Product Formula of Multiple Integrals of Levy Process

TL;DR

This paper derives a product formula for multiple stochastic integrals with respect to Levy processes, utilizing exponential vectors and polarization to simplify the derivation.

Contribution

It introduces a new product formula for multiple integrals of Levy processes using exponential vectors and polarization techniques.

Findings

Simplifies the derivation of product formulas for Levy process integrals

Provides a new mathematical tool for stochastic analysis of Levy processes

Enhances understanding of the structure of multiple integrals in Levy processes

Abstract

We derive a product formula for the multiple stochastic integrals with respect to Levy process. The idea is to use exponential vectors and the polarization technique which greatly simplify the argument.

Equations

$$\Delta \eta (t):=\eta (t)-\eta (t-)if\; \Delta \eta (t)\ne 0.$$

$$N(t,U):=\sum _{0\le s\le t}{\chi }_U(\Delta \eta (s)),$$

$$\nu (U):=\mathbb{E}[N(1,U)]$$

$$\tilde{N}(dt,dz):=N(dt,dz)-\nu (dz)dt.$$

$${\hat{L}}^{2,n}:={\left(L^2(\mathbb{T},\lambda \times \nu )\right)}^{\otimes n}\subseteq L^2\left({\mathbb{T}}^n,{\left(\lambda \times \nu \right)}^n\right)$$

$$\Vert f{\Vert }_{{\hat{L}}^{2,n}}^2={\int }_{{\mathbb{T}}^n}{f}^2(t_1,z_1,\cdots ,t_n,z_n)d{t}_1\nu (d{z}_1)\cdots d{t}_n\nu (d{z}_n)<\infty ,$$

$$f(t_1,z_1,\cdots ,t_i,z_i,\cdots ,t_j,z_j,\cdots ,t_n,z_n)=f(t_1,z_1,\cdots ,t_j,z_j,\cdots ,t_i,z_i,\cdots ,t_n,z_n)$$

$$I_n(f):={\int }_{{\mathbb{T}}^n}f(t_1,z_1,\cdots ,t_n,z_n)\tilde{N}(d{t}_1,d{z}_1)\cdots \tilde{N}(d{t}_n,d{z}_n)$$

$$F=\sum _{n=0}^{\infty }{I}_n(f_n),$$

$$\Vert F{\Vert }_{L^2(P)}^2=\sum _{n=0}^{\infty }n!\Vert f_n{\Vert }_{{\hat{L}}^{2,n}}^2.$$

$$\text{Upupsilon}={\text{Upupsilon}}_m=\left\{\mathbf{i}=(i_1,\cdots ,i_{\text{upalpha}}),\text{upalpha}=2,\cdots ,m,1\le i_1<\cdots <i_{\text{upalpha}}\le m\right\},$$

$$\{\begin{array}{l}\text{Upomega}=\left\{\vec{l}:\vec{l}:\text{Upupsilon}\to {\mathbb{Z}}_{+}\right\}and\; for\; any\vec{l},\vec{n}\in \text{Upomega},\\ \text{upchi}(k,\vec{l},\vec{n})={\sum }_{1\le \text{upalpha}\le {\kappa }_m}\left[l_{{\mathbf{i}}_{\text{upalpha}}}{\chi }_{\left\{{\mathbf{i}}_{\text{upalpha}}containsk\right\}}+n_{{\mathbf{i}}_{\text{upalpha}}}{\chi }_{\left\{{\mathbf{i}}_{\text{upalpha}}containsk\right\}}\right].\end{array}$$

$${\hat{\otimes }}_{\mathbf{i}}^{\text{upmu}}(f_1,\cdots ,f_m)$$

$$V_{\mathbf{j}}^{\text{upnu}}(f_1,\cdots ,f_m)=f_{j_1}((t_1,z_1),\cdots ,(t_{\text{upnu}},z_{\text{upnu}}),\cdots )\hat{\otimes }\cdots$$

$$\hat{\otimes }{f}_{j_{\text{upbeta}}}((t_1,z_1),\cdots ,(t_{\text{upnu}},z_{\text{upnu}}),\cdots )f_1\hat{\otimes }\cdots \hat{\otimes }{\hat{f}}_{j_1}\hat{\otimes }\cdots \hat{\otimes }{\hat{f}}_{j_{\text{upbeta}}}\cdots \hat{\otimes }{f}_m,$$

$${\hat{\otimes }}_{\vec{\mathbf{i}}}^{\vec{l}}={\hat{\otimes }}_{{\mathbf{i}}_1,\cdots ,{\mathbf{i}}_{{\kappa }_m}}^{l_{{\mathbf{i}}_1},\cdots ,l_{{\mathbf{i}}_{{\kappa }_m}}}={\hat{\otimes }}_{{\mathbf{i}}_1}^{l_{{\mathbf{i}}_1}}\cdots {\hat{\otimes }}_{{\mathbf{i}}_{{\kappa }_m}}^{l_{{\mathbf{i}}_{{\kappa }_m}}},V_{\vec{\mathbf{j}}}^{\vec{n}}=V_{{\mathbf{j}}_1,\cdots ,{\mathbf{j}}_{{\kappa }_m}}^{{\mu }_{{\mathbf{j}}_1},\cdots ,n_{j_{{\kappa }_m}}}=V_{{\mathbf{j}}_1}^{{\mu }_{{\mathbf{j}}_1}}\hat{\otimes }\cdots \hat{\otimes }{V}_{{\mathbf{j}}_{{\kappa }_m}}^{n_{{\mathbf{j}}_{{\kappa }_m}}}.$$

$$\prod _{k=1}^m{I}_{q_k}(f_k)=\sum _{\genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{\vec{l},\vec{n}\in \text{Upomega}}{\genfrac{}{}{0ex}{}{\text{upchi}(1,\vec{l},\vec{n})\le q_1}{\cdots }}}{\text{upchi}(m,\vec{l},\vec{n})\le q_m}}\frac{{\prod }_{k=1}^m{q}_k!}{{\prod }_{\text{upalpha}=1}^{{\kappa }_m}{l}_{{\mathbf{i}}_{\text{upalpha}}}!{\prod }_{\text{upbeta}=1}^{{\kappa }_m}{\mu }_{{\mathbf{j}}_{\text{upbeta}}}!{\prod }_{k=1}^m(q_k-\text{upchi}(k,\vec{l},\vec{n}))!}$$

$$I_{|q|+|\vec{n}|-|\text{upchi}(\vec{l},\vec{n})|}({\hat{\otimes }}_{{\mathbf{i}}_1,\cdots ,{\mathbf{i}}_{{\kappa }_m}}^{l_{{\mathbf{i}}_1},\cdots ,l_{{\mathbf{i}}_{{\kappa }_m}}}\hat{\otimes }{V}_{{\mathbf{j}}_1,\cdots ,{\mathbf{j}}_{{\kappa }_m}}^{{\mu }_{{\mathbf{j}}_1},\cdots ,n_{j_{{\kappa }_m}}}(f_1,\cdots ,f_m)),$$

$$I_n(f_n)I_m(g_m)$$

$$f_n{\hat{\otimes }}_{k,l}{g}_m(s_1,z_1,\cdots ,s_{n+m-k-2l},z_{n+m-k-2l})$$

$$I_n(f_n)I_m(g_m)$$

$$Y(T)$$

$$Y(T)=1+\int\limits_{0}^{T}\int\limits_{\mathbb{R}_{0}}Y(s-)\Big{[}\exp{(\rho(s,z))}-1\Big{]}\tilde{N}(ds,dz)\,.$$

$$\mathcal{E}(\rho (s,z))$$

$$f_{n}=f_{n}(s_{1},z_{1},\cdots,s_{n},z_{n})=(e^{\rho}-1)^{\hat{\otimes}n}=\prod_{i=1}^{n}\big{(}e^{\rho(s_{i},z_{i})}-1\big{)}\,.$$

$${\rho }_k(u_k,s,z)=\log (1+u_k{p}_k(s,z)),$$

$$\mathcal{E}({\rho }_k(u_k,s,z))=\sum _{n=0}^{\infty }\frac{1}{n!}{u}_k^n{I}_n(f_{k,n}),$$

$$f_{k,n}=\frac{1}{u_k^n}\prod _{i=0}^n(e^{{\rho }_k(u_k,s_i,z_i)}-1)=p_k^{\otimes n}=\prod _{i=1}^n{p}_k(s_i,z_i)$$

$$\prod _{k=1}^m\mathcal{E}({\rho }_k(u_k,s,z))=\sum _{q_1,\cdots ,q_m=0}^{\infty }\frac{1}{q_1!\cdots q_m!}{u}_1^{q_1}\cdots u_m^{q_m}{I}_{q_1}(f_{1,q_1})\cdots I_{q_m}(f_{m,q_m}),$$

$$\prod _{k=1}^m\mathcal{E}({\rho }_k(u_k,s,z))$$

$$\displaystyle\qquad=\prod_{k=1}^{m}\exp{\Big{\{}\int_{\mathbb{T}}\rho_{k}(u_{k},s,z)\tilde{N}(dz,ds)-\int_{\mathbb{T}}\Big{(}e^{\rho_{k}(u_{k},s,z)}-1-\rho_{k}(u_{k},s,z)\Big{)}\nu(dz)ds\Big{\}}}$$

$$\displaystyle\qquad=\exp\Big{\{}\int_{\mathbb{T}}\sum_{k=1}^{m}\rho_{k}(u_{k},s,z)\tilde{N}(dz,ds)$$

Full text

1. Introduction

Stochastic analysis of nonlinear functionals of Lévy processes (including Brownian motion and Poisson process) have been studied extensively and found many applications. There have been already many standard books on this topic [1, 7, 8]. In the analysis of Brownian nonlinear functional the Wiener-Itô chaos expansion to expand a nonlinear functional of Brownian motion into the sum of multiple Wiener-Itô integrals is a fundamental contribution to the field. The product formula to express the product of two (or more) multiple integrals as linear combinations of some other multiple integrals is one of the important tools ([9]). It plays an important role in stochastic analysis, e.g. Malliavin calculus ([2, 6]).

The product formula for two multiple integrals of Brownian motion is known since the work of [9, Section 4] and the general product formula can be found for instance in [2, chapter 5]. In this paper we give a general formula for the product of $m$ multiple integrals of the Poisson random measure associated with (purely jump) Lévy process. The formula is in a compact form and it reduced to the Shigekawa’s formula when $m=2$ and the Lévy process is reduced to Brownian motion.

When $m=2$ a similar formula was obtained in [3], where the multiple integrals is with respect to the Lévy process itself (ours is with respect to the associated Poisson random measure which has a better properties). To obtain their formula in [3] Lee and Shih use white noise analysis framework. In this work, we only use the classical framework in hope that this work is accessible to a different spectrum of readers.

The product formula for multiple Wiener-Itô formula of Brownian motion plays an important role in many aplications such as U-statistics [4]. We hope similar things may happen. But we are not pursuing this goal in the current paper. Our formula is for purely jump Lévy process. It can be combined with the classical formulas [2, 4, 6, 9] to general case.

This paper is organized as follows. In Section 2, we give some preliminaries on Lévy process, the associated Poisson random measure, multiple integrals. We also state our main result in this section. In Section 3, we give the proof of the formula.

2. Preliminary and main results

Let $T>0$ be a positive number and let $\left\{\eta(t)=\eta(t,\omega)\,,0\leq t\leq T\right\}$ be a Lévy process on some probability space $({\Omega},{\mathcal{F}},P)$ with filtration $\{{\mathcal{F}}_{t}\,,0\leq t\leq T\}$ satisfying the usual condition. This means that $\left\{\eta(t)\right\}$ has independent and stationary increment and the sample path is right continuous with left limit. Without loss of generality, we assume $\eta(0)=0$. If the process $\eta(t)$ has all moments for any time index $t$, then presumably, one can use the polynomials of the process to approximate any nonlinear functional of the process $\left\{\eta(t)\,,0\leq t\leq T\right\}$. However, it is more convenient to use the associated Poisson random measure to carry out the stochastic analysis of these nonlinear functionals.

The jump of the process $\eta$ at time $t$ is defined by

[TABLE]

Denote $\mathbb{R}_{0}:=\mathbb{R}\backslash\{0\}$ and let $\mathcal{B}(\mathbb{R}_{0})$ be the Borel $\sigma$-algebra generated by the family of all Borel subsets $U\subset\mathbb{R}$, such that $\bar{U}\subset\mathbb{R}_{0}$. If $U\in\mathcal{B}(\mathbb{R}_{0})$ with $\bar{U}\subset\mathbb{R}_{0}$ and $t>0$, we then define the Poisson random measure, $N:[0,T]\times\mathcal{B}(\mathbb{R}_{0})\times{\Omega}\rightarrow\mathbb{R}$, associated with $\eta$ by

[TABLE]

where $\chi_{U}$ is the indicator function of $U$. The associated Lévy measure $\nu$ of $\eta$ is defined by

[TABLE]

and compensated jump measure $\tilde{N}$ is defined by

[TABLE]

The stochastic integral $\int_{\mathbb{T}}f(s,z)\tilde{N}(ds,dz)$ is well-defined for a predictable process $f(s,z)$ such that $\int_{\mathbb{T}}\mathbb{E}|f(s,z)|^{2}\nu(dz)ds<\infty$, where and throughout this paper we use $\mathbb{T}$ to represent the domain $[0,T]\times\mathbb{R}_{0}$ to simplify notation.

Let

[TABLE]

be the space of symmetric, deterministic real functions $f$ such that

[TABLE]

where $\lambda(dt)=dt$ is the Lebesgue measure. In the above the symmetry means that

[TABLE]

for all $1\leq i<j\leq n$. For any $f\in\hat{L}^{2,n}$ the multiple Wiener-Itô integral

[TABLE]

is well-defined. The importance of the introduction of the associated Poisson measure and the multiple Wiener-Itô integrals are in the following theorem which means any nonlinear functional $F$ of the Lévy process $\eta$ can be expanded as multiple Wiener-Itô integrals.

Theorem 2.1** (Wiener-Itô chaos expansion for Lévy process).**

Let ${\mathcal{F}}_{T}=\sigma(\eta(t),0\leq t\leq T)$ be $\sigma$-algebra generated the Lévy process $\eta$. Let $F\in L^{2}(\Omega,{\mathcal{F}}_{T},P)$ be an $\mathcal{F}_{T}$ measurable square integrable random variable. Then F admits the following chaos expansion:

[TABLE]

where $f_{n}\in\hat{L}^{2,n},n=1,2,\cdots$ and where we denote $I_{0}(f_{0}):=f_{0}=\mathbb{E}(F)$. Moreover, we have

[TABLE]

This chaos expansion theorem is one of the fundamental result in stochastic analysis of Lévy processes. It has been widely studied in particular when $\eta$ is the Brownian motion (Wiener process). We refer to [2], [6], [7] and references therein for further reading.

To state our main result of this paper, we need some notation. Fix a positive integer $m\geq 2$. Denote

[TABLE]

where ${\upalpha}=|{\bf i}|$ is the length of the multi-index ${\bf i}$ (we shall use ${\upalpha}$, ${\upbeta}$ to denote a natural number). It is easy to see that the cardinality of $\Upupsilon$ is $\kappa_{m}:=2^{m}-1-m$. Denote ${\vec{{\bf i}}}=({\bf i}_{1},\cdots,{\bf i}_{\kappa_{m}})$, which is unordered list of the elements of $\Upupsilon$, where ${\bf i}_{\upbeta}\in\Upupsilon$. We use ${\vec{l}}=(l_{{\bf i}_{1}},\cdots,l_{{\bf i}_{\kappa_{m}}})$ to denote a multi-index of length $\kappa_{m}$ associated with $\Upupsilon$, where $l_{{\bf i}_{\upalpha}}$, $1\leq{\upalpha}\leq\kappa_{m}$ are nonnegative integers. ${\vec{l}}$ can be regarded as a function from $\Upsilon$ to $\mathbb{Z}_{+}=\{0,1,2,\cdots\}$. Denote

[TABLE]

Again the above mentioned $\chi$ refers to the indicator function. The conventional notations such as $|{\vec{l}}|=l_{{\bf i}_{1}}+\cdots+l_{{\bf i}_{\kappa_{m}}}$; ${\vec{l}}!=l_{{\bf i}_{1}}!\cdots l_{{\bf i}_{\kappa_{m}}}\!\!!$ and so on are in use. Notice that we use $l_{{\bf i}_{1}}$ instead of $l_{1}$ to emphasize that the $l_{{\bf i}_{1}}$ corresponds to ${\bf i}_{1}$. For ${\bf i}=(i_{1},\cdots,i_{\upalpha})\,,{\bf j}=(j_{1},\cdots,j_{\upbeta})\in\Upupsilon$, and non negative integers $\upmu$ and $\upnu$ denote

[TABLE]

and

[TABLE]

where $\hat{\otimes}$ denotes the symmetric tensor product and $\hat{f}_{j_{1}}$ means that the function $f_{j_{1}}$ is removed from the list. Let us emphasize that both $\hat{\otimes}_{{\bf i}}^{\upmu}$ and $V_{{\bf j}}^{\upnu}$ are well-defined when the lengths of ${\bf i}$ and ${\bf j}$ are one. However, we shall not use $\hat{\otimes}_{{\bf i}}^{\upmu}$ when $|{\bf i}|=1$ and when $|{\bf j}|=1$, $V_{{\bf j}}^{\upnu}(f_{1},\cdots,f_{m})=f_{1}\hat{\otimes}\cdots\hat{\otimes}f_{m}$ (namely, the identity operator). For any two elements ${\vec{l}}=(l_{{\bf i}_{1}},\cdots,l_{{\bf i}_{\kappa_{m}}})$ and ${\vec{n}}=(\mu_{{\bf j}_{1}},\cdots,\mu_{{\bf j}_{\kappa_{m}}})$ in $\Upomega$, denote

[TABLE]

Now we can state the main result of the paper.

Theorem 2.2**.**

Let $f_{k}\in\left(L^{2}([0,T]\times\mathbb{R}_{0},dt\otimes\nu(dz))\right)^{\hat{\otimes}q_{k}}$, $k=1,\cdots,m$. Then

[TABLE]

where we recall $|q|=q_{1}+\cdots+q_{m}$ and $|\upchi({\vec{l}},{\vec{n}})|=\upchi(1,{\vec{l}},{\vec{n}})+\cdots+\upchi(m,{\vec{l}},{\vec{n}})$.

If $m=2$, then $\kappa_{m}=1$. To shorten the notations we can write $q_{1}=n$, $q_{2}=m$, $f_{1}=f$, $f_{2}=g$, $l_{{\upalpha}_{1}}=l$, $n_{{\upbeta}_{1}}=k$. Thus, $\upchi(1,{\vec{l}},{\vec{n}})=\upchi(2,{\vec{l}},{\vec{n}})=l+k$ and $|q|+|{\vec{n}}|-|\upchi({\vec{l}},{\vec{n}})|=n+m+k-2(l+k)=n+m-2l-k$. Thus for the product of two multiple integrals the above theorem can be written as

Theorem 2.3**.**

Let $f\in\left(L^{2}([0,T]\times\mathbb{R}_{0},dt\otimes\nu(dz))\right)^{\hat{\otimes}n}$ and $g\in\left(L^{2}([0,T]\times\mathbb{R}_{0},dt\otimes\nu(dz))\right)^{\hat{\otimes}m}$. Then

[TABLE]

where $\mathbb{Z}_{+}$ denotes the set of non negative integers and

[TABLE]

Remark 2.4**.**

1. (1)

When $\eta$ is the Brownian motion, the product formula (LABEL:e.2.7) is known since [9] (see e.g. [2, Theorem 5.6] for a formula of the general form (2.12)) and is given by

[TABLE]

It is a “special case” of (LABEL:e.2.7) when $k=0$.

3. Proof of Theorem 2.2

We shall prove the main result (Theorem 2.2) of this paper. We shall prove this by using the polarization technique (see [2, Section 5.2]). First, let us find the Wiener-Itô chaos expansion for the exponential functional (random variable) of the form

[TABLE]

where $\rho(s,z)\in\hat{L}^{2}:=\hat{L}^{2,1}=L^{2}(\mathbb{T},\nu(dz)\otimes{\lambda}(dt))$. An application of Itô formula (see e.g. [7]) yields

[TABLE]

Repeatedly using this formula, we obtain the chaos expansion of $Y(T)$ as follows.

[TABLE]

where the convergence is in $L^{2}({\Omega},{\mathcal{F}}_{T},P)$ and

[TABLE]

We shall first make critical application of the above expansion formula (3.17)-(3.18). For any functions $p_{k}(s,z)\in\hat{L}^{2}$ (in what follows when we write $k$ we always mean $k=1,2,\cdots,m$ and we shall omit $k=1,2,\cdots,m$), we denote

[TABLE]

From (3.17)-(3.18), we have (consider $u_{k}$ as fixed real numbers)

[TABLE]

where

[TABLE]

It is clear that

[TABLE]

where $f_{k,q_{k}}$, $k=1,\cdots,m$ are defined by (3.21).

On the other hand, from the definition of the exponential functional (3), we have

[TABLE]

where $A$ and $B$ denote the above first and second exponential factors.

The first exponential factor $A$ is an exponential functional of the form (3). Thus, again by the chaos expansion formula (3.17)-(3.18), we have

[TABLE]

where

[TABLE]

By the definition of $q_{k}$, we have

[TABLE]

Or

[TABLE]

where $\hat{\otimes}$ denotes the symmetric tensor product and ${\rm Sym}_{(s_{1},z_{1}),\cdots,(s_{n},z_{n})}$ denotes the symmetriization with respect to $(s_{1},z_{1}),\cdots,(s_{n},z_{n})$.

Define

[TABLE]

The cardinality of $S$ is $|S|=\tilde{\kappa}_{m}:=2^{m}-1$. We shall freely use the notations introduced in Section 2. Denote also

[TABLE]

We have

[TABLE]

where ${\vec{\upmu}}:S\rightarrow\mathbb{Z}_{+}$ is a multi-index and we used the notation $u_{\vec{{\bf j}}}^{\vec{\upmu}}=u_{{\bf j}_{1}}^{\mu_{{\bf j}_{1}}}\cdots u_{{\bf j}_{\tilde{\kappa}_{m}}}^{\mu_{{\bf j}_{\tilde{\kappa}_{m}}}}$; and $p_{\vec{{\bf j}}}^{\hat{\otimes}{\vec{\upmu}}}=p_{{\bf j}_{1}}^{\hat{\otimes}\mu_{{\bf j}_{1}}}\hat{\otimes}\cdots\hat{\otimes}p_{{\bf j}_{\tilde{\kappa}_{m}}}^{\hat{\otimes}\mu_{{\bf j}_{\tilde{\kappa}_{m}}}}$. Inserting the above expression into (3.24) we have

[TABLE]

Now we consider the second exponential factor in (3.23):

[TABLE]

where $\Upupsilon$ is defined by (2.7) (which is a subset of $S$ such that $|{\bf j}|\geq 2$). Thus,

[TABLE]

where ${\vec{l}}\in\Upomega$ is a multi-index. Combining (LABEL:e.3.11)-(3.27), we have

[TABLE]

To get an expression for $B_{{\bf i},{\bf j},l_{\bf i},\mu_{\bf j}}$ we use the notations (2.9)-(2.10) and (2.11). Then

[TABLE]

To compare the coefficients of $u_{1}^{n_{{\bf i}_{1}}}\cdots u_{m}^{n_{m}}$, we need to express the right hand side of (3.28) as a power series of $u_{1},\cdots,u_{m}$. For $k=1,\cdots,m$ denote

[TABLE]

Combining (3.23), (3.28) and (3.30), we have

[TABLE]

Comparing the coefficient of $u_{1}^{q_{1}}\cdots u_{m}^{q_{m}}$, we have

[TABLE]

Notice that when $|{\bf j}|=1$, namely, ${\bf j}=(k),k=1,\cdots,m$, then $V_{\bf j}^{\upmu}(f_{1},\cdots,f_{m})=f_{1}\hat{\otimes}\cdots\hat{\otimes}f_{m}$. We can separate these terms from the remaining ones, which will satisfy $|{\bf j}|\geq 2$. Thus, the remaining multi-indices ${\bf j}$’s consists of the set $\Upupsilon$. We can write a multi-index ${\vec{\upmu}}:S\rightarrow\mathbb{Z}_{+}$ as ${\vec{\upmu}}=(n_{(1)},\cdots,n_{(m)},{\vec{n}})$, where ${\vec{n}}\in\Upupsilon$. We also observe $q_{k}=\tilde{\upchi}(k,{\vec{l}},{\vec{\upmu}})=n_{(k)}+\upchi(k,{\vec{l}},{\vec{n}})$. After replacing ${\vec{\upmu}}$ by ${\vec{n}}$, (3.33) gives (2.12). This proves Theorem 2.2 for $f_{k}=p_{k}^{\otimes q_{k}}$, $k=1,\cdots,m$. By polarization technique (see e.g. [2, Section 5.2]), we also know the identity (2.12) holds true for $f_{k}=p_{k,1}\otimes\cdots\otimes p_{k,q_{k}}$, $p_{k,q_{k}}\in L^{2}([0,T]\times\mathbb{R}_{0},ds\times\nu(dz))$, $k=1,\cdots,m$. Because both sides of (2.12) are multi-linear with respect to $f_{k}$, we know (2.12) holds true for

[TABLE]

where $c_{k,\ell}$ are constants, $p_{k,k^{\prime},\ell}\in L^{2}([0,T]\times\mathbb{R}_{0},ds\times\nu(dz))$, $k=1,\cdots,m,k^{\prime}=1,\cdots,q_{k}$ and $\ell=1,\cdots,\upnu_{k}$. Finally, the identity (2.12) is proved by a routine limiting argument.