The It\^{o} integral with respect to an infinite dimensional L\'{e}vy process: A series approach

TL;DR

This paper introduces a new series-based construction of the infinite dimensional Itô integral for Hilbert space valued Lévy processes, connecting it with classical real-valued stochastic integration.

Contribution

It provides an alternative, series-based approach to defining the infinite dimensional Itô integral, aligning it with existing literature.

Findings

The series approach is equivalent to existing definitions.

The construction simplifies understanding of infinite dimensional Lévy integrals.

The method bridges real-valued and infinite dimensional stochastic calculus.

Abstract

We present an alternative construction of the infinite dimensional It\^{o} integral with respect to a Hilbert space valued L\'{e}vy process. This approach is based on the well-known theory of real-valued stochastic integration, and the respective It\^{o} integral is given by a series of It\^{o} integrals with respect to standard L\'{e}vy processes. We also prove that this stochastic integral coincides with the It\^{o} integral that has been developed in the literature.

Equations

$$\displaystyle X\bullet M:=\sum_{k\in\mathbb{N}}\big{(}\langle X,f_{k}\rangle_{H}\bullet M\big{)}f_{k},$$

$$\sum _{j\in \mathbb{N}}{X}^j\bullet M^j.$$

$$X\bullet L:=\sum _{j\in \mathbb{N}}{X}^j\bullet M^j.$$

$$X\bullet L:=\Psi (X)\bullet \Phi (L),$$

$${\mathcal{L}}_T^p(E):={\mathcal{L}}^p(\Omega ,{\mathcal{F}}_T,\mathbb{P};\mathbb{D}([0,T];E)),$$

$$M_T^p(E):={\mathcal{M}}_T^p(E)/N,A_T^p(E):={\mathcal{A}}_T^p(E)/N,L_T^p(E):={\mathcal{L}}_T^p(E)/N,$$

$$\displaystyle\|X\|_{L_{T}^{p}(E)}:=\mathbb{E}\big{[}\|X\|_{E}^{p}\big{]}^{1/p}$$

$$M_T^p(E)\subset A_T^p(E)\subset L_T^p(E).$$

$$\displaystyle\mathbb{E}\big{[}\|M_{\tau}\|_{E}^{p}\big{]}\leq\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|M_{t}\|_{E}^{p}\bigg{]}<\infty,$$

$$\displaystyle\mathbb{E}\big{[}\|M_{\tau}^{n}-M_{\tau}\|_{E}^{p}\big{]}\leq\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|M_{t}^{n}-M_{t}\|_{E}^{p}\bigg{]}\rightarrow 0.$$

$$\mathbb{E}[M_{\tau }]=\underset{n\to \infty }{\lim }\mathbb{E}[M_{\tau }^n]=\underset{n\to \infty }{\lim }\mathbb{E}[M_0^n]=\mathbb{E}[M_0].$$

$$\displaystyle\mathbb{E}\big{[}\|X_{t}^{n}-X_{t}\|_{E}^{p}\big{]}\leq\mathbb{E}\bigg{[}\sup_{s\in[0,T]}\|X_{s}^{n}-X_{s}\|_{E}^{p}\bigg{]}\rightarrow 0,$$

$$\displaystyle\|M\|_{M_{T}^{p}(E)}:=\mathbb{E}\big{[}\|M_{T}\|_{E}^{p}\big{]}^{1/p}.$$

$$\displaystyle\langle M,N\rangle_{M_{T}^{2}(H)}:=\mathbb{E}\big{[}\langle M_{T},N_{T}\rangle_{H}\big{]}.$$

$$\displaystyle\bigg{\|}\sum_{n=1}^{\infty}h_{n}\bigg{\|}_{H}^{2}=\sum_{n=1}^{\infty}\|h_{n}\|_{H}^{2}.$$

$$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\|X_{s}\|_{H}^{2}d\langle M,M\rangle_{s}\bigg{]}<\infty.$$

$$\displaystyle\sum_{k\in\mathbb{N}}\big{(}\langle X,f_{k}\rangle_{H}\bullet M\big{)}f_{k}$$

$$\displaystyle\big{\langle}\big{(}\langle X,f_{j}\rangle_{H}\bullet M\big{)}f_{j},\big{(}\langle X,f_{k}\rangle_{H}\bullet M\big{)}f_{k}\big{\rangle}_{M_{T}^{2}(H)}$$

$$\displaystyle=\mathbb{E}\bigg{[}\Big{\langle}\bigg{(}\int_{0}^{T}\langle X_{s},f_{j}\rangle_{H}dM_{s}\bigg{)}f_{j},\bigg{(}\int_{0}^{T}\langle X_{s},f_{k}\rangle_{H}dM_{s}\bigg{)}f_{k}\Big{\rangle}_{H}\bigg{]}$$

$$\displaystyle=\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{T}\langle X_{s},f_{j}\rangle_{H}dM_{s}\bigg{)}\bigg{(}\int_{0}^{T}\langle X_{s},f_{k}\rangle_{H}dM_{s}\bigg{)}\langle f_{j},f_{k}\rangle_{H}\bigg{]}=0.$$

$$\displaystyle\sum_{k=1}^{\infty}\big{\|}\big{(}\langle X,f_{k}\rangle_{H}\bullet M\big{)}f_{k}\big{\|}_{M_{T}^{2}(H)}^{2}=\sum_{k=1}^{\infty}\mathbb{E}\Bigg{[}\bigg{\|}\bigg{(}\int_{0}^{T}\langle X_{s},f_{k}\rangle_{H}dM_{s}\bigg{)}f_{k}\bigg{\|}_{H}^{2}\Bigg{]}$$

$$\displaystyle=\sum_{k=1}^{\infty}\mathbb{E}\Bigg{[}\bigg{|}\int_{0}^{T}\langle X_{s},f_{k}\rangle_{H}dM_{s}\bigg{|}^{2}\Bigg{]}=\sum_{k=1}^{\infty}\mathbb{E}\bigg{[}\int_{0}^{T}|\langle X_{s},f_{k}\rangle_{H}|^{2}d\langle M,M\rangle_{s}\bigg{]}$$

$$\displaystyle=\mathbb{E}\bigg{[}\int_{0}^{T}\sum_{k=1}^{\infty}|\langle X_{s},f_{k}\rangle_{H}|^{2}d\langle M,M\rangle_{s}\bigg{]}=\mathbb{E}\bigg{[}\int_{0}^{T}\|X_{s}\|_{H}^{2}d\langle M,M\rangle_{s}\bigg{]}.$$

$$\displaystyle\mathbb{J}^{f}:=\sum_{k=1}^{\infty}\big{(}\langle X,f_{k}\rangle_{H}\bullet M\big{)}f_{k}\quad\text{and}\quad\mathbb{J}^{g}:=\sum_{k=1}^{\infty}\big{(}\langle X,g_{k}\rangle_{H}\bullet M\big{)}g_{k}.$$

$$⟨h,{\mathbb{J}}^f{⟩}_H,⟨h,{\mathbb{J}}^g{⟩}_H\in M_T^2(\mathbb{R})$$

$$\Vert ⟨h,{\mathbb{J}}^f{⟩}_H-⟨h,X{⟩}_H\bullet M{\Vert }_{M_T^2(\mathbb{R})}^2$$

$$\displaystyle=\bigg{\|}\Big{\langle}h,\sum_{k=1}^{\infty}\big{(}\langle X,f_{k}\rangle_{H}\bullet M\big{)}f_{k}\Big{\rangle}_{H}-\langle h,X\rangle_{H}\bullet M\bigg{\|}_{M_{T}^{2}(\mathbb{R})}^{2}$$

$$\displaystyle=\bigg{\|}\sum_{k=1}^{\infty}\big{(}\langle h,f_{k}\rangle_{H}\langle f_{k},X\rangle_{H}\bullet M\big{)}-\langle h,X\rangle_{H}\bullet M\bigg{\|}_{M_{T}^{2}(\mathbb{R})}^{2}$$

$$\displaystyle=\lim_{n\rightarrow\infty}\bigg{\|}\sum_{k=1}^{n}\big{(}\langle h,f_{k}\rangle_{H}\langle f_{k},X\rangle_{H}\bullet M\big{)}-\langle h,X\rangle_{H}\bullet M\bigg{\|}_{M_{T}^{2}(\mathbb{R})}^{2}$$

$$\displaystyle=\lim_{n\rightarrow\infty}\bigg{\|}\bigg{(}\sum_{k=1}^{n}\langle h,f_{k}\rangle_{H}\langle f_{k},X\rangle_{H}-\langle h,X\rangle_{H}\bigg{)}\bullet M\bigg{\|}_{M_{T}^{2}(\mathbb{R})}^{2}.$$

$$\displaystyle\bigg{|}\sum_{k=1}^{n}\langle x,f_{k}\rangle_{H}\langle f_{k},h\rangle_{H}-\langle x,h\rangle_{H}\bigg{|}^{2}\rightarrow 0\quad\text{as $n\rightarrow\infty$,}$$

$$\displaystyle\bigg{|}\sum_{k=1}^{n}\langle x,f_{k}\rangle_{H}\langle f_{k},h\rangle_{H}-\langle x,h\rangle_{H}\bigg{|}^{2}=\bigg{|}\sum_{k=n+1}^{\infty}\langle x,f_{k}\rangle_{H}\langle f_{k},h\rangle_{H}\bigg{|}^{2}$$

$$\displaystyle\leq\bigg{(}\sum_{k=1}^{\infty}|\langle x,f_{k}\rangle_{H}|^{2}\bigg{)}\bigg{(}\sum_{k=1}^{\infty}|\langle f_{k},h\rangle_{H}|^{2}\bigg{)}=\|x\|_{H}^{2}\|h\|_{H}^{2}\quad\text{for each $n\in\mathbb{N}$.}$$

$$\Vert ⟨h,{\mathbb{J}}^f{⟩}_H-⟨h,X{⟩}_H\bullet M{\Vert }_{M_T^2(\mathbb{R})}^2$$

$$\displaystyle=\lim_{n\rightarrow\infty}\mathbb{E}\Bigg{[}\bigg{|}\int_{0}^{T}\bigg{(}\sum_{k=1}^{n}\langle h,f_{k}\rangle_{H}\langle f_{k},X_{s}\rangle_{H}-\langle h,X_{s}\rangle_{H}\bigg{)}dM_{s}\bigg{|}^{2}\Bigg{]}$$

Full text

The author is grateful to an anonymous referee for valuable comments and suggestions.

1. Introduction

The Itô integral with respect to an infinite dimensional Wiener process has been developed in [4, 17, 10], and for the more general case of an infinite dimensional square-integrable martingale it has been defined in [13, 16]. In these references, one first constructs the Itô integral for elementary processes, and then extends it via the Itô isometry to a larger space, in which the space of elementary processes is dense.

For stochastic integrals with respect to a Wiener process, series expansions of the Itô integral have been considered, e.g., in [11, 7, 3]. Moreover, in the article [14], series expansions have been used in order to define the Itô integral with respect to a Wiener process for deterministic integrands with values in a Banach space. Later, in [15] this theory has been extended to general integrands with values in UMD Banach spaces.

Best to the author’s knowledge, a series approach for the construction of the Itô integral with respect to an infinite dimensional Lévy process does not exist in the literature so far. The goal of the present paper is to provide such a construction, which is based on the real-valued Itô integral, see, e.g., [1, 12, 18], and where the Itô integral is given by a series of Itô integrals with respect to real-valued Lévy processes. This approach has the advantage that we can use results from the finite dimensional case, and it might also be beneficial for lecturers teaching students who are already aware of the real-valued Itô integral and have some background in Functional Analysis. In particular, it avoids the tedious procedure of proving that elementary processes are dense in the space of integrable processes.

In [8], the stochastic integral with respect to an infinite dimensional Lévy process is defined as a limit of Riemannian sums, and a series expansion is provided. A particular feature of [8] is that stochastic integrals are considered as $L^{2}$-curves. The connection to the usual Itô integral for a finite dimensional Lévy process has been established in [23], see also Appendix B in [6]. Furthermore, we point out the articles [21] and [9], where the theory of stochastic integration with respect to Lévy processes has been extended to Banach spaces.

The idea to use series expansions for the definition of the stochastic integral has also been utilized in the context of cylindrical processes, see [19] for cylindrical Wiener processes and [2] for cylindrical Lévy processes.

The construction of the Itô integral, which we present in this paper, is divided into the following steps:

• •

For a $H$-valued process $X$ (with $H$ denoting a separable Hilbert space) and a real-valued square-integrable martingale $M$ we define the Itô integral

[TABLE]

where $(f_{k})_{k\in\mathbb{N}}$ denotes an orthonormal basis of $H$, and $\langle X,f_{k}\rangle_{H}\bullet M$ denotes the real-valued Itô integral. We will show that this definition does not depend on the choice of the orthonormal basis.

• •

Based on the just defined integral, for a $\ell^{2}(H)$-valued process $X$ and a sequence $(M^{j})_{j\in\mathbb{N}}$ of standard Lévy processes we define the Itô integral as

[TABLE]

For this, we will ensure convergence of the series.

• •

In the next step, let $L$ denote a $\ell_{\lambda}^{2}$-valued Lévy process, where $\ell_{\lambda}^{2}$ is a weighted space of sequences (cf. [5]). From the Lévy process $L$ we can construct a sequence $(M^{j})_{j\in\mathbb{N}}$ of standard Lévy processes, and for a $\ell^{2}(H)$-valued process $X$ we define the Itô integral

[TABLE]

• •

Finally, let $L$ be a general Lévy process on some separable Hilbert space $U$ with covariance operator $Q$. Then, there exist sequences of eigenvalues $(\lambda_{j})_{j\in\mathbb{N}}$ and eigenvectors, which diagonalize the operator $Q$. Denoting by $L_{2}^{0}(H)$ an appropriate space of Hilbert Schmidt operators from $U$ to $H$, our idea is to utilize the integral from the previous step, and to define the Itô integral for a $L_{2}^{0}(H)$-valued process $X$ as

[TABLE]

where $\Phi:U\rightarrow\ell_{\lambda}^{2}$ and $\Psi:L_{2}^{0}(H)\rightarrow\ell^{2}(H)$ are isometric isomorphisms such that $\Phi(L)$ is a $\ell_{\lambda}^{2}$-valued Lévy process. We will show that this definition does not depend on the choice of the eigenvalues and eigenvectors.

The remainder of this text is organized as follows: In Section 2 we provide the required preliminaries and notation. After that, we start with the construction of the Itô integral as outlined above. In Section 3 we define the Itô integral for $H$-valued processes with respect to a real-valued square-integrable martingale, and in Section 4 we define the Itô integral for $\ell^{2}(H)$-valued processes with respect to a sequence of standard Lévy processes. Section 5 gives a brief overview about Lévy processes in Hilbert spaces, together with the required results. Then, in Section 6 we define the Itô integral for $\ell^{2}(H)$-valued processes with respect to a $\ell_{\lambda}^{2}$-valued Lévy process, and in Section 7 we define the Itô integral in the general case, where the integrand is a $L_{2}^{0}(H)$-valued process and the integrator a general Lévy process on some separable Hilbert space $U$. We also prove the mentioned series representation of the stochastic integral, and show that it coincides with the usual Itô integral, which has been developed in [16].

2. Preliminaries and notation

In this section, we provide the required preliminary results and some basic notation. Throughout this text, let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual conditions. For the upcoming results, let $E$ be a separable Banach space, and let $T>0$ be a finite time horizon.

2.1 Definition**.**

Let $p\geq 1$ be arbitrary.

1. (1)

We define the Lebesgue space

[TABLE]

where $\mathbb{D}([0,T];E)$ denotes the Skorokhod space consisting of all càdlàg functions from $[0,T]$ to $E$, equipped with the supremum norm. 2. (2)

We denote by $\mathcal{A}_{T}^{p}(E)$ the space of all $E$-valued adapted processes $X\in\mathcal{L}_{T}^{p}(E)$. 3. (3)

We denote by $\mathcal{M}_{T}^{p}(E)$ the space of all $E$-valued martingales $M\in\mathcal{L}_{T}^{p}(E)$. 4. (4)

We define the factor spaces

[TABLE]

where $N\subset\mathcal{M}_{T}^{p}(E)$ denotes the subspace consisting of all $M\in\mathcal{M}_{T}^{p}(E)$ with $M=0$ up to indistinguishability.

2.2 Remark**.**

Let us emphasize the following:

1. (1)

Since the Skorokhod space $\mathbb{D}([0,T];E)$ equipped with the supremum norm is a Banach space, the Lebesgue space $L_{T}^{p}(E)$ equipped with the standard norm

[TABLE]

is a Banach space, too. 2. (2)

By the completeness of the filtration $(\mathcal{F}_{t})_{t\geq 0}$, adaptedness of an element $X\in L_{T}^{p}(E)$ does not depend on the choice of the representative. This ensures that the factor space $A_{T}^{p}(E)$ of adapted processes is well-defined. 3. (3)

The definition of $E$-valued martingales relies on the existence of conditional expectation in Banach spaces, which has been established in **[4, Prop. 1.10]**.

Note that we have the inclusions

[TABLE]

The following auxiliary result shows that these inclusions are closed.

2.3 Lemma**.**

Let $p\geq 1$ be arbitrary. Then, the following statements are true:

1. (1)

$M_{T}^{p}(E)$* is closed in $A_{T}^{p}(E)$.* 2. (2)

$A_{T}^{p}(E)$* is closed in $L_{T}^{p}(E)$.*

Proof.

Let $(M^{n})_{n\in\mathbb{N}}\subset M_{T}^{p}(E)$ be a sequence and let $M\in A_{T}^{p}(E)$ be such that $M^{n}\rightarrow M$ in $L_{T}^{p}(E)$. Furthermore, let $\tau\leq T$ be a bounded stopping time. Then we have

[TABLE]

showing that $M_{\tau}\in L^{p}(\Omega,\mathcal{F}_{\tau},\mathbb{P};E)$. Furthermore, we have

[TABLE]

By Doob’s optional stopping theorem (which also holds true for $E$-valued martingales, see [17, Remark 2.2.5]), it follows that

[TABLE]

Using Doob’s optional stopping theorem again, we conclude that $M\in M_{T}^{p}(E)$, proving the first statement.

Now, let $(X^{n})_{n\in\mathbb{N}}\subset A_{T}^{p}(E)$ be a sequence and let $X\in L_{T}^{p}(E)$ be such that $X^{n}\rightarrow X$ in $L_{T}^{p}(E)$. Then, for each $t\in[0,T]$ we have

[TABLE]

and hence $\mathbb{P}$–almost surely $X_{t}^{n_{k}}\rightarrow X_{t}$ for some subsequence $(n_{k})_{k\in\mathbb{N}}$, showing that $X_{t}$ is $\mathcal{F}_{t}$-measurable. This proves $X\in A_{T}^{p}(E)$, providing the second statement. ∎

Note that, by Doob’s martingale inequality [17, Thm. 2.2.7], for $p>1$ an equivalent norm on $M_{T}^{p}(E)$ is given by

[TABLE]

Furthermore, if $E=H$ is a separable Hilbert space, then $M_{T}^{2}(H)$ is a separable Hilbert space equipped with the inner product

[TABLE]

Finally, we recall the following result about series of pairwise orthogonal vectors in Hilbert spaces.

2.4 Lemma**.**

Let $H$ be a separable Hilbert space and let $(h_{n})_{n\in\mathbb{N}}\subset H$ be a sequence with $\langle h_{n},h_{m}\rangle_{H}=0$ for $n\neq m$. Then, the following statements are equivalent:

1. (1)

The series $\sum_{n=1}^{\infty}h_{n}$ converges in $H$. 2. (2)

The series $\sum_{n\in\mathbb{N}}h_{n}$ converges unconditionally in $H$. 3. (3)

We have $\sum_{n=1}^{\infty}\|h_{n}\|_{H}^{2}<\infty$.

If the previous conditions are satisfied, then we have

[TABLE]

Proof.

This follows from [20, Thm. 12.6] and [24, Satz V.4.8]. ∎

3. The Itô integral with respect to a real-valued square-integrable martingale

In this section, we define the Itô integral for Hilbert space valued processes with respect to a real-valued, square-integrable martingale, which is based on the real-valued Itô integral.

In what follows, let $H$ be s separable Hilbert space, and let $T>0$ be a finite time horizon. Furthermore, let $M\in\mathcal{M}_{T}^{2}(\mathbb{R})$ be a square-integrable martingale. Recall that the quadratic variation $\langle M,M\rangle$ is the (up to indistinguishability) unique real-valued, non-decreasing, predictable process with $\langle M,M\rangle_{0}=0$ such that $M^{2}-\langle M,M\rangle$ is a martingale.

3.1 Proposition**.**

Let $X$ be a $H$-valued, predictable process with

[TABLE]

Then, for every orthonormal basis $(f_{k})_{k\in\mathbb{N}}$ of $H$ the series

[TABLE]

converges unconditionally in $M_{T}^{2}(H)$, and its value does not depend on the choice of the orthonormal basis $(f_{k})_{k\in\mathbb{N}}$.

Proof.

Let $(f_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$. For $j,k\in\mathbb{N}$ with $j\neq k$ we have

[TABLE]

Moreover, by the Itô isometry for the real-valued Itô integral and the monotone convergence theorem we obtain

[TABLE]

Therefore, by (3.1) and Lemma 2.4, the series (3.2) converges unconditionally in $M_{T}^{2}(H)$.

Now, let $(g_{k})_{k\in\mathbb{N}}$ be another orthonormal basis of $H$. We define $\mathbb{J}^{f},\mathbb{J}^{g}\in M_{T}^{2}(H)$ by

[TABLE]

Let $h\in H$ be arbitrary. Then we have

[TABLE]

and the identity

[TABLE]

For all $x\in H$ we have

[TABLE]

and, by the Cauchy-Schwarz inequality,

[TABLE]

Therefore, by the Itô isometry for the real-valued Itô integral and Lebesgue’s dominated convergence theorem together with (3.1) we obtain

[TABLE]

Analogously, we prove that

[TABLE]

Therefore, denoting by $\tilde{\mathbb{J}}^{f},\tilde{\mathbb{J}}^{g}\in\mathcal{M}_{T}^{2}(H)$ representatives of $\mathbb{J}^{f}$, $\mathbb{J}^{g}$, we obtain

[TABLE]

By separability of $H$, we deduce that

[TABLE]

Consequently, we have

[TABLE]

implying $\mathbb{J}^{f}=\mathbb{J}^{g}$. This proves that the value of the series (3.2) does not depend on the choice of the orthonormal basis. ∎

Now, Proposition 3.1 gives rise to the following definition:

3.2 Definition**.**

For every $H$-valued, predictable process $X$ satisfying (3.1) we define the Itô integral $X\bullet M=(\int_{0}^{t}X_{s}dM_{s})_{t\in[0,T]}$ as

[TABLE]

where $(f_{k})_{k\in\mathbb{N}}$ denotes an orthonormal basis of $H$.

According to Proposition 3.1, the Definition (3.5) of the Itô integral is independent of the choice of the orthonormal basis $(f_{k})_{k\in\mathbb{N}}$, and the integral process $X\bullet M$ belongs to $M_{T}^{2}(H)$.

3.3 Remark**.**

As the proof of Proposition 3.1 shows, the components of the Itô integral $X\bullet M$ are pairwise orthogonal elements of the Hilbert space $M_{T}^{2}(H)$.

3.4 Proposition**.**

For every $H$-valued, predictable process $X$ satisfying (3.1) we have the Itô isometry

[TABLE]

Proof.

Let $(f_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$. According to (3.3) we have

[TABLE]

Thus, by Lemma 2.4 and (3.4) we obtain

[TABLE]

finishing the proof. ∎

3.5 Proposition**.**

Let $X$ be a $H$-valued simple process of the form

[TABLE]

with $0=t_{1}<\ldots<t_{n+1}=T$ and $\mathcal{F}_{t_{i}}$-measurable random variables $X_{i}:\Omega\rightarrow H$ for $i=0,\ldots,n$. Then, we have

[TABLE]

Proof.

Let $(f_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$. Then, for each $k\in\mathbb{N}$ the process $\langle X,f_{k}\rangle$ is a real-valued simple process with representation

[TABLE]

Thus, by the definition of the real-valued Itô integral for simple processes we obtain

[TABLE]

finishing the proof. ∎

3.6 Lemma**.**

Let $X$ be a $H$-valued, predictable process satisfying (3.1). Then, for every orthonormal basis $(f_{k})_{k\in\mathbb{N}}$ of $H$ we have

[TABLE]

where the convergence takes place in $A_{T}^{1}(\mathbb{R})$.

Proof.

We define the integral process

[TABLE]

and the sequence $(\mathbb{I}^{n})_{n\in\mathbb{N}}$ of partial sums by

[TABLE]

By (3.1) we have $\mathbb{I}\in A_{T}^{1}(\mathbb{R})$ and $(\mathbb{I}^{n})_{n\in\mathbb{N}}\subset A_{T}^{1}(\mathbb{R})$. Furthermore, by Lebesgue’s dominated convergence theorem we have

[TABLE]

which concludes the proof. ∎

3.7 Remark**.**

As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales $X,Y\in\mathcal{M}_{T}^{2}(H)$ there exists a (up to indistinguishability) unique real-valued, predictable process $\langle X,Y\rangle$ with finite variation paths and $\langle X,Y\rangle_{0}=0$ such that $\langle X,Y\rangle_{H}-\langle X,Y\rangle$ is a martingale.

3.8 Proposition**.**

For every $H$-valued, predictable process $X$ satisfying (3.1) we have

[TABLE]

Proof.

Let $(f_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$. We define the process $\mathbb{J}:=X\bullet M$ and the sequence $(\mathbb{J}^{n})_{n\in\mathbb{N}}$ of partial sums by

[TABLE]

By Proposition 3.1 we have

[TABLE]

Defining the integral process $\mathbb{I}$ by (3.6) and the sequence $(\mathbb{I}^{n})_{n\in\mathbb{N}}$ of partial sums by (3.7), using Lemma 3.6 we have

[TABLE]

Furthermore, we define the process $M\in A_{T}^{1}(\mathbb{R})$ and the sequence $(M^{n})_{n\in\mathbb{N}}\subset A_{T}^{1}(\mathbb{R})$ as

[TABLE]

Then we have $(M^{n})_{n\in\mathbb{N}}\subset M_{T}^{1}(\mathbb{R})$. Indeed, for each $n\in\mathbb{N}$ we have

[TABLE]

For every $k\in\mathbb{N}$ the quadratic variation of the real-valued process $\langle X,f_{k}\rangle_{H}\bullet M$ is given by

[TABLE]

see, e.g. [12, Thm. I.4.40.d], which shows that $M^{n}$ is a martingale. Since $M^{n}\in A_{T}^{1}(\mathbb{R})$, we deduce that $M^{n}\in M_{T}^{1}(\mathbb{R})$.

Next, we prove that $M^{n}\rightarrow M$ in $A_{T}^{1}(\mathbb{R})$. Indeed, since

[TABLE]

by the Cauchy-Schwarz inequality and (3.8) we obtain

[TABLE]

Therefore, together with (3.9) we get

[TABLE]

showing that $M^{n}\rightarrow M$ in $A_{T}^{1}(\mathbb{R})$. Now, Lemma 2.3 yields that $M\in M_{T}^{1}(\mathbb{R})$, which concludes the proof. ∎

3.9 Theorem**.**

Let $N\in\mathcal{M}_{T}^{2}(\mathbb{R})$ be another square-integrable martingale, and let $X,Y$ be two $H$-valued, predictable processes satisfying (3.1) and

[TABLE]

Then we have

[TABLE]

Proof.

Using Proposition 3.8 and the identities

[TABLE]

identity (3.11) follows from a straightforward calculation. ∎

3.10 Proposition**.**

Let $N\in\mathcal{M}_{T}^{2}(\mathbb{R})$ be another square-integrable martingale such that $\langle M,N\rangle=0$, and let $X,Y$ be two $H$-valued, predictable processes satisfying (3.1) and (3.10). Then we have

[TABLE]

Proof.

Using Remark 3.7, Theorem 3.9 and the hypothesis $\langle M,N\rangle=0$ we obtain

[TABLE]

completing the proof. ∎

4. The Itô integral with respect to a sequence of standard Lévy processes

In this section, we introduce the Itô integral for $\ell^{2}(H)$-valued processes with respect to a sequence of standard Lévy processes, which is based on the Itô integral (3.5) from the previous section. We define the space of sequences

[TABLE]

which, equipped with the inner product

[TABLE]

is a separable Hilbert space.

4.1 Definition**.**

A sequence $(M^{j})_{j\in\mathbb{N}}$ of real-valued Lévy processes is called a sequence of standard Lévy processes if it consists of square-integrable martingales with $\langle M^{j},M^{k}\rangle_{t}=\delta_{jk}\cdot t$ for all $j,k\in\mathbb{N}$. Here $\delta_{jk}$ denotes the Kronecker delta

[TABLE]

For the rest of this section, let $(M^{j})_{j\in\mathbb{N}}$ be a sequence of standard Lévy processes.

4.2 Proposition**.**

For every $\ell^{2}(H)$-valued, predictable process $X$ with

[TABLE]

the series

[TABLE]

converges unconditionally in $M_{T}^{2}(H)$.

Proof.

For $j,k\in\mathbb{N}$ with $j\neq k$ we have $\langle M^{j},M^{k}\rangle=0$, and hence, by Proposition 3.10 we obtain

[TABLE]

Moreover, by the Itô isometry (Proposition 3.4) and the monotone convergence theorem we have

[TABLE]

Thus, by (4.1) and Lemma 2.4, the series (4.2) converges unconditionally in $M_{T}^{2}(H)$. ∎

Therefore, for a $\ell^{2}(H)$-valued, predictable process $X$ satisfying (4.1) we can define the Itô integral as the series (4.2).

4.3 Remark**.**

As the proof of Proposition 4.2 shows, the components of the Itô integral $\sum_{j\in\mathbb{N}}X^{j}\bullet M^{j}$ are pairwise orthogonal elements of the Hilbert space $M_{T}^{2}(H)$.

4.4 Proposition**.**

For each $\ell^{2}(H)$-valued, predictable process $X$ satisfying (4.1) we have the Itô isometry

[TABLE]

Proof.

Using (4.3), Lemma 2.4 and identity (4.4) we obtain

[TABLE]

completing the proof. ∎

4.5 Proposition**.**

Let $X$ be a $\ell^{2}(H)$-valued simple process of the form

[TABLE]

with $0=t_{1}<\ldots<t_{n+1}=T$ and $\mathcal{F}_{t_{i}}$-measurable random variables $X_{i}:\Omega\rightarrow\ell^{2}(H)$ for $i=0,\ldots,n$. Then we have

[TABLE]

Proof.

For each $j\in\mathbb{N}$ the process $X^{j}$ is a $H$-valued simple process having the representation

[TABLE]

Hence, by Proposition 3.5 we obtain

[TABLE]

which finishes the proof. ∎

5. Lévy processes in Hilbert spaces

In this section, we provide the required results about Lévy processes in Hilbert spaces. Let $U$ be a separable Hilbert space.

5.1 Definition**.**

An $U$-valued càdlàg, adapted process $L$ is called a Lévy process if the following conditions are satisfied:

1. (1)

We have $L_{0}=0$. 2. (2)

$L_{t}-L_{s}$* is independent of $\mathcal{F}_{s}$ for all $s\leq t$.* 3. (3)

We have $L_{t}-L_{s}\overset{{\rm d}}{=}L_{t-s}$ for all $s\leq t$.

5.2 Definition**.**

An $U$-valued Lévy process $L$ with $\mathbb{E}[\|L_{t}\|_{U}^{2}]<\infty$ and $\mathbb{E}[L_{t}]=0$ for all $t\geq 0$ is called a square-integrable Lévy martingale.

Note that any square-integrable Lévy martingale $L$ is indeed a martingale, that is

[TABLE]

see [16, Prop. 3.25]. According to [16, Thm. 4.44], for each square-integrable Lévy martingale $L$ there exists a unique self-adjoint, nonnegative definite trace class operator $Q\in L(U)$, called the covariance operator of $L$, such that for all $t,s\in\mathbb{R}_{+}$ and $u_{1},u_{2}\in U$ we have

[TABLE]

Moreover, for all $u_{1},u_{2}\in U$ the angle bracket process is given by

[TABLE]

see [16, Thm. 4.49].

5.3 Lemma**.**

Let $L$ be an $U$-valued square-integrable Lévy martingale with covariance operator $Q$, let $V$ be another separable Hilbert space and let $\Phi:U\rightarrow V$ be an isometric isomorphism. Then the process $\Phi(L)$ is a $V$-valued square-integrable Lévy martingale with covariance operator $Q_{\Phi}:=\Phi Q\Phi^{-1}$.

Proof.

The process $\Phi(L)$ is a $V$-valued càdlàg, adapted process with $\Phi(L_{0})=\Phi(0)=0$. Let $s\leq t$ be arbitrary. Then the random variable $\Phi(L_{t})-\Phi(L_{s})=\Phi(L_{t}-L_{s})$ is independent of $\mathcal{F}_{s}$, and we have

[TABLE]

Moreover, for each $t\in\mathbb{R}_{+}$ we have

[TABLE]

showing that $\Phi(L)$ is a $V$-valued square-integrable Lévy martingale.

Let $t,s\in\mathbb{R}_{+}$ and $v_{i}\in V$, $i=1,2$ be arbitrary, and set $u_{i}:=\Phi^{-1}v_{i}\in U$, $i=1,2$. Then we have

[TABLE]

showing that the Lévy martingale $\Phi(L)$ has the covariance operator $Q_{\Phi}$. ∎

Now, let $Q\in L(U)$ be a self-adjoint, positive definite trace class operator. Then there exist a sequence $(\lambda_{j})_{j\in\mathbb{N}}\subset(0,\infty)$ with $\sum_{j=1}^{\infty}\lambda_{j}<\infty$ and an orthonormal basis $(e_{j}^{(\lambda)})_{j\in\mathbb{N}}$ of $U$ and such that

[TABLE]

We define the sequence of pairwise orthogonal vectors $(e_{j})_{j\in\mathbb{N}}$ as

[TABLE]

5.4 Proposition**.**

Let $L$ be an $U$-valued square-integrable Lévy martingale with covariance operator $Q$. Then the sequence $(M^{j})_{j\in\mathbb{N}}$ given by

[TABLE]

is a sequence of standard Lévy processes.

Proof.

For each $j\in\mathbb{N}$ the process $M^{j}$ is a real-valued square-integrable Lévy martingale. By (5.1), for all $j,k\in\mathbb{N}$ we obtain

[TABLE]

showing that $(M^{j})_{j\in\mathbb{N}}$ is a sequence of standard Lévy processes. ∎

6. The Itô integral with respect to a $\ell_{\lambda}^{2}$-valued Lévy process

In this section, we introduce the Itô integral for $\ell^{2}(H)$-valued processes with respect to a $\ell_{\lambda}^{2}$-valued Lévy process, which is based on the Itô integral (4.2) from Section 4.

Let $(\lambda_{j})_{j\in\mathbb{N}}\subset(0,\infty)$ be a sequence with $\sum_{j=1}^{\infty}\lambda_{j}<\infty$ and denote by $\ell_{\lambda}^{2}$ the weighted space of sequences

[TABLE]

which, equipped with the inner product

[TABLE]

is a separable Hilbert space. Note that we have the strict inclusion $\ell^{2}\subsetneqq\ell_{\lambda}^{2}$, where $\ell^{2}$ denotes the space of sequences

[TABLE]

We denote by $(g_{j})_{j\in\mathbb{N}}$ the standard orthonormal basis of $\ell^{2}$, which is given by

[TABLE]

Then the system $(g_{j}^{(\lambda)})_{j\in\mathbb{N}}$ defined as

[TABLE]

is an orthonormal basis of $\ell_{\lambda}^{2}$. Let $Q\in L(\ell_{\lambda}^{2})$ be a linear operator such that

[TABLE]

Then $Q$ is a nuclear, self-adjoint, positive definite operator. Let $L$ be a $\ell_{\lambda}^{2}$-valued, square-integrable Lévy martingale with covariance operator $Q$. According to Proposition 5.4, the sequence $(M^{j})_{j\in\mathbb{N}}$ given by

[TABLE]

is a sequence of standard Lévy processes.

6.1 Definition**.**

For every $\ell^{2}(H)$-valued, predictable process $X$ satisfying (4.1) we define the Itô integral $X\bullet L:=(\int_{0}^{t}X_{s}dL_{s})_{t\in[0,T]}$ as

[TABLE]

6.2 Remark**.**

Note that $L_{2}^{0}(H)\cong\ell^{2}(H)$, where $L_{2}^{0}(H)$ denotes the space of Hilbert-Schmidt operators from $\ell^{2}$ to $H$. In [5], the Itô integral for $L_{2}^{0}(H)$-valued processes with respect to a $\ell_{\lambda}^{2}$-valued Wiener process has been constructed in the usual fashion (first for elementary and afterwards for general processes), and then the series representation (6.3) has been proven, see [5, Prop. 2.2.1].

Now, let $(\mu_{k})_{k\in\mathbb{N}}$ be another sequence with $\sum_{k=1}^{\infty}\mu_{k}<\infty$, and let $\Phi:\ell_{\lambda}^{2}\rightarrow\ell_{\mu}^{2}$ be an isometric isomorphism such that

[TABLE]

By Lemma 5.3, the process $\Phi(L)$ is a $\ell_{\mu}^{2}$-valued, square integrable Lévy martingale with covariance operator $Q_{\Phi}$, and by Proposition 5.4, the sequence $(N^{k})_{k\in\mathbb{N}}$ given by

[TABLE]

is a sequence of standard Lévy processes.

6.3 Theorem**.**

Let $\Psi\in L(\ell^{2}(H))$ be an isometric isomorphism such that

[TABLE]

Then for every $\ell^{2}(H)$-valued, predictable process $X$ satisfying (4.1) we have

[TABLE]

and the identity

[TABLE]

Proof.

Since $\Psi$ is an isometry, by (4.1) we have

[TABLE]

showing (6.6). Moreover, by (6.4) we have

[TABLE]

and hence, we get

[TABLE]

By (6.2) and (6.8), the vectors $(g_{j}^{(\lambda)})_{j\in\mathbb{N}}$ and $(\Phi^{-1}g_{k}^{(\mu)})_{k\in\mathbb{N}}$ are eigenvectors of $Q$ with corresponding eigenvalues $(\lambda_{j})_{j\in\mathbb{N}}$ and $(\mu_{k})_{k\in\mathbb{N}}$. Therefore, and since $\Phi$ is an isometry, for $j,k\in\mathbb{N}$ with $\lambda_{j}\neq\mu_{k}$ we obtain

[TABLE]

Let $h\in H$ be arbitrary. Then we have

[TABLE]

Since $(\lambda_{j})_{j\in\mathbb{N}}$ and $(\mu_{k})_{k\in\mathbb{N}}$ are eigenvalues of $Q$, for each $j\in\mathbb{N}$ there are only finitely many $k\in\mathbb{N}$ such that $\lambda_{j}=\mu_{k}$. Therefore, by (6.9), and since $(\Phi(g_{j}^{(\lambda)}))_{j\in\mathbb{N}}$ is an orthonormal basis of $\ell_{\mu}^{2}$, we obtain

[TABLE]

Thus, taking into account (6.5) gives us

[TABLE]

Since $h\in H$ was arbitrary, using the separability of $H$ as in the proof of Proposition 3.1, we arrive at (6.7). ∎

6.4 Remark**.**

From a geometric point of view, Theorem 6.3 says that the “angle” measured by the Itô integral is preserved under isometries.

7. The Itô integral with respect to a general Lévy process

In this section, we define the Itô integral with respect to a general Lévy process, which is based on the Itô integral (6.3) from the previous section.

Let $U$ be a separable Hilbert space and let $Q\in L(U)$ be a nuclear, self-adjoint, positive definite linear operator. Then there exist a sequence $(\lambda_{j})_{j\in\mathbb{N}}\subset(0,\infty)$ with $\sum_{j=1}^{\infty}\lambda_{j}<\infty$ and an orthonormal basis $(e_{j}^{(\lambda)})_{j\in\mathbb{N}}$ of $U$ and such that

[TABLE]

namely, the $\lambda_{j}$ are the eigenvalues of $Q$, and each $e_{j}^{(\lambda)}$ is an eigenvector corresponding to $\lambda_{j}$. The space $U_{0}:=Q^{1/2}(U)$, equipped with the inner product

[TABLE]

is another separable Hilbert space and the sequence $(e_{j})_{j\in\mathbb{N}}$ given by

[TABLE]

is an orthonormal basis of $U_{0}$. We denote by $L_{2}^{0}(H):=L_{2}(U_{0},H)$ the space of Hilbert-Schmidt operators from $U_{0}$ into $H$, which, endowed with the Hilbert-Schmidt norm

[TABLE]

itself is a separable Hilbert space. We define the isometric isomorphisms

[TABLE]

Recall that $(g_{j}^{(\lambda)})_{j\in\mathbb{N}}$ denotes the orthonormal basis of $\ell_{\lambda}^{2}$, which we have defined in (6.1). Let $L$ be an $U$-valued square-integrable Lévy martingale with covariance operator $Q$.

7.1 Lemma**.**

The following statements are true:

1. (1)

The process $\Phi_{\lambda}(L)$ is a $\ell_{\lambda}^{2}$-valued square-integrable Lévy martingale with covariance operator $Q_{\Phi_{\lambda}}$. 2. (2)

We have

[TABLE]

Proof.

By Lemma 5.3, the process $\Phi_{\lambda}(L)$ is a $\ell_{\lambda}^{2}$-valued square-integrable Lévy martingale with covariance operator $Q_{\Phi_{\lambda}}$. Furthermore, by (7.2) and (7.1), for all $j\in\mathbb{N}$ we obtain

[TABLE]

showing (7.4). ∎

Now, our idea is to the define the Itô integral for a $L_{2}^{0}(H)$-valued, predictable process $X$ with

[TABLE]

by setting

[TABLE]

where the right-hand side of (7.6) denotes the Itô integral (6.3) from Definition 6.1. One might suspect that this definition depends on the choice of the eigenvalues $(\lambda_{j})_{j\in\mathbb{N}}$ and eigenvectors $(e_{j}^{(\lambda)})_{j\in\mathbb{N}}$. In order to prove that this is not the case, let $(\mu_{k})_{k\in\mathbb{N}}\subset(0,\infty)$ be another sequence with $\sum_{k=1}^{\infty}\mu_{k}<\infty$ and let $(f_{k}^{(\mu)})_{k\in\mathbb{N}}$ be another orthonormal basis of $U$ such that

[TABLE]

Then the sequence $(f_{k})_{k\in\mathbb{N}}$ given by

[TABLE]

is an orthonormal basis of $U_{0}$. Analogous to (7.2) and (7.3), we define the isometric isomorphisms

[TABLE]

Furthermore, we define the isometric isomorphisms

[TABLE]

The following diagram illustrates the situation:

[TABLE]

In order to show that the Itô integral (7.6) is well-defined, we have to show that

[TABLE]

For this, we prepare the following auxiliary result:

7.2 Lemma**.**

For all $h\in H$ and $w\in\ell^{2}(H)$ we have

[TABLE]

Proof.

By (7.1) and (7.7), the vectors $(e_{j}^{(\lambda)})_{j\in\mathbb{N}}$ and $(f_{k}^{(\mu)})_{k\in\mathbb{N}}$ are eigenvectors of $Q$ with corresponding eigenvalues $(\lambda_{j})_{j\in\mathbb{N}}$ and $(\mu_{k})_{k\in\mathbb{N}}$. Therefore, for $j,k\in\mathbb{N}$ with $\lambda_{j}\neq\mu_{k}$ we have $\langle e_{j}^{(\lambda)},f_{k}^{(\mu)}\rangle_{U}=0$. For each $v\in\ell_{\lambda}^{2}$ we obtain

[TABLE]

Let $w\in\ell^{2}(H)$ be arbitrary. By (7.3) we have

[TABLE]

and hence

[TABLE]

Therefore, for all $h\in H$ and $w\in\ell^{2}(H)$ we obtain

[TABLE]

finishing the proof. ∎

7.3 Proposition**.**

The following statements are true:

1. (1)

$\Phi_{\lambda}(L)$* is a $\ell_{\lambda}^{2}$-valued Lévy process with covariance operator $Q_{\Phi_{\lambda}}$, and we have*

[TABLE] 2. (2)

$\Phi_{\mu}(L)$* is a $\ell_{\mu}^{2}$-valued Lévy process with covariance operator $Q_{\Phi_{\mu}}$, and we have*

[TABLE] 3. (3)

For every $L_{2}^{0}(H)$-valued, predictable process $X$ with (7.5) we have

[TABLE]

and the identity (7.8).

Proof.

The first two statements follow from Lemma 7.1. Since $\Psi_{\lambda}$ and $\Psi_{\mu}$ are isometries, we obtain

[TABLE]

which, together with (7.5), yields (7.9). Now, Theorem 6.3 applies by virtue of Lemma 7.2, and yields

[TABLE]

proving (7.8). ∎

7.4 Definition**.**

For every $L_{2}^{0}(H)$-valued process $X$ satisfying (7.5) we define the Itô-Integral $X\bullet L=(\int_{0}^{t}X_{s}dL_{s})_{t\in[0,T]}$ by (7.6).

By virtue of Proposition 7.3, the Definition (7.6) of the Itô integral neither depends on the choice of the eigenvalues $(\lambda_{j})_{j\in\mathbb{N}}$ nor on the eigenvectors $(e_{j}^{(\lambda)})_{j\in\mathbb{N}}$.

Now, we shall prove the announced series representation of the Itô integral. According to Proposition 5.4, the sequences $(M^{j})_{j\in\mathbb{N}}$ and $(N^{j})_{j\in\mathbb{N}}$ of real-valued processes given by

[TABLE]

are sequences of standard Lévy processes.

7.5 Proposition**.**

For every $L_{2}^{0}(H)$-valued, predictable process $X$ satisfying (7.5) the process $(\xi^{j})_{j\in\mathbb{N}}$ given by

[TABLE]

is a $\ell^{2}(H)$-valued, predictable process, and we have

[TABLE]

where the right-hand side of (7.10) converges unconditionally in $M_{T}^{2}(H)$.

Proof.

Since $\Phi_{\lambda}$ is an isometry, for each $j\in\mathbb{N}$ we obtain

[TABLE]

Thus, by Definitions 7.4 and 6.1 we obtain

[TABLE]

and, by Proposition 4.2, the series converges unconditionally in $M_{T}^{2}(H)$. ∎

7.6 Remark**.**

By Remark 4.3 and the proof of Proposition 7.5, the components of the Itô integral $\sum_{j\in\mathbb{N}}\xi^{j}\bullet M^{j}$ are pairwise orthogonal elements of the Hilbert space $M_{T}^{2}(H)$.

7.7 Proposition**.**

For every $L_{2}^{0}(H)$-valued process $X$ satisfying (7.5) we have the Itô isometry

[TABLE]

Proof.

By the Itô isometry (Proposition 4.4), and since $\Psi_{\lambda}$ is an isometry, we obtain

[TABLE]

completing the proof. ∎

We shall now prove that the stochastic integral, which we have defined so far, coincides with the Itô integral developed in [16]. For this purpose, it suffices to consider elementary processes. Note that for each operator $S\in L(U,H)$ the restriction $S|_{U_{0}}$ belongs to $L_{2}^{0}(H)$, because

[TABLE]

7.8 Proposition**.**

Let $X$ be a $L(U,H)$-valued simple process of the form

[TABLE]

with $0=t_{1}<\ldots<t_{n+1}=T$ and $\mathcal{F}_{t_{i}}$-measurable random variables $X_{i}:\Omega\rightarrow L(U,H)$ for $i=0,\ldots,n$. Then we have

[TABLE]

Proof.

The process $\Psi_{\lambda}(X|_{U_{0}})$ is a $\ell^{2}(H)$-valued simple process having the representation

[TABLE]

Thus, by Proposition 4.5, and since $\Phi_{\lambda}$ is an isometry, we obtain

[TABLE]

completing the proof. ∎

Therefore, and since the space of simple processes is dense in the space of all predictable processes satisfying (7.5), see, e.g. [16, Cor. 8.17], the Itô integral (7.6) coincides with that in [16] for every $L_{2}^{0}(H)$-valued, predictable process $X$ satisfying (7.5). In particular, for a driving Wiener process, it coincides with the Itô integral from [4, 17, 10].

By a standard localization argument, we can extend the definition of the Itô integral to all predictable processes $X$ satisfying

[TABLE]

Since the respective spaces of predictable and adapted, measurable processes are isomorphic (see [22]), proceeding as in the [22, Sec. 3.2], we can further extend the definition of the Itô integral to all adapted, measurable processes $X$ satisfying (7.11).