6.4 and Proposition 5.9.4.10, we obtain
the decomposition of the Brownian motion in the enlarged filtration
W
t
=
W
t
−
t
0
1
[0,τ]
(s)
Φ
1 − Φ
|W
s
|
√
1 − s
sgn(W
s
)
√
1 − s
ds
+ 1
{τ≤t}
sgn(W
1
)
t
τ
Φ
Φ
|W
s
|
√
1 − s
ds
where Φ(x)=
2
π
x
0
exp(−u
2
/2)du.
Comments 5.9.4.12 (a) The (H) hypothesis was studied by Br´emaud and
Yor [126] and Mazziotto and Szpirglas [632], and in a financial setting by
Kusuoka [552], Elliott et al. [315] and Jeanblanc and Rutkowski [486, 487].
(b) An incomplete list of authors concerned with enlargement of filtration
in finance for insider trading is: Amendinger [12], Amendinger et al. [13],
Baudoin [61], Corcuera et al. [194], Eyraud-Loisel [338], Florens and Foug`ere
[347], Gasbarra et al. [374], Grorud and Pontier [410], Hillairet [436], Imkeller
[457], Imkeller et al. [458], Karatzas and Pikovsky [512], Kohatsu-Higa [532,
533] and Kohatsu-Higa and Øksendal [534].
(c) Enlargement theory is also used to study asymmetric information, see
e. g. F¨ollmer et al. [353] and progressive enlargement is an important tool for
the study of default in the reduced form approach by Bielecki et al. [91, 92, 93],
Elliott et al.[315] and Kusuoka [552](see Chapter 7).
(d) See also the papers of Ankirchner et al. [19] and Yoeurp [858].
(e) Note that the random time τ presented in Subsection 5.6.5 is not the
end of a predictable set, hence, is not honest. However, F-martingales are
semi-martingales in the progressive enlarged filtration: it suffices to note that
F-martingales are semi-martingales in the filtration initially enlarged with W
1
.
5.10 Filtering the Information 329
5.10 Filtering the Information
A priori, one might think somewhat na¨ıvely that the drift term in the
historical dynamics of the asset plays no rˆole in contingent claims valuation.
Nevertheless, working in the filtration generated by the asset shows the
importance of this parameter. We present here some results, linked with
filtering theory. However, we do not present the theory in detail, and the
reader can refer to Lipster and Shiryaev [598]andBr´emaud [124] for processes
with jumps.
5.10.1 Independent Drift
Suppose that dB
(Y )
t
= Ydt+dB
t
,B
(Y )
0
= 0 where Y is some r.v. independent
of B and with law ν. The following proposition describes the distribution
of B
(Y )
.
Proposition 5.10.1.1 The law of B
(Y )
is W
h
ν
defined as
W
h
ν
|
F
t
= h
ν
(X
t
,t) W|
F
t
.
Here, h
ν
(x, t)=
ν(dy)exp(yx −
y
2
2
t).
Proof: Let F be a functional on C([0,t], R). Using the independence between
Y and B, and the Cameron-Martin theorem, we get
E[F (B
(Y )
s
,s≤ t)] = E[F (sY + B
s
,s≤ t)] =
ν(dy)E[F (sy + B
s
,s≤ t)]
=
ν(dy)E
F (B
s
,s≤ t)exp
yB
t
−
y
2
2
t
= E[F (B
s
; s ≤ t)h
ν
(B
t
,t)] .
We now give the canonical decomposition of B
(Y )
in its own filtration. Let
W
h
ν
|
F
t
= h
ν
(X
t
,t) W|
F
t
= L
t
W|
F
t
. Therefore, the bracket X, L
t
is equal
to
t
0
∂
x
h
ν
(X
s
,s) ds, and, from Girsanov’s theorem,
β
t
= X
t
−
t
0
ds
∂
x
h
ν
h
ν
(X
s
,s)
is a W
h
ν
-martingale, more precisely a W
h
ν
-Brownian motion and
X
t
= β
t
+
t
0
ds
∂
x
h
ν
h
ν
(X
s
,s) .
The canonical decomposition of B
(Y )
is
330 5 Complements on Continuous Path Processes
B
(Y )
t
= γ
t
+
t
0
ds
∂
x
h
ν
h
ν
(B
(Y )
s
,s) .
where γ is a BM with respect to the natural filtration of B
(Y )
.
The next proposition describes the conditional law of Y , given B
(Y )
.
Proposition 5.10.1.2 If f : R → R
+
is a Borel function, then
π
t
(f):=E(f(Y )|B
(Y )
s
,s≤ t)=
h
(f·ν)
(B
(Y )
t
,t)
h
ν
(B
(Y )
t
,t)
where h
(f·ν)
(x, t)=
ν(dy)f(y)exp(yx −
y
2
2
t) and
π
t
(f)=1+
t
0
∂
x
h
(f·ν)
h
ν
(B
(Y )
s
,s)dγ
s
.
Proof: On the one hand
E(f(Y )F (B
(Y )
s
,s≤ t)=E(F (B
s
,s≤ t) h
(f·ν)
(B
t
,t)) . (5.10.1)
On the other hand, if
Φ(B
(Y )
s
,s≤ t)=E(f (Y )|B
(Y )
s
,s≤ t) ,
the left-hand side of (5.10.1)isequalto
E
Φ(B
(Y )
s
,s≤ t)F (B
(Y )
s
,s≤ t)
= E(Φ(B
s
,s≤ t)F (B
s
,s≤ t)h
ν
(B
t
,t)) .
(5.10.2)
It follows that
π
t
(f)=Φ(B
(Y )
s
,s≤ t)=
h
(f·ν)
(B
(Y )
t
,t)
h
ν
(B
(Y )
t
,t)
.
The expression of π
t
(f) as a stochastic integral follows directly from this
expression of π
t
(f) (and the martingale property of π
t
(f)).
5.10.2 Other Examples of Canonical Decomposition
The above result can be generalized to the case where
dX
t
= dW
t
+(f(t)
˜
W
t
+ h(t)X
t
)dt
where
˜
W is independent of W . In that case, studied by F¨ollmer et al. [353],
the canonical decomposition of X is
5.10 Filtering the Information 331
X
t
= β
t
+
t
0
(f(u)k
u
(X
v
; v ≤ u)+h(u)X
u
) du
where
k
u
(X
s
; s ≤ u)=
1
Ψ
(u)
u
0
Ψ(v)(f(v)dX
v
− f (v)h(v)X
v
dv)
with Ψ the fundamental solution of the Sturm-Liouville equation
Ψ
(t)=f
2
(t)Ψ(t)
with boundary conditions Ψ(0) = 0,Ψ
(0) = 1.
5.10.3 Innovation Process
The following formula plays an important rˆole in filtering theory and will be
illustrated below.
Proposition 5.10.3.1 Let dX
t
= Y
t
dt + dW
t
,whereW is an F-Brownian
motion and Y an F-adapted process. Define
Y
t
= E(Y
t
|F
X
t
), the optional
projection of Y on F
X
. Then, the process
Z
t
:=X
t
−
t
0
Y
s
ds
is an F
X
-Brownian motion, called the innovation process.
Proof: Note that, for t>s,
E(Z
t
|F
X
s
)=E(X
t
|F
X
s
) − E
t
0
Y
u
du|F
X
s
= E(W
t
|F
X
s
)+E
t
0
Y
u
du|F
X
s
−
s
0
Y
u
du − E
t
s
Y
u
du|F
X
s
.
From the inclusion F
X
t
⊂F
t
and the fact that W is an F-martingale, we
obtain E(W
t
|F
X
s
)=E(W
s
|F
X
s
). Therefore, by using
t
s
E(Y
u
|F
X
s
)du =
t
s
E(
Y
u
|F
X
s
)du
we obtain
E(Z
t
|F
X
s
)=E(W
s
|F
X
s
)+E
t
0
Y
u
du|F
X
s
−
s
0
Y
u
du − E
t
s
Y
u
du|F
X
s
= E(X
s
|F
X
s
)+
t
s
E(Y
u
|F
X
s
)du −
s
0
Y
u
du − E
t
s
Y
u
du|F
X
s
= X
s
+
t
s
E(
Y
u
|F
X
s
)du −
s
0
Y
u
du − E
t
s
Y
u
du|F
X
s
= X
s
−
s
0
Y
u
du .
332 5 Complements on Continuous Path Processes
Proposition 5.10.3.1 is in fact a particular case of the more general result
that follows, which is of interest if Z is not F-adapted.
Proposition 5.10.3.2 Let Z be a measurable process such that E(
t
0
|Z
u
|du)
is finite for every t.Then,E(
t
0
Z
u
du|F
t
) is an F-semi-martingale which
decomposes as M
t
+
t
0
du E(Z
u
|F
u
),whereM is a martingale.
Proof: We leave the proof to the reader.
Example 5.10.3.3 As an example, take Z
u
= B
1
, ∀u, with B a Brownian
motion. Then
E
t
0
duB
1
|F
t
= tB
t
= M
t
+
t
0
duB
u
.
Comment 5.10.3.4 The paper of Pham and Quenez [711]andthepaper
of Lefebvre et al. [574] study the problem of optimal consumption under
partial observation, by means of filtering theory. See also Nakagawa [665]
for an application to default risk.
6
A Special Family of Diffusions:
Bessel Processes
Bessel processes are intensively used in finance, to model the dynamics of asset
prices, of the spot rate and of the stochastic volatility, or as a computational
tool. In particular, computations for the celebrated Cox-Ingersoll-Ross (CIR)
and Constant Elasticity Variance (CEV) models can be carried out using
Bessel processes.
We present here some main facts about Bessel, CIR, and CEV processes in
the spirit of the survey in G¨oing-Jaeschke and Yor [398], and we apply these
facts to Asian options; more generally, applications to finance can be found
in, among others, Aquilina and Rogers [22], Chen and Scott [164], Dassios
and Nagaradjasarma [214], Deelstra and Parker [229], Delbaen [230], Davydov
and Linetsky [225], Delbaen and Shirakawa [238], Dufresne [279], Duffie and
Singleton [275], Grasselli [402], Heath and Platen [428], Leblanc [571, 572],
Linetsky [594], Shirakawa [790] and in Szatzschneider [816, 817].
6.1 Definitions and First Properties
6.1.1 The Euclidean Norm of the n-Dimensional Brownian Motion
Let β =(β
1
,β
2
,...,β
n
)beann-dimensional BM where n ∈ N and define
the process R as R
t
= ||β
t
||, i.e., R
2
t
=
n
i=1
(β
i
)
2
(t). Itˆo’s formula leads to
dR
2
t
=
n
i=1
2β
i
(t)dβ
i
(t)+ndt.
Note that for any t>0, P(R
t
= 0) = 0, hence the process W defined as
dW
t
=
1
R
t
n
i=1
β
i
(t)dβ
i
(t)
is a real-valued Brownian motion (see Exercise 1.5.3.5) and the process R
satisfies
d(R
2
t
)=2R
t
dW
t
+ ndt.
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial
Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4
6,
c
Springer-Verlag London Limited 2009
333
334 6 A Special Family of Diffusions: Bessel Processes
Hence, setting ρ
t
= R
2
t
dρ
t
=2
√
ρ
t
dW
t
+ ndt,
and, using Itˆo’s formula, which may be easily justified for n>1, one obtains
dR
t
= dW
t
+
n − 1
2
dt
R
t
. (6.1.1)
The case n = 1 requires more care: if B is a one dimensional Brownian motion,
then Tanaka’s formula (4.1.8) asserts that |B
t
| =
t
0
sgn(B
s
)dB
s
+ L
0
t
, i.e.,
dR
t
= dβ
t
+ dL
0
t
, which is the analog of equation (6.1.1).
We shall say that R is a Bessel process (BES) of dimension n,andρ
is a squared Bessel process (BESQ) of dimension n. The reason for the
Bessel terminology is that many quantities involving Bessel processes may
be expressed in terms of Bessel functions.
Exercise 6.1.1.1 Develop ( + B
2
t
)
1/2
using Itˆo’s formula. Letting go to 0,
prove that |B
t
| = W
t
+ L
0
t
,whereL
0
is the local time at 0 of B.Prove(6.1.1)
using the same method for n>1.
Hint:
( + B
2
t
)
1/2
=
1/2
+
1
2
t
0
2B
s
dB
s
+ B
2
s
+
2
t
0
ds
( + B
2
s
)
3/2
.
6.1.2 General Definitions
Let W be a real-valued Brownian motion. Using the elementary inequality
|
√
x −
√
y|≤
|x − y|,forx, y ≥ 0, the existence Theorem 1.5.5.1 proves that
for every δ ≥ 0, not necessarily an integer, and x ≥ 0, the equation
dρ
t
= δdt+2
|ρ
t
|dW
t
,ρ
0
= x (6.1.2)
admits a unique strong solution. The solution is called the squared Bessel
process of dimension δ, in short BESQ
δ
. In the particular case when x =0and
δ = 0, the obvious solution ρ ≡ 0 is the unique solution. From the comparison
Theorem 1.5.5.9,if0≤ δ ≤ δ
and if ρ and ρ
are squared Bessel processes
with dimensions δ and δ
starting at the same point, then 0 ≤ ρ
t
≤ ρ
t
a.s..
Therefore, ρ satisfies ρ
t
≥ 0 for all t and a posteriori the absolute value under
thesquarerootin(6.1.2) is not needed.
Definition 6.1.2.1 (BESQ
δ
) For every δ ≥ 0 and x ≥ 0, the unique strong
solution to the equation
ρ
t
= x + δt +2
t
0
√
ρ
s
dW
s
,ρ
t
≥ 0
is called a squared Bessel process with dimension δ, starting at x and is
denoted by BESQ
δ
x
.
6.1 Definitions and First Properties 335
In particular, this process is a diffusion, and so is any positive power of this
process.
Definition 6.1.2.2 (BES
δ
) Let ρ be a BESQ
δ
x
. The process R =
√
ρ is called
a Bessel process of dimension δ, starting at r =
√
x and is denoted BES
δ
r
.
We also parametrize the family of Bessel processes with the index ν given by
ν =(δ/2) − 1 instead of the dimension δ. We shall write BES
(ν)
instead of
BES
δ
.
It follows from the note after the previous definition that BES
δ
is a diffusion.
• For δ>1, the BES
δ
r
is the solution of
R
t
= r + W
t
+
δ − 1
2
t
0
1
R
s
ds . (6.1.3)
In terms of the index, the BES
(ν)
process is the solution of
R
t
= r + W
t
+
ν +
1
2
t
0
1
R
s
ds .
• For δ<1, the integral
t
0
ds
R
s
does not converge and
R
t
= r + W
t
+
δ − 1
2
p.v.
t
0
1
R
s
ds ,
where the principal value is defined as
p.v.
t
0
1
R
s
ds =
∞
0
x
δ−2
(
x
t
−
0
t
)dx
and the family of diffusion local times is defined via the occupation time
formula and the speed measure (see Section 5.5):
t
0
φ(R
s
)ds =
∞
0
φ(x)
x
t
x
δ−1
dx .
For a study of principal values, see Chapter 10 in Yor [868].
• For δ =1
R
t
= r + W
t
+
1
2
L
0
t
(R) ,
where L
0
(R) is the local time of R at level 0.
We use the superscript
(ν)
for the index ν,whereasthereisnobracketin
the superscript when we refer to the dimension δ. Important particular cases
are ν =1/2 which corresponds to δ =3,ν = 0 which corresponds to δ =2
and ν = −
1
2
which corresponds to δ = 1 (reflected BM).
336 6 A Special Family of Diffusions: Bessel Processes
We now present scale functions of the Bessel processes as diffusions.
Definition 6.1.2.3 Let B be a one-dimensional Brownian motion. A process
with the same law as |B| is called a reflected Brownian motion.
Proposition 6.1.2.4 Let ρ be a squared Bessel process with dimension δ.A
scale function is
− x
1−(δ/2)
for δ>2; ln x for δ =2; x
1−(δ/2)
for δ<2 .
Let R be a Bessel process with dimension δ.Ascalefunctionis
− x
2−δ
for δ>2; ln x for δ =2; x
2−δ
for δ<2 .
Proof: The result can be checked from an application of Itˆo’s lemma.
Warning 6.1.2.5 A blind application of the result “the scale function of a
Bessel process with dimension 1 is s(x)=x” would lead to the false conclusion
that |W | is a local martingale.
Example 6.1.2.6 Strict Local Martingale. Here, we give an example
of a local martingale which is not a martingale, i.e., a strict local mar-
tingale.LetM be a continuous martingale such that M
0
= 1 and define
T
0
=inf{t : M
t
=0}. We assume that P (T
0
< ∞)=1. We introduce the
probability measure Q as Q|
F
t
= M
t∧T
0
P|
F
t
. It follows that
Q(T
0
<t)=E
P
(1
T
0
<t
M
t∧T
0
)=0, (6.1.4)
i.e., Q (T
0
= ∞)=1. The process X defined by (X
t
= M
−1
t
,t ≥ 0) is a Q-
local martingale and is positive. It is not a martingale: indeed its expectation
is not constant
E
Q
(X
t
)=E
P
M
t∧T
0
M
t
= P(t<T
0
) =1=X
0
.
From Girsanov’s theorem, the process
M
t
= M
t
−
t
0
d M
s
M
s
is a Q-local
martingale. In the case M
t
= B
t
, we get
B
t
= β
t
+
t
0
ds
B
s
where β is a Q-Brownian motion. Hence, the process B is a Q-Bessel process
of dimension 3. See Wong and Heyde [848], Elworthy et al. [319], Kotani
[539], Kotani and Sin [799] and Watanabe [837] for comments on strict local
martingales.