Elect. Comm. in Probab. 14 (2009), 202–209
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
AN OPTIMAL ITÔ FORMULA FOR LÉVY PROCESSES
NATHALIE EISENBAUM
LPMA, UMR 7599, CNRS-Université Paris 6 - 4, place Jussieu, 75252 Paris Cedex 05, France
email: nathalie.eisenbaum@upmc.fr
ALEXANDER WALSH
LPMA, UMR 7599, CNRS-Université Paris 6 - 4, place Jussieu, 75252 Paris Cedex 05, France
email: awalshz@gmail.com
Submitted December 15, 2008, accepted in final form April 20, 2009
AMS 2000 Subject classification: 60G44, 60H05, 60J55, 60J65
Keywords: stochastic calculus, Lévy process, local time, Itô formula.
Abstract
Several Itô formulas have been already established for Lévy processes. We explain according to
which criteria they are not optimal and establish an extended Itô formula that satisfies that criteria.
The interest, in particular, of this formula, is to obtain the explicit decomposition of F (X t , t), for
X Lévy process and F deterministic function with locally bounded first order Radon-Nikodym
derivatives, as the sum of a Dirichlet process and a bounded variation process.
1 Introduction and main results
Let X be a general real-valued Lévy process with characteristic triplet (a, σ, ν), i.e. its characteristic
exponent is equal to
Z
u2
ψ(u) = iua − σ2 + (e iu y − 1 − iu y1{| y|≤1} )ν(d y)
2
R
where a and σ are real numbers and ν is a Lévy measure. We will denote by (σB t , t ≥ 0) the
Brownian component of X . Let F be a C 2,1 function from R × R+ to R. The classical Itô formula
gives
F (X t , t)
=
+
+
F (X 0 , 0) +
Z
t
∂F
Z
t
0
∂F
∂t
(X s− , s)ds
σ2
Z
t
∂ 2F
(X s , s)ds
∂x
2 0 ∂ x2
X
∂F
{F (X s , s) − F (X s− , s) −
(X s− , s)∆X s }
∂x
0<s≤t
(X s− , s)d X s +
0
202
(1.1)
An optimal Itô formula for Lévy processes
203
This formula can be rewritten under the following form (see [8]): (F (X t , t), t ≥ 0) is a semimartingale admitting the decomposition
F (X t , t) = F (X 0 , 0) + M t + Vt
(1.2)
where the local martingale M and the adapted with bounded variation process V are given by
Mt = σ
Z
t
∂F
∂x
0
Vt =
(X s− , s)d Bs +
Z tZ
0
X
{F (X s− + y, s) − F (X s− , s)}µ̃X (d y, ds)
(1.3)
{| y|<1}
{F (X s , s) − F (X s− , s)}1{|∆X s |≥1} +
0<s≤t
Z
t
A F (X s , s)ds
(1.4)
0
where µ̃X (d y, ds) denotes the compensated Poisson measure associated to the jumps of X , and A
is the operator associated to X defined by
A G(x, t)
=
1 ∂ 2G
(x, t)
(x, s) + σ2
∂Z
t
∂x
2 ∂ x2
∂G
+ {G(x + y, t) − G(x, t) − y
(x, t)}1(| y|<1) ν(d y)
∂x
R
∂G
(x, t) + a
∂G
2
for any function G defined on R × R+ , such that ∂∂ Gx , ∂∂ Gt and ∂∂ xG2 exist as Radon-Nikodym derivatives with respect to the Lebesgue measure and the integral is well defined. The later condition is
2
satisfied when ∂∂ xG2 is locally bounded.
Note that the existence of locally bounded first order Radon-Nikodym derivatives alone guarantees
the existence of
Z t
Z t
∂F
∂F
F (X t , t) − F (X 0 , 0) −
(X s− , s)ds −
(X s− , s)d X s
(1.5)
∂t
∂x
0
0
but then to say that this expression coincides with
σ2
2
Z
t
0
∂ 2F
∂
x2
(X s , s)ds +
X
{F (X s , s) − F (X s− , s) −
0<s≤t
∂F
∂x
(X s− , s)∆X s }
we need to assume much more on F .
In that sense one might say that the classical Itô formula is not optimal. The interest of an optimal
formula is two-fold. It allows to expand F (X t , t) under minimal conditions on F but also to know
explicitly the structure of the process F (X t , t). Such an optimal formula has been established in
the particular case when X is a Brownian motion [4]. Indeed in that case, under the minimal
assumption on F for the existence of (1.5), namely that F admits locally bounded first order
Radon-Nikodym derivatives, we know that this expression coincides with
−
1
2
Z tZ
0
∂F
R
∂x
(x, s)d Lsx
where (Lsx , x ∈ R, s ≥ 0) is the local time process of X .
RtR
( 0 R ∂∂ Fx (x, s)d Lsx , t ≥ 0) has a 0-quadratic energy.
Moreover the process
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Electronic Communications in Probability
In the general case, various extensions of (1.1) have been established. We will quote here only the
extensions exploiting the notion of local times, we send to [4] for a more exhaustive bibliography.
Meyer [9] has been the first to relax the assumption on F by introducing an integral with respect to
local time, followed then by Bouleau and Yor [3], Azéma et al [1], Eisenbaum [4], [5], Ghomrasni
and Peskir [7], Eisenbaum and Kyprianou [6]. In the discontinuous
P case, none of the obtained
Itô formulas is optimal because of the presence of the expression 0<s≤t {F (X s , s) − F (X s− , s) −
∂F
(X s− , s)∆X s }.
∂x
The Itô formula for Lévy processes presented below in Theorem 1.1, is available for X admitting
a Brownian component. It lightens the condition on the jumps of X required by [5], and it also
lightens the condition on the first order derivatives of F required by [6]. Besides it is optimal. To
introduce it we need the operator I defined on the set of locally bounded measurable functions G
on R × R+ by
Z
x
I G(x, t) =
G( y, t)d y.
0
We will denote the Markov local time process of X by (L tx , x ∈ R, t ≥ 0).
Theorem 1.1. : Assume that σ 6= 0. Let F be a function from R × R+ to R such that ∂∂ Fx and ∂∂ Ft exist
as Radon-Nikodym derivatives with respect to the Lebesgue measure and are locally bounded. Then
the process (F (X t , t), t ≥ 0) admits the following decomposition
F (X t , t) = F (X 0 , 0) + M t + Vt + Q t
with M the local martingale given by (1.3), V is the bounded variation process
X
Vt =
{F (X s , s) − F (X s− , s)}1{|∆X s |≥1}
0≤s≤t
and Q the following adapted process with 0-quadratic variation
Z tZ
A I F (x, s)d Lsx .
Qt = −
0
R
As a simple application of Theorem 1.1 consider the example of the function F (x, s) = |x| in the
R1
case 0 yν(d y) = +∞. This function does not satisfy the assumption of Theorem 3 of [6] nor X
does satisfy the assumption of Theorem 2.2 in [5]. But, thanks to Theorem 1.1, we immediately
obtain Tanaka’s formula.
The proofs are presented in Section 2.
2 Proofs
We first remind the meaning of integration with respect to the semimartingale local time process
of X denoted (ℓsx , x ∈ R, s ≥ 0). Theorem 1.1 is expressed in terms of the Markov local time
process (Lsx , x ∈ R, s ≥ 0). The two processes are connected by:
(Lsx , x ∈ R, s ≥ 0) = ( σ12 ℓsx , x ∈ R, s ≥ 0).
Let σB be the Brownian component of X . Defined the norm ||.|| of a measurable function f from
R × R+ to R by
Z1
Z1
|Bs |
2
1/2
ds).
|| f || = 2IE(
f (X s , s)ds) + IE(
| f (X s , s)|
s
0
0
An optimal Itô formula for Lévy processes
205
In [6], integration with respect to ℓ of locally bounded mesurable function f has been defined by
Z tZ
Z t
Z1
f (x, s)dℓsx = σ
0
f (X s− , s)d Bs + σ
0
R
f (X̂ s− , 1 − s)d B̂s , 0 ≤ t ≤ 1
(2.1)
1−t
where B̂ and X̂ are the time reversal at 1 of B and X .
We have the following properties:
RtR
(i) IE(| 0 R f (x, s)dℓsx |) ≤ |σ||| f ||.
(ii) If f admits a locally bounded Radon-Nikodym derivative with respect to x, then:
Rt ∂f
−σ2 0 ∂ x (X s , s)ds.
RtR
(iii) The process ( 0 R f (x, s)dℓsx , 0 ≤ t ≤ 1) has 0-quadratic variation.
Proof of Theorem 1.1 : We start by assuming that F and
Z Z
∂F
∂x
RtR
0
R
f (x, s)dℓsx =
are bounded. We set
F (x − y/n, t − s/n) f ( y)h(s)d y ds
Fn (x, t) =
R2
R
where f and h are nonnegative C ∞ functions with compact supports such that : R f (x)d x =
R
h(x)d x = 1. Thanks to the usual Itô formula we have:
R
Z t
Z t
∂ Fn
∂ Fn
(X s− , s)d Bs +
(X s , s)ds
Fn (X t , t) =
Fn (0, 0) + σ
∂x
∂t
0
0
Z t
X
∂ Fn
+a
(X s , s)ds +
{Fn (X s , s) − Fn (X s− , s)}1{|∆X s |≥1}
∂x
0
0≤s≤t
Z tZ
+
{Fn (X s− + y, s) − Fn (X s− , s)}1{| y|<1} µ̃(ds, d y)
0
2
+
+
σ
R
Z t
∂ 2 Fn
2 0 ∂ x2
Z tZ 1
(X s , s)ds
{Fn (X s + y, s) − Fn (X s , s) −
0
−1
(2.2)
∂ Fn
∂x
(X s , s) y}ν(d y)ds
With the same arguments as in the proof of Theorem 2.2 of [5], we see that as n tends to ∞,
Fn (X t , t) and each of the first five terms of the RHS of (2.2) converges at least in probability to the
corresponding expression with F replacing Fn . Besides we note that
Z t
Z tZ Z x
∂F
1
∂F
(X s , s)ds = − 2
( y, s)d y)dℓsx
(
∂
t
∂t
σ
0
0
R
0
Z tZ
Z x
∂
= −
(
F ( y, s)d y)d Lsx
∂
t
0
R
0
since
∂F
∂t
is locally bounded. Hence we have:
Z t
Z tZ
∂F
∂ (I F )
(X s , s)ds = −
(x, s)d Lsx .
∂
t
∂t
0
0
R
(2.3)
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Electronic Communications in Probability
Since : F (x, s) =
∂ (I F )
(x, s),
∂x
a
we immediately obtain:
Z
t
0
∂F
(X s , s)ds = −
∂x
Z tZ
0
a
∂ (I F )
R
∂x
(x, s)d Lsx .
(2.4)
The convergence in L 2 of the sixth term of the RHS is obtained with the same proof as in [6]. The
limit is equal to
Z tZ
{F (X s− + y, s) − F (X s− , s)}1{| y|<1} µ̃(ds, d y)
0
(2.5)
R
For the seventh term of the RHS of (2.2), we note that :
R t ∂ 2F
Rt R ∂F
σ2
n
(X s , s)ds = − 12 0 R ∂ xn (x, s)dℓsx . Thanks to the properties (i) and (ii) of the integration
2
0 ∂ x2
RtR
with respect to the local times, this expression converges in L 1 to − 12 0 R ∂∂ Fx (x, s)dℓsx . We can
obviously write:
Z tZ
Z tZ
1
σ2
∂F
∂ 2 (I F )
(x, s)d Lsx .
(2.6)
−
(x, s)dℓsx = −
2 0 R∂x
2 0 R ∂ x2
We now study the convergence of the last term of the RHS of (2.2). We have:
Z tZ
0
1
{Fn (X s + y, s) − Fn (X s , s) −
−1
=
−
Z tZ
0
Rx R1
where: H n (x, s) =
−1
0
∂ 2 Fn
(x, t)
∂ x2
= n2
∂ Fn
(z, s) y|1{| y|<1}
∂x
Fn (z, s) −
Consequently :
H n (x, s)
=
=
=
Z
Z
1
Z
−1 0
1 Z x+ y
{
−1
RR
R2
∂ 2 Fn
∂ x2
Z
|1{| y|<1} .
∂ Fn
∂x
(z, s) y}dzν(d y)
x
Fn (z, s)dz − y Fn (x, s) + y Fn (0, s) −
0
1
( y Fn (0, s) −
−1
We have:
≤ cst e n2 y 2 1{| y|<1} sup|F |
Fn (z, s)dz −
Gn (x, s) +
∂ Fn
(z, s) y}ν(d y)dz.
∂x
F (x − y/n, t − s/n) f ′′ ( y)h(s)d y ds, we obtain |Fn (z + y, s) −
{Fn (z + y, s) − Fn (z, s) −
Z
(2.7)
∂ Fn
− Fn (z, s) −
(z, s) y|1{| y|<1}
∂x
Z z+ y
∂ Fn
∂ Fn
(v, t) −
(z, t)d v|1{| y|<1}
=|
∂
x
∂x
z
x
0
(X s , s) y}ν(d y)ds
R
≤ y 2 sup|
Noting that:
∂x
H n (x, s)d Lsx
{Fn (z + y, s) − Fn (z, s) −
|Fn (z + y, s)
∂ Fn
Z
y
Fn (z, s)dz)ν(d y)
0
Z
y
Fn (z, s)dz}ν(d y)
0
An optimal Itô formula for Lévy processes
where Gn (x, s) =
know that
R1
−1
207
(I Fn (x + y, s) − I Fn (x, s) − y Fn (x, s))ν(d y). Thanks to Corollary 8 of [6], we
Z tZ
0
H n (x, s)d Lsx
=
Z tZ
0
R
Gn (x, s)d Lsx .
(2.8)
R
By dominated convergence, we have as n tends to ∞ for every (x, s)
I Fn (x + y, s) − I Fn (x, s) − y Fn (x, s) → I F (x + y, s) − I F (x, s) − y F (x, s).
Besides, for every n : |I Fn (x + y, s) − I Fn (x, s) − y Fn (x, s)| ≤ y 2 1{| y|<1} sup| ∂∂ Fx |, hence for every
(x, s) : Gn (x, s) tends to G(x, s), where
Z
G(x, s) =
(I F (x + y, s) − I F (x, s) − y F (x, s))1{| y|<1} ν(d y).
R
By dominated convergence, (Gn )n>0 converges for the norm ||.|| to G. Consequently the limit of
the last term of the RHS of (2.2) is equal by (2.7) and (2.8) to
Z tZ Z
(I F (x + y, s) − I F (x, s) − y F (x, s))1{| y|<1} ν(d y)d Lsx .
−
0
R
(2.9)
R
Summing all the limits (2.3), (2.4), (2.5), (2.6) and (2.9), we finally obtain
F (X t , t)
=
+
+
−
F (X 0 , 0) + σ
Z tZ
0
X
Z tZ
0
t
∂F
∂x
0
(X s− , s)d Bs
(2.10)
{F (X s− + y, s) − F (X s− , s)}1{| y|<1|} µ̃(ds, d y)
R
{F (X s , s) − F (X s− , s)}1{|∆X s |≥1}
0<s≤t
Z tZ
0
−
Z
{
∂ (I F )
∂t
R
(x, s) + a
∂ (I F )
∂x
(x, s) +
σ2 ∂ 2 (I F )
2
∂ x2
(x, s)}d Lsx
Z
{ {I F (x + y, s) − I F (x, s) − y F (x, s)}1{| y|<1} ν(d y)}d Lsx .
R
which summarizes in
F (X t , t)
=
F (X 0 , 0) + σ
+
Z tZ
0
+
X
0<s≤t
Z
t
0
∂F
∂x
(X s− , s)d Bs
{F (X s− + y, s) − F (X s− , s)}1{| y|<1|} µ̃(ds, d y)
R
{F (X s , s) − F (X s− , s)}1{|∆X s |≥1} −
Z tZ
0
A I F (x, s)d Lsx .
R
In the general case, we set:
F̃n (x, s) = F (x, s)1[an ,bn ] (x) + F (an , s)1(−∞,an ) (x) + F (bn , s)1(bn ,∞) (x)
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Electronic Communications in Probability
where (−an )n∈N and (bn )n∈N are two positive real sequences increasing to ∞.
We write (2.10) for F̃n and stop the process ( F̃n (X s , s), 0 ≤ s ≤ 1) at Tm = 1 ∧ inf{s ≥ 0 : |X s | >
m}. We let n tend to ∞ and then m tend to ∞. The behavior of two terms deserves specific
explanations, the other terms converging respectively to the expected expressions.
R t∧Tm R R
The first one is : 0
{ {I F̃n (x + y, s)− I F̃n (x, s)− y F̃n (x, s)}1{| y|<1} ν(d y)}d Lsx . Thanks to the
R
definition of the integral with respect to local time (2.1), it is equal to
1
σ
Z
t∧Tm
H̃ n (X s− , s)d Bs +
0
1
σ
Z
1
H̃ n (X̂ s− , s)d B̂s
(2.11)
1−(t∧Tm )
R
where H̃ n (x, s) = {I F̃n (x + y, s) − I F̃n (x, s) − y F̃n (x, s)}1{| y|<1} ν(d y).
R
We set H(x, s) = {I F (x + y, s) − I F (x, s) − y F (x, s)}1{| y|<1} ν(d y).
We can choose n big enough to have |an | and bn bigger than m + 1. Hence (2.11) is equal to
1
σ
Z
t∧Tm
H(X s− , s)d Bs +
0
1
σ
Z
1
H(X̂ s− , s)d B̂s .
1−(t∧Tm )
For every ε > 0
IP( sup |
0≤t≤1
Z
1
H(X̂ s− , s)d B̂s
−
1−(t∧Tm )
Z
1
H(X̂ s− , s)d B̂s | ≥ ε)
1−t
≤
IP(Tm < 1)
=
IP( sup |X t | > m)
0≤t≤1
R1
which shows that as m tends to ∞, 1−(t∧T ) H(X̂ s− , s)d B̂s converges in probability uniformly on
R t∧Tm m
Rt
R1
H(X s− , s)d Bs converges in probability to 0 H(X s− , s)d Bs .
[0, 1] to 1−t H(X̂ s− , s)d B̂s . Similarly 0
Consequently as m tends to ∞, (2.11) converges to
Z tZ
0
Z
{ {I F (x + y, s) − I F (x, s) − y F (x, s)}1{| y|<1} ν(d y)}d Lsx .
R
RtR
The second term is : 0 R { F̃n (X s− + y, s) − F̃n (X s− , s)}1{s<Tm } 1{| y|<1|} µ̃(ds, d y). For n big enough
such that |an |, bn > m, this term is equal to
RtR
{F (X s− + y, s) − F (X s− , s)}1{s<Tm } 1{| y|<1|} µ̃(ds, d y). As Ikeda and Watanabe [8], we then
0 R
RtR
denote by ( 0 R {F (X s− + y, s) − F (X s− , s)}1{| y|<1|} µ̃(ds, d y), 0 ≤ t ≤ 1) the local martingale
(Yt , 0 ≤ t ≤ 1) defined by :
Yt∧Tm =
Z tZ
0
{ F̃ (X s− + y, s) − F̃ (X s− , s)}1{s<Tm } 1{| y|<1|} µ̃(ds, d y).ƒ
R
An optimal Itô formula for Lévy processes
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