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In this article, we give a new proof of the It
ô formula for some integral processes related to the space-time Lévy noise introduced in [1] [2] as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the
*p*-th moment of the stochastic integral, and the It
ô representation theorem leading to a chaos expansion similar to the Gaussian case.

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 [

Borel sets in

In the recent articles [

Let N be a Poisson random measure (PRM) on

We denote by

for some

(In particular, Z can be an a-stable random measure with

The stochastic integral with respect to

Assume that N is defined on a probability space

where

An elementary process on

where

As in the classical theory, for any predictable process H such that

we can define the stochastic integral of H with respect to

is a zero-mean square-integrable martingale which satisfies

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

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

Theorem 1 (Ito Formula I). Let

where G, K and H are predictable processes which satisfy

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

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 [

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

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

The method that we use for proving Theorems 1 and 2 is similar to the one described in Section 4.4.2 of [

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

We consider first processes of the form (4). We start by examining the case when both integrands H and K

vanish outside a set

(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

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

with respect to the measure

Lemma 1. Let

where

Then, there exists a càdlàg modification

where

for some sequence

Proof: We use the same argument as in the proof of Theorem 4.3.4 of [

where

Note that

Note that

By Chebyshev’s inequality,

We consider now the case when the at least one of the integrands H and K do not vanish outside a set

Assumption B. It is not possible to find

with respect to the measure

Assumption

with respect to the measure

We consider bounded Borel sets in

Theorem 3 (Interlacing I). Let

where

with

Proof: Fix

Note that

We denote by

By Chebyshev’s inequality,

Note that

and hence, using relation (3),

By Markov’s inequality,

Let

We consider next processes of the form (9) with G = 0. Note that if H vanishes a.e. outside a set

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

with respect to the measure

Theorem 4 (Interlacing II). Let Y be a process given by (9) with

with

Proof: We proceed as in the proof of Theorem 3. Fix

By Assumption C,_{n}(t) as the sum of two integrals, corresponding to the regions

and

be the càdlàg modification of

Let

and the conclusion follows as in the proof of Lemma 1. ,

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 [

Lemma 2. Let

where G is a predictable process which satisfies (5),

Proof: We denote

Case 1: G = 0. By the representation of N,

and the conclusion follows since N has points

Case 2: G is arbitrary. The map

where A and B are defined as follows: if

Note that

It remains to prove that

For this, we assume that

So it suffices to prove that

for all

We first prove (13). Fix

We extend

where for the last equality we used the fact that

This proves (13).

Next, we prove (14). Note that if

Arguing as above, we see that

where for the last equality we used the fact that

This concludes the proof of (14). ,

Proof of Theorem 1: We fix

Case 1: H and K vanish outside a fixed set

If H vanishes a.e. on

where the process

Note that

where

Cauchy-Schwarz inequality,

After using the definitions of

we obtain that:

We denote by

We treat separately the four terms in the right-hand side. By the dominated convergence theorem,

Since

and

and the fact that

Finally,

and

a.s., by (16) and the continuity of f. By the dominated convergence theorem,

and the fact that

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

The conclusion follows letting

Proof of Theorem 2: We assume that

Case 1. H vanishes outside a set

where

is a bounded set). By Theorem 1, there exists a càdlàg modification of Y (denoted also by Y) such that

We add and subtract

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

In this section, we assume that the Lévy measure

As in [

For any predictable process

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 [

The following maximal inequality is due to Kunita (see Theorem 2.11 of [

Theorem 5 (Kunita Inequality). Let

where X is a predictable process which satisfies (20).

If

where

Proof: We apply Theorem 2 with

Remark 1. Kunita’s constant

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 [

For any

fication of the process

where

Hence

The following result is the analogue of Lemma 5.3.3 of [

Lemma 3. For any

Proof: We apply Theorem 2 to the function

Hence,

Since the sum of the last two integrals is 0, the conclusion follows. ,

We fix

of C-valued square-integrable random variables which are measurable with respect to

Lemma 4. The linear span of the set

Proof: The proof is similar to that of Lemma 5.3.4 of [

Theorem 6 (Ito Representation Theorem). For any

such that

Proof: By Lemma 3, relation (22) holds for

The multiple (and iterated) integral with respect

More precisely, we consider the Hilbert space

For any integer

Moreover, if

We have the following result.

Theorem 7 (Chaos Expansion). For any

In particular,

Proof: We use the same argument as in the classical case, when

is a square-integrable Lévy process (see Theorem 5.4.6 of [

By (21),

Satisfying

such that

We substitute this into (23) and iterate the procedure. We omit the details. ,

Research of R. M. Balan is funded by a grant from the Natural Sciences and Engineering Research Council of Canada.

Raluca M.Balan,Cheikh B.Ndongo, (2015) Itô Formula for Integral Processes Related to Space-Time Lévy Noise. Applied Mathematics,06,1755-1768. doi: 10.4236/am.2015.610156