Itô Formula for Integral Processes Related to Space-Time Lévy Noise ()
1. Introduction
Random processes indexed by sets in the space-time domain are useful objects in stochastic analysis, since they can be viewed as mathematical models for the noise perturbing a stochastic partial differential equation (SPDE). In the recent years, a lot of effort has been dedicated to studying the behaviour of the solution of basic equations (like the heat or wave equations), driven by a Gaussian white noise. This type of noise was introduced by Walsh in [3] and is defined as a zero-mean Gaussian process
, with covariance
, where
denotes the Lebesgue measure and
is the class of bounded
Borel sets in
.
In the recent articles [1] [2] , a new process has been introduced as an alternative for the Gaussian white noise perturbing an SPDE, which has a structure similar to a Lévy process. We introduce briefly the definition of this process below.
Let N be a Poisson random measure (PRM) on
of intensity
where
and
is a Lévy measure on
:

We denote by
the compensated PRM defined by
for any Borel set A in
with
. The Lévy-type noise process mentioned above is defined as
, where
![]()
for some
. It was shown in [2] that Z is an “independently scattered random measure” (in the sense of [4] ) with characteristic function:
![]()
(In particular, Z can be an a-stable random measure with
, as in Definition 3.3.1 of [5] .) One can define the stochastic integral of a process
with respect to Z and for a certain integrands,
![]()
The stochastic integral with respect to
(or N) can be defined using classical methods (see e.g. [6] ). We review briefly this definition here.
Assume that N is defined on a probability space
. On this space, we consider the filtration
![]()
where
is the class of bounded Borel sets in
and
is the class of Borel sets in
which are bounded away from 0.
An elementary process on
is a process of the form
![]()
where
, X is an
-measurable bounded random variable,
and
. A pro- cess
is called predictable if it is measurable with respect to the s-field
generated by all linear combinations of elementary processes.
As in the classical theory, for any predictable process H such that
(1)
we can define the stochastic integral of H with respect to
and the process
![]()
is a zero-mean square-integrable martingale which satisfies
(2)
On the other hand, for any predictable process K such that
![]()
we can define the integral of K with respect to N and this integral satisfies
(3)
In this article, we work with processes whose trajectories are right-continuous with left limits. If x is a right
continuous function with left limits, we denote by
the left limit at time t and
the jump size at time t. We will prove the following result.
Theorem 1 (Ito Formula I). Let
be a process defined by
(4)
where G, K and H are predictable processes which satisfy
(5)
(6)
(7)
Then there exists a modification of Y (denoted also by Y) whose sample paths are right-continuous with left limits, such that for any function
and for any
, with probability 1,
(8)
Note that since the first two terms on the right-hand side of (4) are processes of finite variation and the last term is a square-integrable martingale, Y is a semimartingale. Therefore, the Itô formula given by Theorem 1 can be derived from the corresponding result for a general semimartingale, assuming that Y has sample paths which are right-continuous with left limits (see e.g. Theorem 2.5 of [7] ).
The goal of the present article is to give an alternative proof of this result which contains the explicit construction of the modification of Y for which the Itô formula holds.
We will also give the proof of the following variant of the Itô formula, which will be useful for the applications related to the (finite-variance) Lévy white noise, discussed in Section 4.
Theorem 2 (Ito Formula II). Let
be a process defined by
(9)
where G and H are predictable processes which satisfy (5), respectively (1). Then there exists a càdlàg modification of Y (denoted also by Y) such that for any
, with probability 1,
![]()
The method that we use for proving Theorems 1 and 2 is similar to the one described in Section 4.4.2 of [6] in the case of classical Lévy processes, the difference being that in our case, N is a PRM on
instead of
. This method relies on a double “interlacing” technique, which consists in first approximating the set
of small jumps by sets of the form
with
(in the case when H and K vanish outside a bounded Borel set
), and then approximating the spatial domain
by regions of the form
with
. This approximation method is described in Section 2. Section 3 is dedicated to the proofs of Theorems 1 and 2. Finally, in Section 4 we discuss two applications of Theorem 2 in the case of the (finite-variance) Lévy white noise introduced in [1] .
2. Approximation by Right-Continuous Processes with Left Limits
In this section, we show that the Lévy-type integral processes given by (4) and (9) have right-continuous modifications with left limits, which are constructed by approximation. These modifications will play an important role in the proof of Itô’s formula. Since the process
is continuous, we assume that
.
We consider first processes of the form (4). We start by examining the case when both integrands H and K
vanish outside a set
. Since the process
is clearly càdlàg
(the integral being a sum with finitely many terms), we need to consider only the integral process which depends on H.
Note that if H vanishes a.e. on
for some
and
, then
![]()
is a process whose sample paths are right-continuous with left limits (the first term is a sum with finitely many terms and the second term in continuous). Therefore, we will suppose that H satisfies the following assumption:
Assumption A. It is not possible to find
and
such that
![]()
with respect to the measure
.
Lemma 1. Let
be a process defined by
![]()
where
and H is a predictable process which satisfies Assumption A and
(10)
Then, there exists a càdlàg modification
of Y such that for all
,
![]()
where
![]()
for some sequence
(depending on T) such that
.
Proof: We use the same argument as in the proof of Theorem 4.3.4 of [6] . Fix
. Let
![]()
where
![]()
Note that
is non-increasing and
. (If
then
for all n. Hence
, which contradicts Assumption A.)
Note that
is a càdlàg martingale. By Doob’s submartingale inequality and relation (2),
![]()
By Chebyshev’s inequality,
. By Borel-Cantelli lemma, with probability 1, the sequence
is Cauchy in the space
of càdlàg functions on
equipped with the sup-norm. Its limit
is a modification of Y since for any
,
also converges to
in
. Finally, we note that the process
does not depend on T (although the approximation sequence
does). If
is the modification of Y on
and
is the modification of Y on
with
, then
a.s. for any
. Hence,
can be extended to
. ,
We consider now the case when the at least one of the integrands H and K do not vanish outside a set
. More precisely, we introduce the following assumptions:
Assumption B. It is not possible to find
and
such that
![]()
with respect to the measure
.
Assumption
. It is not possible to find
and
such that
![]()
with respect to the measure
.
We consider bounded Borel sets in
of the form
.
Theorem 3 (Interlacing I). Let
be a process defined by (4) with
, where H and K are predictable processes which satisfy conditions (7), respectively (6), such that either H satisfies Assumption B, or K satisfies Assumption
. Then, there exists a càdlàg modification
of Y such that for all T > 0,
(11)
where
is a càdlàg modification of the process
defined by
![]()
with
for some sequence
(depending on T) such that
.
Proof: Fix
. Let
where
![]()
Note that
is non-decreasing and
. (If
then
for all n, and hence
, which contradicts Assumptions B or
.) Let
be the process given in the statement of the theorem with
. We denote by
and
the two integrals which compose
, depending on H, respectively K.
We denote by
the càdlàg modification of
given by Lemma 1. By Doob’s submartingale inequality and relation (2),
![]()
By Chebyshev’s inequality,
.
Note that
is a càdlàg process. For any
,
![]()
and hence, using relation (3),
![]()
By Markov’s inequality,
.
Let
. Then
, and the conclusion follows by the Borel-Cantelli Lemma, as in the proof of Lemma 1. ,
We consider next processes of the form (9) with G = 0. Note that if H vanishes a.e. outside a set
then
![]()
where the first term has a càdlàg modification given by Lemma 1, the second term is càdlàg, and the third term is continuous. Therefore, we will suppose that H satisfies the following assumption:
Assumption C. It is not possible to find
and
such that
![]()
with respect to the measure
.
Theorem 4 (Interlacing II). Let Y be a process given by (9) with
, where H is a predictable process which satisfies (1) and Assumption C. Then, there exists a càdlàg modification
of Y such that (11) holds, where
is a càdlàg modification of the process
defined by:
![]()
with
for some sequence
(depending on T) such that
.
Proof: We proceed as in the proof of Theorem 3. Fix
. Let
where
![]()
By Assumption C,
. We write Yn(t) as the sum of two integrals, corresponding to the regions
,
and
. We denote these integrals by
, respectively
. Note that
is càdlàg. Let ![]()
be the càdlàg modification of
given by Lemma 1.
Let
. By Doob’s submartingale inequality,
![]()
and the conclusion follows as in the proof of Lemma 1. ,
3. Proof of Itô Formula
In this section, we give the proofs of Theorem 1 and Theorem 2.
We start with the simpler case when there are no small jumps (the analogue of Lemma 4.4.6 of [6] ).
Lemma 2. Let
![]()
where G is a predictable process which satisfies (5),
,
and K is a predictable process. Then, for any function
and for any
,
![]()
Proof: We denote
. By Proposition 5.3 of [8] , we may assume that the restriction of N to the set
has points
, where
are the points of a Poisson process on
of intensity
and
are i.i.d. on
with distribution
, independent of
. We consider two cases.
Case 1: G = 0. By the representation of N,
. So
is a step function which has a jump of size
at each point
and
. Hence
![]()
and the conclusion follows since N has points
in
.
Case 2: G is arbitrary. The map
is a step function which has a jump of size
at time
. Since
is continuous, the jump times and the jump sizes of Y coincide with those of
, i.e.
. We use the decomposition
![]()
where A and B are defined as follows: if
, we let
![]()
![]()
Note that
![]()
It remains to prove that
(12)
For this, we assume that
and we write
![]()
So it suffices to prove that
(13)
for all
, and
(14)
We first prove (13). Fix
. For any
,
and
.
We extend
by continuity to
. Hence
![]()
where for the last equality we used the fact that
and hence
.
This proves (13).
Next, we prove (14). Note that if
, both terms are zero. So, we assume that
. For any
,
and
.
Arguing as above, we see that
![]()
where for the last equality we used the fact that
and hence
.
This concludes the proof of (14). ,
Proof of Theorem 1: We fix
. We assume that
and
are bounded. (Otherwise, we use
for
.)
Case 1: H and K vanish outside a fixed set
.
If H vanishes a.e. on
for some
and
, the conclusion follows from Lemma 2. Therefore, we suppose that H satisfies Assumption A. By Lemma 1, there exists a càdlàg modification of Y (denoted also by Y) such that
(15)
where the process
is defined by
![]()
being the sequence given by Lemma 1 with
. Consequently,
(16)
Note that
![]()
where
and
. By the
Cauchy-Schwarz inequality,
satisfies (5) (since B is a bounded set and H satisfies (10)). We apply Lemma 2 to
:
![]()
After using the definitions of
and
, as well as adding and subtracting
![]()
we obtain that:
(17)
We denote by
, respectively
the four terms on the right-hand side of (8). The conclusion will follow by taking the limit as
in (17). The left-hand side converges to
, by (15).
We treat separately the four terms in the right-hand side. By the dominated convergence theorem,
![]()
Since
is a sum with a finite number of terms, using (15) and the continuity of f, we see that
a.s. For the third term, note that
, where
![]()
![]()
and
a.s., by (15) and the continuity of f. By the dominated convergence theorem,
and
. To justify the application of this theorem, we use Taylor’s formula of the first order:
(18)
and the fact that
is bounded. This proves that
in
.
Finally,
, where
![]()
![]()
and
![]()
a.s., by (16) and the continuity of f. By the dominated convergence theorem,
and
. To justify the application of this theorem, we use Taylor’s formula of second order:
(19)
and the fact that
is bounded. This proves that
in
.
Case 2. H satisfies Assumption B or K satisfies Assumption
.
By Theorem 3, there exists a càdlàg approximation of Y (denoted also by Y) such that (15) holds, where
is a càdlàg modification of
![]()
being the sequence given by Theorem 3 with
. Using the result of Case 1 for the pro- cess
, we obtain
![]()
The conclusion follows letting
as in Case 1. ,
Proof of Theorem 2: We assume that
and
are bounded. We fix t.
Case 1. H vanishes outside a set
. We write
![]()
where
. By the Cauchy-Schwarz inequality,
satisfies (5) (since B
is a bounded set). By Theorem 1, there exists a càdlàg modification of Y (denoted also by Y) such that
![]()
We add and subtract
. The conclusion follows by rear-
ranging the terms.
Case 2. H satisfies Assumption C.
By Theorem 4, there exists a càdlàg modification of Y (denoted also by Y) such that (15) holds, where
is a càdlàg modification of
![]()
being the sequence given by Theorem 4 with
. We write the Itô formula for the process
(using Case 1) and we let
. ,
4. Applications
In this section, we assume that the Lévy measure
satisfies the condition:
![]()
As in [1] , we consider the process
defined by:
![]()
For any predictable process
such that
(20)
we can define the stochastic integral of X with respect to L and this integral satisfies:
![]()
By (2), this integral has the following isometry property:
![]()
When used as a noise process perturbing an SPDE, L behaves very similarly to the Gaussian white noise. For this reason, L was called a Lévy white noise in [1] .
4.1. Kunita Inequality
The following maximal inequality is due to Kunita (see Theorem 2.11 of [7] ). In problems related to SPDEs with noise L, this result plays the same role as the Burkholder-Davis-Gundy inequality for SPDEs with Gaussian white noise.
Theorem 5 (Kunita Inequality). Let
be a process given by
![]()
where X is a predictable process which satisfies (20).
If
for some
, then for any
,
![]()
where
and
is the constant in Theorem 2.11 of [7] .
Proof: We apply Theorem 2 with
and
. The proof is identical to that of Theorem 2.11 of [7] . We omit the details. ,
Remark 1. Kunita’s constant
cannot be computed explicitly. Theorem 5 is proved in [9] using a different method which shows that
is directly related to the constant
in Rosenthal’s inequality, which is
.
4.2. Itô Representation Theorem and Chaos Expansion
In this section, we give an application to Theorem 2 to exponential martingales, which leads to Itô representation theorem and a chaos expansion (similarly to Sections 5.3 and 5.4 of [6] ).
For any
we let
for
. We work with the càdlàg modi-
fication of the process
given by Theorem 4. By Lemma 2.4 of [1] ,
![]()
where
![]()
Hence
for all
, where
![]()
The following result is the analogue of Lemma 5.3.3 of [6] .
Lemma 3. For any
and
, with probability 1,
![]()
Proof: We apply Theorem 2 to the function
and the process
![]()
Hence,
and
. We obtain:
![]()
Since the sum of the last two integrals is 0, the conclusion follows. ,
We fix
. We let
. We denote by
be the space
of C-valued square-integrable random variables which are measurable with respect to
.
Lemma 4. The linear span of the set
is dense in
.
Proof: The proof is similar to that of Lemma 5.3.4 of [6] . We omit the details. ,
Theorem 6 (Ito Representation Theorem). For any
, there exists a unique predictable C-valued process
satisfying
(21)
such that
(22)
Proof: By Lemma 3, relation (22) holds for
with
. The conclusion follows by an approximation argument using Lemma 4. ,
The multiple (and iterated) integral with respect
can be defined similarly to the Gaussian white-noise case (see e.g. Section 5.4 of [6] ).
More precisely, we consider the Hilbert space
, where
,
and
.
For any integer
, we consider the n-th tensor product space
. The n-th multiple integral
with respect to
can be constructed for any function
, and this integral has the isometry property:
![]()
Moreover, if
, then
for all
and
.
We have the following result.
Theorem 7 (Chaos Expansion). For any
, there exist some symmetric functions
,
such that
![]()
In particular,
![]()
Proof: We use the same argument as in the classical case, when
is a PRM on
and
![]()
is a square-integrable Lévy process (see Theorem 5.4.6 of [6] or Theorem 10.2 of [10] ). By Theorem 6, there exists a predictable process
satisfying (1) such that
(23)
By (21),
for almost all
. For such
fixed, we apply Theorem 6 again to the variable
. Hence, there exists a predictable process
![]()
Satisfying
![]()
such that
![]()
We substitute this into (23) and iterate the procedure. We omit the details. ,
Acknowledgements
Research of R. M. Balan is funded by a grant from the Natural Sciences and Engineering Research Council of Canada.
NOTES
*Corresponding author.