Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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620 11 L´evy Processes
Ψ(a)=
σ
2
2
a
2
+ am +
0
−∞
ν(dx)
e
ax
− 1 − ax1
{|x|≤1}
is the Laplace exponent of X.
Proof: We apply the previous proposition for f(x)=e
−ax
.SinceX is a
spectrally negative L´evy process, the process Σ is continuous. We obtain that
e
−a(Σ
t
−X
t
)
+ aΣ
t
−
σ
2
2
t
0
a
2
e
−a(Σ
s
−X
s
)
ds − m
t
0
ae
−a(Σ
s
−X
s
)
ds
−
t
0
ds
ν(dx)
e
−a(Σ
s
−X
s
−x)
− e
−a(Σ
s
−X
s
)
− ae
−a(Σ
s
−X
s
)
x1
{|x|≤1}
is a local martingale. This last expression is equal to
e
−a(Σ
t
−X
t
)
+ aΣ
t
−
t
0
ds e
−a(Σ
s
−X
s
)
Ψ(a) .
In the case where X = B is a Brownian motion, we obtain that
e
−a(Σ
t
−B
t
)
+ aΣ
t
−
a
2
2
t
0
ds e
−a(Σ
s
−B
s
)
is a martingale, or
e
−a|B
t
|
+ aL
t
−
a
2
2
t
0
ds e
−a|B
s
|
is a martingale, where L is the local time at 0 for B.
11.2.7 Harness Property
Proposition 11.2.7.1 Any L´evy process such that E(|X
1
|) < ∞ enjoys the
harness property given in Definition 8.5.2.1.
Proof: Let X be a L´evy process. Then, for s<t
E[exp(i(λX
s
+ μX
t
))] = exp(−sΦ(λ + μ) − (t − s)Φ(μ)) .
Therefore, by differentiation with respect to λ, and taking λ =0,
iE[X
s
exp(iμX
t
)] = −sΦ
(μ)exp(−tΦ(μ))
which implies
tE[X
s
exp(iμX
t
)] = sE[X
t
exp(iμX
t
)] .
It follows that
11.2 L´evy Processes 621
E
X
s
s
%
%
%
σ(X
u
,u≥ t)
=
X
t
t
.
Then, using the homogeneity of the increments and recalling the notation
F
s], [T
= σ(X
u
,u≤ s, u ≥ T ) already given in Definition 8.5.2.1
E
X
t
− X
s
t − s
%
%
F
s], [T
=
X
T
− X
s
T − s
.
See Proposition 8.5.2.2, and Mansuy and Yor [621] for more comments on
the harness property.
Exercise 11.2.7.2 The following amplification of Proposition 11.2.7.1 is due
to Pal. Let X be a L´evy process and τ =(t
k
,k ∈ N) a sequence of subdivisions
of R
+
and define F
(τ)
= σ(X
t
k
,k ∈ N).
Prove that E(X
t
|F
(τ)
)=X
(τ)
t
where
X
(τ)
t
=
k
X
t
k
+
t − t
k
t
k+1
− t
k
X
t
k+1
− X
t
k
1
t
k
<t≤t
k+1
is the linear interpolation of X along the subdivision τ.Ifτ ⊂ τ
,provethat
E(X
(τ)
t
|F
(τ
)
)=X
(τ
)
t
.
Hint: Write, for t
k
<t<t
k+1
E
X
t
− X
t
k
t − t
k
|X
t
k
,X
t
k+1
=
1
t
k+1
− t
k
X
t
k+1
− X
t
k
.
11.2.8 Representation Theorem of Martingales in a L´evy Setting
Proposition 11.2.8.1 Let X be an R
d
-valued L´evy process and F
X
its
natural filtration. Let M be an F
X
-local martingale. Then, there exist an R
d
-
valued predictable process ϕ =(ϕ
i
) and ψ : R
+
× Ω × R
d
→ R a predictable
function such that
t
0
(ϕ
i
s
)
2
ds < ∞, a.s.,
t
0
|x|≤1
|ψ(s, x)|ds ν(dx) < ∞, a.s.,
t
0
|x|>1
ψ
2
(s, x)ds ν(dx) < ∞, a.s.
and
M
t
= M
0
+
d
i=1
t
0
ϕ
i
s
dW
i
s
+
t
0
R
d
ψ(s, x)
N(ds, dx) .
622 11 L´evy Processes
Moreover, if (M
t
,t≤ T ) is a square integrable martingale, then
E
⎡
⎣
T
0
ϕ
i
s
dW
i
s
2
⎤
⎦
= E
T
0
(ϕ
i
s
)
2
ds
< ∞,
E
⎡
⎣
T
0
ψ(s, x)
N(ds, dx)
2
⎤
⎦
= E
T
0
ds
ψ
2
(s, x)ν(dx)
< ∞.
The processes ϕ
i
,ψ are essentially unique.
Proof: See Kunita and Watanabe [550] and Kunita [549].
Proposition 11.2.8.2 If L is a strictly positive local martingale such that
L
0
=1, then there exist
a predictable process f =(f
1
,...,f
d
) such that
T
0
|f
s
|
2
ds < ∞,
a predictable function g where
T
0
|x|≤1
|e
g(s,x)
− 1|ds ν(dx) < ∞,
T
0
|x|>1
g
2
(s, x) ds ν(dx) < ∞,
such that
dL
t
= L
t
−
d
i=1
f
i
(t)dW
i
t
+
(e
g(t,x)
− 1)
N(dt, dx)
.
In a closed form, we obtain
L
t
=exp
d
i=1
t
0
f
i
(s)dW
i
s
−
1
2
t
0
|f
s
|
2
ds
× exp
N
t
(g
0
) −
t
0
ds
|x|≤1
(e
g
0
(s,x)
− 1) ν(dx)
× exp
N
t
(g
1
) −
t
0
ds
|x|>1
(e
g
1
(s,x)
− 1 − g
1
(s, x)) ν(dx)
,
with g
0
(s, x)=1
{|x|≤1}
g(s, x),g
1
(s, x)=1
{|x|>1}
g(s, x).
Comment 11.2.8.3 Nualart and Schoutens [682] have established the fol-
lowing predictable representation theorem for L´evy processes which satisfy
1
{|x|≥1}
e
δ|x|
ν(dx) < ∞ for some δ>0. These processes have moments of
all orders. The processes
X
(1)
s
= X
s
X
(i)
s
=
0<u≤s
(ΔX
u
)
i
,i≥ 2
are L´evy processes and, from Proposition 11.2.2.3, E(X
(i)
s
)=s
x
i
ν(dx).
11.3 Absolutely Continuous Changes of Measures 623
Let Y
(i)
be martingales defined as Y
(1)
s
= X
s
− E(X
s
) and, for i ≥ 2,
Y
(i)
s
=
0<u≤s
(ΔX
u
)
i
− E
⎛
⎝
0<u≤s
(ΔX
u
)
i
⎞
⎠
.
Let H
(i)
be a set of pairwise strongly orthogonal martingales, obtained by the
Gram-Schmidt orthogonalization procedure of Y
(i)
. Then, for F ∈ L
2
(F
T
),
there exist predictable processes (ϕ
(i)
) such that
F = E(F )+
∞
i=1
T
0
ϕ
(i)
s
dH
(i)
s
.
See Corcuera et al. [195] and Leon et al. [577] for applications.
11.3 Absolutely Continuous Changes of Measures
11.3.1 Esscher Transform
Let X be a L´evy process, and assume that E(e
θ X
t
) < ∞ for some θ ∈ R
d
.
We define a probability P
(θ)
, locally equivalent to P by the formula
P
(θ)
|
F
t
=
e
θ X
t
E(e
θ X
t
)
P|
F
t
. (11.3.1)
Note that, from Proposition 11.2.6.3, the process
L
t
=
e
θ X
t
E(e
θ X
t
)
= e
θ X
t
−tΨ(θ)
is a martingale. This particular choice of measure transformation, (called a θ-
Esscher transform, or exponential tilting) preserves the L´evy process property
as we now prove.
Proposition 11.3.1.1 Let X be a P-L´evy process with parameters (m, A, ν).
Let θ be such that E(e
θ X
t
) < ∞ and suppose that P
(θ)
is defined by (11.3.1).
Then X is a P
(θ)
-L´evy process and the L´evy-Khintchine representation of X
under P
(θ)
is
E
P
(θ)
(e
iu X
t
) = exp
iu m
(θ)
−
u Au
2
+
R
d
(e
iu x
− 1 − iu x1
|x|≤1
)ν
(θ)
(dx)
with
m
(θ)
= m +
1
2
(A + A
T
)θ +
|x|≤1
x(e
θ x
− 1)ν(dx) ,
ν
(θ)
(dx)=e
θ x
ν(dx) .
624 11 L´evy Processes
The characteristic exponent of X under P
(θ)
is
Φ
(θ)
(u)=Φ(u − iθ) − Φ(−iθ) ,
and the Laplace exponent is Ψ
(θ)
(u)=Ψ(u + θ) − Ψ (θ) for u ≥ min(−θ, 0).
Proof: It is not difficult to prove that X has independent and stationary
increments under P
(θ)
. The characteristic exponent of X under P
(θ)
is Φ
(θ)
such that
e
−tΦ
(θ)
(u)
= E
P
(θ)
(e
iu X
t
)=E(e
iu X
t
+θ X
t
)e
tΦ(−iθ)
= e
−t(Φ(u−iθ)−Φ(−iθ))
.
A simple computation leads to
Φ(u − iθ) − Φ(−iθ)=−iu m +
1
2
u Au −
1
2
iu Aθ −
1
2
iθ Au
−
e
θ x
(e
iu x
− 1) − iu x1
{|x|≤1}
ν(dx)
= −iu
m +
1
2
(A + A
T
)θ +
(e
θ x
− 1)x1
{|x|≤1}
ν(dx)
+
1
2
u Au +
e
θ x
(e
iu x
− 1 − iu x1
{|x|≤1}
)ν(dx) .
Hence, X
1
has the required L´evy-Khintchine representation under P
(θ)
.
Corollary 11.3.1.2 Let X be a (m, σ
2
,ν) real-valued L´evy process and β
such that
|e
βx
(e
x
− 1) − x1
{|x|≤1}
|ν(dx) < ∞
and
m +
β +
1
2
σ
2
+
e
βx
(e
x
− 1) − x1
{|x|≤1}
ν(dx)=0.
Let P
(β)
|
F
t
=
e
βX
t
E(e
βX
t
)
P|
F
t
. Then, the process (e
X
t
,t ≥ 0) is a P
(β)
-
martingale.
Proof: This is a consequence of Proposition 11.3.1.1 and Corollary 11.2.6.4.
Comment 11.3.1.3 The Esscher transform was introduced by Esscher [335]
and again by Gerber and Shiu [388]. See B¨uhlmann et al. [136] for a discussion
on Esscher transforms in finance and Fujiwara and Miyahara [369]forsome
relations between the Esscher transform and minimal entropy measure.
11.3 Absolutely Continuous Changes of Measures 625
Exercise 11.3.1.4 Let X be a real-valued L´evy process such that, for any
λ, E(e
λX
t
) < ∞ and define
P
(λ)
|
F
t
=
e
λX
t
E(e
λX
t
)
P|
F
t
.
Prove that (P
(λ)
)
(μ)
= P
(λ+μ)
and that if S
t
= S
0
e
X
t
, then, for any μ, for any
positive Borel function g,
E
(λ)
[S
μ
t
g(S
t
)] = E
(λ)
[S
μ
t
] E
(λ+μ)
[g(S
t
)] .
Extend this formula to R
d
-valued L´evy processes.
Exercise 11.3.1.5 Prove that the Esscher transforms of the 1/2-stable
process are the inverse Gaussian processes. (See Example 11.6.1.2 for the
definition of Inverse Gaussian processes.)
Exercise 11.3.1.6 Prove that the Esscher transforms of a double exponential
process are double exponential processes.
Exercise 11.3.1.7 Let γ be a Γ (1, 1) process:
E(e
−λγ
t
) = exp
−t
∞
0
dx
x
e
−x
(1 − e
−λx
)
=
1
(1 + λ)
t
.
(a) Prove the double identity, for 1 + μ>0
E(e
−λγ
t
e
−μγ
t
(1 + μ)
t
)=
1
(1 +
λ
1+μ
)
t
= E(e
−
λ
1+μ
γ
t
) .
(b) Conclude that the (−μ)-Esscher transform of γ
t
is
1
1+μ
γ
t
,inother
terms, any multiple (aγ
t
,t≥ 0) is an Esscher transform of (γ
t
,t≥ 0).
(c) Prove that, for a>0, the process (a
γ
t
γ
T
,t≤ T ) is distributed as the γ
bridge process on [0,T], starting from 0 and ending at a,attimeT .
11.3.2 Preserving the L´evy Property with Absolute Continuity
Given a L´evy process X under P with generating triple (m, A, ν), we define,
following Sato [761], its compensated jump part as
X
ν
t
= lim
→0
(s,ΔX
s
)∈(0,t]×{|x|>}
ΔX
s
− t
<|x|≤1
xν(dx) .
We now examine which other Girsanov-type transformations than Esscher
transforms preserve the L´evy property of a L´evy process.
626 11 L´evy Processes
Proposition 11.3.2.1 Let (X, P) (resp. (X, P
∗
))beL´evy processes with
generating triples (m, A, ν) (resp. (m
∗
,A
∗
,ν
∗
)). The following statements are
equivalent:
(i) P and P
∗
are locally equivalent,
(ii) A = A
∗
, dν
∗
(x)=e
ϕ(x)
dν(x) with the function ϕ satisfying
R
d
(e
ϕ(x)/2
− 1)
2
ν(dx) < ∞
and
m
∗
− m −
|x|≤1
x (ν
∗
− ν)(dx) ∈R(A)
where R(A)={Ax, x ∈ R
d
}.
Furthermore, if the previous conditions (ii) are satisfied
P
∗
|
F
t
= e
U
t
P|
F
t
,
where
U
t
= η (X
t
− X
ν
t
) −
t
2
η Aη − tm η
+ lim
→0
s≤t, |ΔX
s
|>
ϕ(ΔX
s
) − t
<|x|≤1
(e
ϕ(x)
− 1)ν(dx) ,
and η is such that m
∗
− m −
|x|≤1
x (ν
∗
− ν)(dx)=Aη.
Proof: See Sato [761], Theorem 33.1. Note that (e
U
t
,t ≥ 0) is a martingale
and that U is a L´evy process with generating triple
σ
2
U
= η Aη,
m
U
= −
1
2
η Aη −
R
(e
y
− 1 − y1
{|y|≤1}
(y)) ν
U
(dy) ,
ν
U
=(ϕ(ν)) |
R\{0}
i.e., ν
U
is the image of ν by ϕ.
Example 11.3.2.2 Let X
1
and X
2
be CGMY L´evy processes (see Section
11.8) with parameters (C, G
i
,M
i
,Y),i =1, 2. Then, it is easy to check that
the hypotheses of Proposition 11.3.2.1 are satisfied. See Cont and Tankov [192]
Chapter 9, Example 9.2 for the case of tempered stable processes.
11.3 Absolutely Continuous Changes of Measures 627
11.3.3 General Case
To complete this discussion, we study the Girsanov transformation of
L´evy processes using a general positive martingale density L. In this generality,
the L´evy property will be lost under the new probability. From the previous
martingale representation Theorem 11.2.8.2, we can and do associate with the
martingale L a pair (f,g) such that
dL
t
= L
t
−
d
i=1
f
i
t
dW
i
t
+
(e
g(t,x)
− 1)[N(dt, dx) − ν(dx) dt]
. (11.3.2)
Sometimes, we shall denote this process L by L(f,g).
Proposition 11.3.3.1 Let Q|
F
t
= L
t
(f,g) P|
F
t
where L(f,g) is defined in
(11.3.2). Then, with respect to Q:
(i) The process W
f
defined by W
f
t
:=W
t
−
t
0
f
s
ds is a Brownian motion.
(ii) The random measure N is compensated by e
g(s,x)
ds ν(dx) meaning that
for any Borel function h such that
T
0
ds
R
|h(s, x)|e
g(s,x)
ν(dx) < ∞,
the process
M
h
t
:=
t
0
R
h(s, x)
N(ds, dx) − e
g(s,x)
ν(dx) ds
is a local martingale.
Proof: Using Itˆo’s calculus, it is easy to check that W
f
L and M
h
L are P-
martingales.
The P-L´evy process X is a Q-semi-martingale; in general, it is not a
L´evy process, nor an additive process. One should note the important gap
between the framework of Propositions 11.3.2.1 and 11.3.3.1: with the latter,
the Markovian property may be lost.
Comment 11.3.3.2 A frequently asked question is to find a condition for
a local exponential martingale to be a martingale. In order that L(f, g)isa
martingale, Kunita [549] gives the following condition: if (f,g) satisfies
E
exp
t
0
af
2
+
|g|>δ
e
2ag
+
dν +2ae
2aδ
|g|≤δ
g
2
dν
ds
< ∞
for some a>1andsomeδ>0, then L(f, g) is a martingale.
628 11 L´evy Processes
11.4 Fluctuation Theory
Here, X is a real-valued L´evy process. We are interested in the law of the
running maximum.
11.4.1 Maximum and Minimum
We start with a simple lemma.
Lemma 11.4.1.1 Let X be a L´evy process and Θ be an exponential variable
with parameter q, independent of X. Then, the pair (Θ, X
Θ
) has an infinitely
divisible law, and its L´evy measure is μ(dt, dx)=t
−1
e
−qt
P(X
t
∈ dx)dt.
Proof: For any pair α, β, with α>0, we have
E(e
−αΘ+iβX
Θ
)=
∞
0
qe
−qt
e
−tα−tΦ(β)
dt
=exp
∞
0
(e
−tα−tΦ(β)
− 1)t
−1
e
−qt
dt
where in the second equality, we have used the Frullani integral
∞
0
(1 − e
−λt
)t
−1
e
−bt
dt =ln
1+
λ
b
.
It remains to write
∞
0
(e
−tα−tΦ(β)
− 1)t
−1
e
−qt
dt =
∞
0
R
(e
−tα+iβx
− 1) P(X
t
∈ dx) t
−1
e
−qt
dt .
Let Σ
t
=sup
s≤t
X
s
be the running maximum of the L´evy process X.The
reflected process Σ − X enjoys the strong Markov property.
Proposition 11.4.1.2 (Wiener-Hopf Factorization.) Let Θ be an ex-
ponential variable with parameter q, independent of X. Then, the random
variables Σ
Θ
and X
Θ
− Σ
Θ
are independent and
E(e
iuΣ
Θ
)E(e
iu(X
Θ
−Σ
Θ
)
)=
q
q + Φ(u)
. (11.4.1)
Sketch of the proof: Note that
E(e
iuX
Θ
)=q
∞
0
E(e
iuX
t
)e
−qt
dt = q
∞
0
e
−tΦ(u)
e
−qt
dt =
q
q + Φ(u)
.
Using excursion theory, the random variables Σ
Θ
and X
Θ
−Σ
Θ
are shown to
be independent (see Bertoin [78]), hence
11.4 Fluctuation Theory 629
E(e
iuΣ
Θ
)E(e
iu(X
Θ
−Σ
Θ
)
)=E(e
iuX
Θ
)=
q
q + Φ(u)
.
The equality (11.4.1) is known as the Wiener-Hopf factorization,the
factors being the characteristic functions E(e
iuΣ
Θ
)andE(e
iu(X
Θ
−Σ
Θ
)
). We
now prepare for the computation of these factors.
There exists a family (L
x
t
) of local times of the reflected process Σ − X
which satisfies
t
0
dsf(Σ
s
− X
s
)=
∞
0
dxf(x)L
x
t
for all positive functions f. We then consider L = L
0
and τ its right continuous
inverse τ
=inf{u>0:L
u
>}. Introduce
H
= Σ
τ
for τ
< ∞,H
= ∞ otherwise . (11.4.2)
The two-dimensional process (τ, H) is called the ladder process and is a
L´evy process.
Proposition 11.4.1.3 Let κ be the Laplace exponent of the ladder process
defined as
e
−κ(α,β)
= E(exp(−ατ
− βH
)) .
There exists a constant k>0 such that
κ(α, β)=k exp
∞
0
dt
∞
0
t
−1
(e
−t
− e
−αt−βx
) P(X
t
∈ dx)
.
Sketch of the proof:
Using excursion theory, and setting G
Θ
=sup{t<Θ : X
t
= Σ
t
},itcanbe
proved that
E(e
−αG
Θ
−βΣ
Θ
)=
κ(q, 0)
κ(α + q, β)
. (11.4.3)
The pair of random variables (Θ, X
Θ
) can be decomposed as the sum of
(G
Θ
,Σ
Θ
)and(Θ − G
Θ
,X
Θ
− Σ
Θ
), which are shown to be independent
infinitely divisible random variables. Let μ, μ
+
and μ
−
denote the respective
L´evy measures of these three two-dimensional variables. The L´evy measure
μ
+
(resp μ
−
) has support in [0, ∞[×[0, ∞[(resp.[0, ∞[×] −∞, 0]) and
μ = μ
+
+ μ
−
. From Lemma 11.4.1.1,theL´evy measure of (Θ, X
Θ
)is
t
−1
e
−qt
P(X
t
∈ dx) dt and noting that this quantity is
t
−1
e
−qt
P(X
t
∈ dx) dt1
{x>0}
+ t
−1
e
−qt
P(X
t
∈ dx) dt1
{x<0}
one establishes that μ
+
(dt, dx)=t
−1
e
−qt
P(X
t
∈ dx) dt 1
{x>0}
.
It follows from (11.4.3)that
κ(q, 0)
κ(α + q, β)
=exp
−
∞
0
∞
0
(1 − e
−αt−βx
) μ
+
(dt, dx)
.