Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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610 11 L´evy Processes
This result is often used to yield the following representation:
X
t
= mt + Z
t
+
t
0
x1
{|x|≤1}
N(ds, dx)+
t
0
x1
{|x|>1}
N(ds, dx)
(11.2.4)
where Z is an R
d
-valued Brownian motion with correlation matrix A, N the
random measure
1
of the jumps of X,and
N(dt, dx)=N(dt, dx) −dt ν(dx)is
the compensated martingale measure. In other words,
X
t
= mt + Z
t
+
t
0
x1
{|x|≤1}
N(ds, dx)+
s≤t
ΔX
s
1
{|ΔX
s
|>1}
.
If
{|x|≤1}
|x|ν(dx) < ∞, the process X can be represented as
X
t
= m
0
t + Z
t
+
t
0
xN(ds, dx)
where m
0
= m −
x1
{|x|≤1}
ν(dx). If ν is a finite measure, the process
t
0
xN(ds, dx) is a compound Poisson process: indeed, let T
k
,k ≥ 1bethe
jump times of X and N
t
=
1
{T
k
≤t}
the associated counting process. The
process N is a Poisson process with intensity λ = ν(R
d
\ 0). Furthermore,
the random variables Y
k
=
xN({T
k
},dx) are i.i.d. with law ν(dx)/λ and
N
t
1
Y
k
=
t
0
xN(ds, dx).
If
{|x|≤1}
|x|ν(dx)=∞, it is not possible to separate the integral
|x|≤1
x(N
t
(·,dx) − tν(dx))
into the two parts
|x|≤1
xN
t
(·,dx)andt
|x|≤1
xν(dx)whichmaybothbe
ill-defined.
Corollary 11.2.3.8 AL´evy process is a semi-martingale.
Proof: This is a consequence of the L´evy-Itˆo decomposition.
Comment 11.2.3.9 As a direct consequence, the series
s≤t
(ΔX
s
)
2
con-
verges, and for any constant a the process
s≤t
(ΔX
s
)
2
∧ a
− t
(x
2
∧ a) ν(dx)
1
We indicate these random measures in boldface to avoid possible confusion with
Poisson processes which are always denoted by N.
11.2 L´evy Processes 611
is a martingale. It follows that
(x
2
∧ a) ν(dx) < ∞, which justifies the
integrability condition on the L´evy measures.
Lemma 11.2.3.10 Let F be a C
2
function and X aL´evy process. Then, the
series
s≤t
|F (ΔX
s
) − F (0) − F
(0)ΔX
s
|
converges.
Proof: The series
0<s≤t
1
{|ΔX
s
|>1}
(F (ΔX
s
) − F (0) − F
(0)ΔX
s
)isobvi-
ously convergent. Furthermore, the inequality
0 ≤|F (x) − F (0) − xF
(0)|≤cx
2
for |x|≤1 and the convergence of the series
s≤t
(ΔX
s
)
2
imply that the
series
0<s≤t
1
{|ΔX
s
|≤1}
|F (ΔX
s
) − F (0) − F
(0)ΔX
s
| is also convergent.
Proposition 11.2.3.11 Let X be a (m, σ
2
,ν)-L´evy process taking values in
R.Then,X is increasing if and only if
σ =0,ν(−∞, 0) = 0,
1
{x≤1}
xν(dx) < ∞,m
0
= m−
1
{x≤1}
xν(dx) ≥ 0 .
Proof: If ν(−∞, 0) = 0, the process X has no negative jumps. Hence,
assuming moreover that σ = 0, and the convergence of
xν(dx), we obtain
X
t
=
t
0
xN(ds, dx)+tm
0
and, if m
0
≥ 0, X is increasing.
For the converse, since X has no negative jumps, ν(−∞, 0) = 0. The proof
of
1
{x≤1}
xν(dx) < ∞ relies on the remark that an increasing function
remains increasing after deleting a finite number of its jumps. We shall delete
jumps of size greater than . Then, setting X
t
=
t
0
],∞[
xN(dx, ds), the
process X
t
− X
t
is increasing, hence X
t
− X
t
≥ 0. It follows that
lim
→0
X
t
=
t
0
R
+
xN(dx, ds)=:
X
t
exists and is bounded by X
t
.Now,
E(e
−λX
t
) = exp
t
],∞[
(e
−λx
− 1)ν(dx)
=exp
t
],∞[
(e
−λx
− 1+λx1
{x≤1}
)ν(dx) − tλ
],1]
xν(dx)
.
As tends to 0, the quantity E(e
−λX
t
) tends to E(e
−λ
b
X
t
) and the quantity
],∞[
(e
−λx
− 1+λx1
{x≤1}
)ν(dx) tends to
R
+
(e
−λx
− 1+λx1
{x≤1}
)ν(dx)
which is finite. It follows that
]0,1]
xν(dx) < ∞. The conclusion of the proof
can be found in Sato [761], Theorem 21.5 page 137.
612 11 L´evy Processes
Exercise 11.2.3.12 Let X be a (m, σ
2
,ν) real-valued L´evy process. Check
that the quadratic variation of the semi-martingale X is
[X]
t
= σ
2
t +
t
0
R
x
2
N(ds, dx) .
The predictable quadratic variation of X is
X
t
= σ
2
t + t
R
x
2
ν(dx) ,
as long as
R
x
2
ν(dx) < ∞.
Exercise 11.2.3.13 Let X be a L´evy process without drift or continuous
part, and f ∈ L
1
(ν). Prove that M
f
t
:=
s≤t
f(ΔX
s
) − t
ν(dx)f(x)is
a martingale. Prove that, for f, g satisfying some integrability conditions,
M
f
,M
g
t
= t
ν(dx) f(x) g(x).
11.2.4 Itˆo’s Formulae for a One-dimensional L´evy Process
Let X be a (m, σ
2
,ν) real-valued L´evy process and f ∈ C
1,2
(R
+
×R, R). Then,
since X is a semi-martingale, we can apply Itˆo’s formula in the optional or
predictable form:
The optional Itˆo’s formula is
f(t, X
t
)=f(0,X
0
)+
t
0
∂
t
f(s, X
s
)ds
+
σ
2
2
t
0
∂
xx
f(s, X
s
)ds +
t
0
∂
x
f(s, X
s
−
)dX
s
+
s≤t
f(s, X
s
−
+ ΔX
s
) − f (s, X
s
−
) − (ΔX
s
) ∂
x
f(s, X
s
−
) .
If f and its first and second derivatives w.r.t. x are bounded, the predictable
Itˆo’s formula is
df (t, X
t
)=∂
x
f(t, X
t
)σdW
t
+
R
(f(t, X
t
−
+ x) − f (t, X
t
−
))
N(dt, dx)
+
1
2
σ
2
∂
xx
f(t, X
t
)+m∂
x
f(t, X
t
)+∂
t
f(t, X
t
)
+
R
f(t, X
t
−
+ x) − f (t, X
t
−
) − x∂
x
f(t, X
t
−
)1
|x|≤1
ν(dx)
dt .
The quantity ∂
x
f(t, X
t
)σdW
t
+
x∈R
(f(t, X
t
−
+ x) − f (t, X
t
−
))
N(dt, dx)is
the local martingale part of the semi-martingale f(t, X
t
), written in “differ-
ential ” form, and
11.2 L´evy Processes 613
1
2
σ
2
∂
xx
f(t, X
t
)+m∂
x
f(t, X
t
)+∂
t
f(t, X
t
)
+
R
"
f(t, X
t
−
+ x) − f (t, X
t
−
) − x∂
x
f(t, X
t
−
)1
|x|≤1
#
ν(dx)
dt
is the continuous finite variation part, again written in differential form. One
may note that this formula is a generalization of the formula obtained for
compound Poisson processes (see Subsection 8.6.4).
11.2.5 Itˆo’s Formula for L´evy-Itˆo Processes
In this section, we step out of the framework of L´evy processes, by considering
the more general class of L´evy-Itˆo processes
dX
t
= a
t
dt + σ
t
dW
t
+
1
{|x|≤1}
γ
t
(x)
N(dt, dx)+
1
{|x|>1}
γ
t
(x)N(dt, dx) ,
(11.2.5)
where we assume that a and σ are adapted processes, γ(x) is predictable and
that the integrals are well defined. These processes are semi-martingales. Itˆo’s
formula can be generalized as follows
Proposition 11.2.5.1 (Itˆo’s Formula for L´evy-Itˆo Processes.) Let X
be defined as in (11.2.5).Iff is bounded and f ∈ C
1,2
b
, then the process
Y
t
:= f(t, X
t
) is a semi-martingale:
dY
t
= ∂
x
f(t, X
t
)σ
t
dW
t
+
|x|≤1
(f(t, X
t−
+ γ
t
(x)) − f(t, X
t−
))
N(dt, dx)
+
∂
t
f(t, X
t
)+∂
x
f(t, X
t
)a
t
+
1
2
σ
2
t
∂
xx
f(t, X
t
)
dt
+
|x|>1
(f(t, X
t
−
+ γ
t
(x)) − f (t, X
t
−
)) N(dt, dx)
+
|x|≤1
(f(t, X
t
−
+ γ
t
(x)) − f(t, X
t
−
) − γ
t
(x)∂
x
f(t, X
t
−
)) ν(dx)
dt .
Note that in the last integral, we do not need to write (X
t
−
); (X
t
) would also
do, but the minus sign may help to understand the origin of this quantity.
More generally, let X
i
,i=1,...,n be n processes with dynamics
X
i
t
= X
i
0
+ V
i
t
+
m
k=1
t
0
f
i,k
(s)dW
k
s
+
t
0
|x|>1
g
i
(s, x)N(ds, dx)
+
t
0
|x|≤1
h
i
(s, x)(N(ds, dx) − ν(dx) ds)
614 11 L´evy Processes
where V is a continuous bounded variation process. Then, for Θ a C
2
b
function
Θ(X
t
)=Θ(X
0
)+
n
i=1
t
0
∂
x
i
Θ(X
s
)dV
i
s
+
n
i=1
m
k=1
t
0
f
i,k
(s)∂
x
i
Θ(X
s
)dW
k
s
+
1
2
n
i,j=1
t
0
m
k=1
f
i,k
(s)f
j,k
(s)∂
x
i
x
j
Θ(X
s
)ds
+
t
0
|x|>1
(Θ(X
s
−
+ g(s, x)) − Θ(X
s
−
))N(ds, dx)
+
t
0
|x|≤1
(Θ(X
s
−
+ h(s, x)) − Θ(X
s
−
))
N(ds, dx)
+
t
0
|x|≤1
(Θ(X
s
−
+ h(s, x)) − Θ(X
s
−
) −
n
i=1
h
i
(s, x)∂
x
i
Θ(X
s
−
))ds ν(dx) .
Exercise 11.2.5.2 Let W be a Brownian motion and N a random Poisson
measure. Let X
i
t
=
t
0
ϕ
i
s
dW
s
+
t
0
ψ
i
(s, x)
N(ds, dx),i =1, 2 be two real-
valued martingales.
Prove that [X
1
,X
2
]
t
=
t
0
ϕ
1
s
ϕ
2
s
ds +
t
0
ψ
1
(s, x)ψ
2
(s, x)N(ds, dx)and
that, under suitable conditions
X
1
,X
2
t
=
t
0
ϕ
1
s
ϕ
2
s
ds +
t
0
ψ
1
(s, x)ψ
2
(s, x) ν(dx) ds .
Exercise 11.2.5.3 Prove that, if
S
t
= S
0
exp
bt + σW
t
+
t
0
|x|≤1
x
N(ds, dx)+
t
0
|x|>1
xN(ds, dx)
then
dS
t
= S
t
−
b +
1
2
σ
2
dt + σdW
t
+
|x|≤1
(e
x
− 1 − x)ν(dx) dt
+
|x|≤1
(e
x
− 1)
N(dt, dx)+
|x|>1
(e
x
− 1)N(dt, dx)
.
Exercise 11.2.5.4 Geometric Itˆo-L´evy process This is a generalization
of the previous exercise. Prove that the solution of
dX
t
= X
t
−
adt + σdW
t
+
|x|≤1
γ(t, x)
N(dt, dx)+
|x|>1
γ(t, x)N(dt, dx)
11.2 L´evy Processes 615
and X
0
= 1 where γ(t, x) ≥−1anda, σ are constant, is
X
t
=exp
a −
1
2
σ
2
t + σW
t
+
t
0
ds
|x|≤1
(ln(1 + γ(s, x)) − γ(s, x)) ν(dx)
+
t
0
|x|≤1
ln(1 + γ(s, x))
N(ds, dx)+
t
0
|x|>1
ln(1 + γ(s, x))N(ds, dx)
.
Exercise 11.2.5.5 Let f be a predictable (bounded) process and g(t, x)a
predictable function. Let g
1
= g1
|x|>1
and g
2
= g1
|x|≤1
. We assume that
e
g
1
−1 is integrable w.r.t.
N and that g
2
is square integrable w.r.t.
N.Prove
that the solution of
dX
t
= X
t
−
f
t
dW
t
+
(e
g(t,x)
− 1)
N(dt, dx)
is
X
t
= E(fW)
t
exp
N
t
(g
1
) −
t
0
(e
g
1
(s,x)
− 1)ν(dx) ds
exp
N
t
(g
2
) −
t
0
(e
g
2
(s,x)
− 1 − g
2
(s, x))ν(dx) ds
where N
t
(g)=
t
0
g(s, x)N(ds, dx)and
N
t
(g)=
t
0
g(s, x)
N(ds, dx).
Exercise 11.2.5.6 Let S
t
= e
X
t
where X is a (m, σ
2
,ν)-one dimensional
L´evy process and λ a constant such that E(e
λX
T
)=E(S
λ
T
) < ∞.Usingthe
fact that
(e
−tΨ(λ)
S
λ
t
= e
λX
t
−tΨ(λ)
,t≥ 0)
is a martingale, prove that S
λ
is a special-semimartingale with canonical
decomposition
S
λ
t
= S
λ
0
+ M
(λ)
t
+ A
(λ)
t
where A
(λ)
t
= Ψ(λ)
t
0
S
λ
s
ds.Provethat
M
(λ)
,M
(μ)
t
=(Ψ(λ + μ) − Ψ(λ) − Ψ (μ))
t
0
S
λ+μ
u
du .
11.2.6 Martingales
We now come back to the L´evy processes framework. Let X be a real-valued
(m, σ
2
,ν)L´evy process.
616 11 L´evy Processes
Proposition 11.2.6.1 (a) An F-L´evy process is a martingale if and only if
it is an F-local martingale.
(b) Let X be a L´evy process such that X is a martingale. Then, the process
E(X) is a martingale.
Proof: See He et al. [427], Theorem 11.4.6. for part (a) and Cont and Tankov
[192] for part (b).
Proposition 11.2.6.2 We assume that, for any t, E(|X
t
|) < ∞,whichis
equivalent to
1
{|x|≥1}
|x|ν(dx) < ∞. Then, the process (X
t
− E(X
t
),t≥ 0)
is a martingale; hence, the process (X
t
,t ≥ 0) is a martingale if and only if
E(X
t
)=0, i.e., m +
1
{|x|≥1}
xν(dx)=0.
Proof: The first part is obvious. The second part of the proposition follows
from the computation of E(X
t
) which is obtained by differentiation of the
characteristic function E(e
iuX
t
)atu = 0. The condition
1
{|x|≥1}
|x|ν(dx) <
∞ is needed for X
t
to be integrable.
Proposition 11.2.6.3 (Wald Martingale.)
For any λ such that Ψ(λ)=lnE(e
λX
1
) < ∞, the process (e
λX
t
−tΨ(λ)
,t ≥ 0)
is a martingale .
Proof: Obvious from the independence of increments.
Note that Ψ(λ) is well defined for every λ>0 in the case where the
L´evy measure has support in (−∞, 0[. In that case, the L´evy process is said
to be spectrally negative (see Section 11.5).
Corollary 11.2.6.4 The process (e
X
t
,t ≥ 0) is a martingale if and only if
|x|≥1
e
x
ν(dx) < ∞ and
1
2
σ
2
+ m +
(e
x
− 1 − x1
{|x|≤1}
)ν(dx)=0.
Proof: This follows from the above proposition and the expression of Ψ(1).
Proposition 11.2.6.5 (Dol´eans-Dade Exponential.) Let X be a real-
valued (m, σ
2
,ν)-L´evy process and Z the Dol´eans-Dade exponential of X, i.e.,
the solution of dZ
t
= Z
t
−
dX
t
,Z
0
=1.Then
Z
t
= e
X
t
−σ
2
t/2
0<s≤t
(1 + ΔX
s
)e
−ΔX
s
:= E(X)
t
.
It is important to note that the product
0<s≤t
(1 + |ΔX
s
|)e
−ΔX
s
=
0<s≤t
e
−ΔX
s
+ln(1+|ΔX
s
|)
11.2 L´evy Processes 617
is convergent: indeed, the product
0<s≤t
(1 + |ΔX
s
|1
{|ΔX
s
|>1/2}
) e
−ΔX
s
is obviously convergent, and the inequality 0 <x− ln(1 + x) <x
2
for
|x| < 1/2 together with the convergence of the series
s≤t
(ΔX
s
)
2
prove
that the product
$
s≤t
(1 + |ΔX
s
|1
{|ΔX
s
|>1/2}
)e
−ΔX
s
is convergent too. Note
that, with obvious definition, (E(X)
t
,t≥ 0) is a multiplicative L´evy process.
We now investigate, as in Goll and Kallsen [400] the link between Dol´eans-
Dade exponentials and ordinary exponentials (this is very close to the previous
Exercises 11.2.5.3 and 11.2.5.4).
Proposition 11.2.6.6 Let X be a real-valued (m, σ
2
,ν)-L´evy process.
(i) Let S
t
= e
X
t
be the ordinary exponential of the process X.The
stochastic logarithm of S (i.e., the process Y which satisfies S
t
= E(Y )
t
)
is a L´evy process and is given by
Y
t
:= L(S)
t
= X
t
+
1
2
σ
2
t −
0<s≤t
1+ΔX
s
− e
ΔX
s
.
The L´evy characteristics of Y are
m
Y
= m +
1
2
σ
2
+
(e
x
− 1)1
{|e
x
−1|≤1}
− x1
{|x|≤1}
ν(dx) ,
σ
2
Y
= σ
2
,
ν
Y
(A)=ν({x : e
x
− 1 ∈ A})=
1
A
(e
x
− 1) ν(dx) .
(ii) Let Z
t
= E(X)
t
the Dol´eans-Dade exponential of X.IfZ>0, the
ordinary logarithm of Z is a L´evy process L given by
L
t
:= ln(Z
t
)=X
t
−
1
2
σ
2
t +
0<s≤t
(ln(1 + ΔX
s
) − ΔX
s
) .
Its L´evy characteristics are
m
L
= m −
1
2
σ
2
+
ln(1 + x)1
{|ln(1+x)|≤1}
− x1
{|x|≤1}
ν(dx) ,
σ
2
L
= σ
2
,
ν
L
(A)=ν({x :ln(1+x) ∈ A} )=
1
A
(ln(1 + x)) ν(dx) .
Proof: We only prove part (i) and leave part (ii) to the reader. Note that
the series
0<s≤t
(1 + ΔX
s
− e
ΔX
s
) is absolutely convergent by application
of Lemma 11.2.3.10.
618 11 L´evy Processes
The process Y
t
= X
t
+
1
2
σ
2
t−
0<s≤t
1+ΔX
s
− e
ΔX
s
is a L´evy process,
σ
2
Y
= σ
2
,andΔY
t
= e
ΔX
t
− 1. This implies ν
Y
(dx)=(e
x
− 1) ν(dx). Using
the equality
s≤t
1+ΔX
s
− e
ΔX
s
=
t
0
(1 + x − e
x
)N(ds, dx)
we obtain that the L´evy-Itˆo decomposition of Y is (where m
Y
is defined in
Proposition 11.2.6.6)
Y
t
= mt + σB
t
+
t
0
{|x|≤1}
x
˜
N(ds, dx)+
t
0
{|x|>1}
xN(ds, dx)+
1
2
σ
2
t
−
t
0
(1 + x − e
x
)N(ds, dx)
= m
Y
t + σB
t
+
t
0
(e
x
− 1)1
{|e
x
−1|≤1}
˜
N(ds, dx)
+
t
0
(e
x
− 1)1
{|e
x
−1|>1}
N(ds, dx)
= m
Y
t + σB
t
+
t
0
y1
{|y|≤1}
˜
N
Y
(ds, dy)+
t
0
y1
{|y|>1}
N
Y
(ds, dy) .
The result follows.
The following proposition may help the reader to become more familiar
with another class of martingales for L´evy processes.
Proposition 11.2.6.7 (Asmussen-Kelly-Whitt Martingale.) Let X be
a real-valued (m, σ
2
,ν)-L´evy process:
X
t
= mt + σB
t
+
t
0
{|x|≤1}
x
N(ds, dx)+
t
0
{|x|>1}
xN(ds, dx) .
Let Z
t
= Σ
t
−X
t
,whereΣ
t
=sup
s≤t
X
s
and Σ
c
is the continuous part of the
increasing process Σ.Letf be a C
2
function. Then, the process
f(Z
t
) − f (Z
0
) − f
(0)Σ
c
t
−
σ
2
2
t
0
f
(Z
s
)ds + m
t
0
f
(Z
s
)ds
−
t
0
ds
ν(dx)[f(Z
s
−
+ h
s
(x)) − f (Z
s
−
)+x1
{|x|≤1}
f
(Z
s
−
)]
is a local martingale, where h
t
(x)=−(x ∧ Z
t
−
) is a predictable function.
Proof: Note that if, at time t, the process X has a jump of size x smaller
than Z
t
−
then Z
t
= Z
t
−
− x; if the process X has a jump of size x greater
than Z
t
−
then Σ
t
= X
t
and Z
t
= 0. In other words, the jumps of the process
11.2 L´evy Processes 619
Z are ΔZ
t
= h
t
(ΔX
t
) with h
t
(x)=−(x ∧ Z
t
−
) a predictable function. From
Itˆo’s formula, the process
Y
t
= f (Z
t
) − f (Z
0
) −
t
0
f
(Z
s
)dΣ
s
−
σ
2
2
t
0
f
(Z
s
)ds + m
t
0
f
(Z
s
)ds
−
t
0
ds
ν(dx)[f(Z
s
−
+ h
s
(x)) − f (Z
s
−
) − h
s
(x)1
{|x|≤1}
f
(Z
s
−
)]
is a local martingale.
We split the last integral into two parts
t
0
ds
Z
s−
−∞
ν(dx)[f(Z
s
−
+ h
s
(x)) − f (Z
s
−
)+x1
{|x|≤1}
f
(Z
s
−
)]
+
t
0
ds
∞
Z
s
−
ν(dx)[f(Z
s
−
+ h
s
(x)) − f (Z
s
−
)+Z
s
−
1
{|x|≤1}
f
(Z
s
−
)] .
We denote by Σ
c
the continuous part of the increasing process Σ and by
Σ
d
t
=
s≤t
ΔΣ
s
its discontinuous part, then
t
0
f
(Z
s
−
)dΣ
s
=
t
0
f
(Z
s
)dΣ
c
s
+
t
0
f
(Z
s
−
)dΣ
d
s
.
The support of the measure dΣ
c
t
is
{t : X
t
−
= X
t
= Σ
t
= Σ
t
−
} = {t : Z
t
= Z
t
−
=0}
hence
t
0
f
(Z
s
−
)dΣ
c
s
= f
(0)Σ
c
t
. It remains to use
−
t
0
f
(Z
s
)dΣ
d
s
+
t
0
ds f
(Z
s
)
∞
Z
s
ν(dx)Z
s
1
{|x|≤1}
=
t
0
ds f
(Z
s
)
∞
Z
s
ν(dx)x1
{|x|≤1}
.
We recall that a spectrally negative L´evy process is a L´evy process whose
L´evy measure ν has its support in ] −∞, 0[.
Corollary 11.2.6.8 Let X be a spectrally negative L´evy process. Then, the
one-sided maximum of X, i.e., Σ
t
=sup
s≤t
X
s
is continuous. For a>0, the
process
e
−a(Σ
t
−X
t
)
+ aΣ
t
− Ψ(a)
t
0
e
−a(Σ
s
−X
s
)
ds
is a local martingale, where