Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2013, Article ID 703769, 14 pages
http://dx.doi.org/10.1155/2013/703769
Research Article
The Itô Integral with respect to an Infinite Dimensional
Lévy Process: A Series Approach
Stefan Tappe
Institut für Mathematische Stochastik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Correspondence should be addressed to Stefan Tappe; tappe@stochastik.uni-hannover.de
Received 6 November 2012; Accepted 20 February 2013
Academic Editor: Josefa Linares-Perez
Copyright © 2013 Stefan Tappe. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present an alternative construction of the infinite dimensional Itô integral with respect to a Hilbert space valued Lévy process.
This approach is based on the well-known theory of real-valued stochastic integration, and the respective Itô integral is given by
a series of Itô integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the Itô
integral that has been developed in the literature.
1. Introduction
The Itô integral with respect to an infinite dimensional
Wiener process has been developed in [1–3], and for the more
general case of an infinite dimensional square-integrable
martingale, it has been defined in [4, 5]. 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, for
example, in [6–8]. Moreover, in [9], 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 [10], this theory has been extended to
general integrands with values in UMD Banach spaces.
To the best of 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, for example, [11–13], 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 [14], 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 [14] is that stochastic integrals are considered as 𝐿2 curves. The connection to the usual Itô integral for a finite
dimensional Lévy process has been established in [15]; see
also Appendix B in [16]. Furthermore, we point out [17, 18],
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 [20] for cylindrical Lévy processes.
The construction of the Itô integral, which we present in
this paper, is divided into the following steps.
(i) For an 𝐻-valued process 𝑋 (with 𝐻 denoting a
separable Hilbert space) and a real-valued squareintegrable martingale 𝑀, we define the Itô integral
𝑋 ⋅ 𝑀 := ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 ,
𝑘∈N
(1)
where (𝑓𝑘 )𝑘∈N denotes an orthonormal basis of 𝐻,
and ⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀 denotes the real-valued Itô integral.
2
International Journal of Stochastic Analysis
We will show that this definition does not depend on
the choice of the orthonormal basis.
(ii) Based on the just defined integral, for an ℓ2 (𝐻)valued process 𝑋 and a sequence (𝑀𝑗 )𝑗∈N of standard
Lévy processes, we define the Itô integral as
𝑗
𝑗
∑𝑋 ⋅ 𝑀 .
(2)
𝑗∈N
(Ω, F, (F𝑡 )𝑡≥0 , 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.
Definition 1. Let 𝑝 ≥ 1 be arbitrary.
(1) We define the Lebesgue space
𝑝
L𝑇 (𝐸) := L𝑝 (Ω, F𝑇 , P; D ([0, 𝑇] ; 𝐸)) ,
For this, we will ensure convergence of the series.
(iii) In the next step, let 𝐿 denote an ℓ𝜆2 -valued Lévy
process, where ℓ𝜆2 is a weighted space of sequences
(cf. [21]). From the Lévy process 𝐿, we can construct a
sequence (𝑀𝑗 )𝑗∈N of standard Lévy processes, and for
a ℓ2 (𝐻)-valued process 𝑋, we define the Itô integral
𝑋 ⋅ 𝐿 := ∑ 𝑋𝑗 ⋅ 𝑀𝑗 .
(3)
𝑗∈N
where D([0, 𝑇]; 𝐸) denotes the Skorokhod space consisting of all càdlàg functions from [0, 𝑇] to 𝐸,
equipped with the supremum norm.
𝑝
(2) We denote by A𝑇 (𝐸) the space of all 𝐸-valued adapted
𝑝
processes 𝑋 ∈ L𝑇 (𝐸).
𝑝
(3) We denote by M𝑇 (𝐸) the space of all 𝐸-valued
𝑝
martingales 𝑀 ∈ L𝑇 (𝐸).
(4) We define the factor spaces
(iv) Finally, let 𝐿 be a general Lévy process on some
separable Hilbert space 𝑈 with covariance operator
𝑄. Then, there exist sequences of eigenvalues (𝜆 𝑗 )𝑗∈N
and eigenvectors, which diagonalize the operator 𝑄.
Denoting by 𝐿02 (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 𝐿02 (𝐻)-valued process 𝑋 as
𝑋 ⋅ 𝐿 := Ψ (𝑋) ⋅ Φ (𝐿) ,
ℓ𝜆2
(5)
(4)
𝐿02 (𝐻)
2
and Ψ :
→ ℓ (𝐻)
where Φ : 𝑈 →
are isometric isomorphisms such that Φ(𝐿) is an ℓ𝜆2 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 earlier. 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 ℓ2 (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 ℓ2 (𝐻)-valued processes with respect to an
ℓ𝜆2 -valued Lévy process, and in Section 7, we define the Itô
integral in the general case, where the integrand is an 𝐿02 (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 [5].
2. Preliminaries and Notation
In this section, we provide the required preliminary
results and some basic notation. Throughout this text, let
𝑝
𝑝
𝑀𝑇 (𝐸) :=
M𝑇 (𝐸)
,
𝑁
𝑝
𝑝
𝐴 𝑇 (𝐸) :=
A𝑇 (𝐸)
,
𝑁
(6)
𝑝
𝑝
𝐿 𝑇 (𝐸) :=
L𝑇 (𝐸)
,
𝑁
(7)
𝑝
where 𝑁 ⊂ M𝑇 (𝐸) denotes the subspace consisting of
𝑝
all 𝑀 ∈ M𝑇 (𝐸) with 𝑀 = 0 up to indistinguishability.
Remark 2. Let us emphasize the following.
(1) Since the Skorokhod space D([0, 𝑇]; 𝐸) equipped with
the supremum norm is a Banach space, the Lebesgue
𝑝
space 𝐿 𝑇 (𝐸) equipped with the standard norm
𝑝 1/𝑝
‖𝑋‖𝐿𝑝 (𝐸) := E[‖𝑋‖𝐸 ]
𝑇
(8)
is a Banach space too.
(2) By the completeness of the filtration (F𝑡 )𝑡≥0 , adapt𝑝
edness of an element 𝑋 ∈ 𝐿 𝑇 (𝐸) does not depend
on the choice of the representative. This ensures that
𝑝
the factor space 𝐴 𝑇 (𝐸) of adapted processes is well
defined.
(3) The definition of 𝐸-valued martingales relies on the
existence of conditional expectation in Banach spaces,
which has been established in [1, Proposition 1.10].
Note that we have the inclusions
𝑝
𝑝
𝑝
𝑀𝑇 (𝐸) ⊂ 𝐴 𝑇 (𝐸) ⊂ 𝐿 𝑇 (𝐸) .
(9)
The following auxiliary result shows that these inclusions
are closed.
Lemma 3. Let 𝑝 ≥ 1 be arbitrary. Then, the following
statements are true:
𝑝
𝑝
(1) 𝑀𝑇 (𝐸) is closed in 𝐴 𝑇 (𝐸);
𝑝
𝑝
(2) 𝐴 𝑇 (𝐸) is closed in 𝐿 𝑇 (𝐸).
International Journal of Stochastic Analysis
3
𝑝
Proof. Let (𝑀𝑛 )𝑛∈N ⊂ 𝑀𝑇 (𝐸) be a sequence, and let 𝑀 ∈
𝑝
𝑝
𝐴 𝑇 (𝐸) be such that 𝑀𝑛 → 𝑀 in 𝐿 𝑇 (𝐸). Furthermore, let
𝜏 ≤ 𝑇 be a bounded stopping time. Then, we have
󵄩 󵄩𝑝
󵄩 󵄩𝑝
E [󵄩󵄩󵄩𝑀𝜏 󵄩󵄩󵄩𝐸 ] ≤ E [ sup 󵄩󵄩󵄩𝑀𝑡 󵄩󵄩󵄩𝐸 ] < ∞,
𝑡∈[0,𝑇]
(10)
showing that 𝑀𝜏 ∈ 𝐿𝑝 (Ω, F𝜏 , P; 𝐸). Furthermore, we have
󵄩𝑝
󵄩
󵄩𝑝
󵄩
E [󵄩󵄩󵄩𝑀𝜏𝑛 − 𝑀𝜏 󵄩󵄩󵄩𝐸 ] ≤ E [ sup 󵄩󵄩󵄩𝑀𝑡𝑛 − 𝑀𝑡 󵄩󵄩󵄩𝐸 ] 󳨀→ 0.
(11)
𝑡∈[0,𝑇]
By Doob’s optional stopping theorem (which also holds true
for 𝐸-valued martingales; see [2, Remark 2.2.5]), it follows
that
E [𝑀𝜏 ] =
lim E [𝑀𝜏𝑛 ]
𝑛→∞
=
lim E [𝑀0𝑛 ]
𝑛→∞
= E [𝑀0 ] .
𝑠∈[0,𝑇]
(13)
𝑛
and, hence, P-almost surely 𝑋𝑡 𝑘 → 𝑋𝑡 for some subsequence (𝑛𝑘 )𝑘∈N , showing that 𝑋𝑡 is F𝑡 -measurable. This
𝑝
proves that 𝑋 ∈ 𝐴 𝑇 (𝐸), providing the second statement.
Note that, by Doob’s martingale inequality [2, Theorem
𝑝
2.2.7], for 𝑝 > 1, an equivalent norm on 𝑀𝑇 (𝐸) is given by
󵄩 󵄩𝑝 1/𝑝
‖𝑀‖𝑀𝑝 (𝐸) := E[󵄩󵄩󵄩𝑀𝑇 󵄩󵄩󵄩𝐸 ] .
𝑇
(14)
Furthermore, if 𝐸 = 𝐻 is a separable Hilbert space, then
𝑀𝑇2 (𝐻) is a separable Hilbert space equipped with the inner
product
⟨𝑀, 𝑁⟩𝑀𝑇2 (𝐻) := E [⟨𝑀𝑇 , 𝑁𝑇 ⟩𝐻] .
(15)
Finally, we recall the following result about series of pairwise
orthogonal vectors in Hilbert spaces.
Lemma 4. Let 𝐻 be a separable Hilbert space, and let
(ℎ𝑛 )𝑛∈N ⊂ 𝐻 be a sequence with ⟨ℎ𝑛 , ℎ𝑚 ⟩𝐻 = 0 for 𝑛 ≠ 𝑚. Then,
the following statements are equivalent.
(1) The series ∑∞
𝑛=1 ℎ𝑛 converges in 𝐻.
(2) The series ∑𝑛∈N ℎ𝑛 converges unconditionally in 𝐻.
2
(3) One has ∑∞
𝑛=1 ‖ℎ𝑛 ‖𝐻 < ∞.
Proposition 5. Let 𝑋 be an 𝐻-valued, predictable process with
𝑇
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐻𝑑⟨𝑀, 𝑀⟩𝑠 ] < ∞.
0
∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘
𝑘∈N
Proof. Let (𝑓𝑘 )𝑘∈N be an orthonormal basis of 𝐻. For 𝑗, 𝑘 ∈ N
with 𝑗 ≠ 𝑘, we have
⟨(⟨𝑋, 𝑓𝑗 ⟩𝐻 ⋅ 𝑀) 𝑓𝑗 , (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 ⟩
𝑀𝑇2 (𝐻)
𝑇
𝑇
0
0
= E [⟨(∫ ⟨𝑋𝑠 , 𝑓𝑗 ⟩𝐻𝑑𝑀𝑠 ) 𝑓𝑗 , (∫ ⟨𝑋𝑠 , 𝑓𝑘 ⟩𝐻𝑑𝑀𝑠 ) 𝑓𝑘 ⟩ ]
𝑇
𝑇
0
0
𝐻
= E [(∫ ⟨𝑋𝑠 , 𝑓𝑗 ⟩𝐻𝑑𝑀𝑠 ) (∫ ⟨𝑋𝑠 , 𝑓𝑘 ⟩𝐻𝑑𝑀𝑠 ) ⟨𝑓𝑗 , 𝑓𝑘 ⟩𝐻]
= 0.
(19)
Moreover, by the Itô isometry for the real-valued Itô integral
and the monotone convergence theorem, we obtain
∞
󵄩
󵄩2
∑ 󵄩󵄩󵄩(⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 󵄩󵄩󵄩𝑀2 (𝐻)
𝑇
𝑘=1
󵄩󵄩 𝑇
󵄩󵄩2
∞
󵄩
󵄩
= ∑ E [󵄩󵄩󵄩󵄩(∫ ⟨𝑋𝑠 , 𝑓𝑘 ⟩𝐻𝑑𝑀𝑠 ) 𝑓𝑘 󵄩󵄩󵄩󵄩 ]
󵄩
󵄩󵄩𝐻
0
󵄩
𝑘=1
󵄨󵄨 𝑇
󵄨󵄨2
∞
󵄨
󵄨
= ∑ E [󵄨󵄨󵄨󵄨∫ ⟨𝑋𝑠 , 𝑓𝑘 ⟩𝐻𝑑𝑀𝑠 󵄨󵄨󵄨󵄨 ]
󵄨󵄨 0
󵄨󵄨
𝑘=1
𝑇
𝑘=1
𝑇 ∞
Proof. This follows from [22, Theorem 12.6] and [23,
Satz V.4.8].
(18)
converges unconditionally in 𝑀𝑇2 (𝐻), and its value does not
depend on the choice of the orthonormal basis (𝑓𝑘 )𝑘∈N .
󵄨
󵄨2
= ∑ E [∫ 󵄨󵄨󵄨⟨𝑋𝑠 , 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 𝑑⟨𝑀, 𝑀⟩𝑠 ]
0
(16)
(17)
Then, for every orthonormal basis (𝑓𝑘 )𝑘∈N of 𝐻, the series
∞
If the previous conditions are satisfied, then one has
󵄩󵄩 ∞ 󵄩󵄩2
∞
󵄩󵄩
󵄩
󵄩󵄩 ∑ ℎ𝑛 󵄩󵄩󵄩 = ∑ 󵄩󵄩󵄩ℎ𝑛 󵄩󵄩󵄩2 .
󵄩󵄩
󵄩󵄩
󵄩 󵄩𝐻
󵄩󵄩𝑛=1 󵄩󵄩𝐻 𝑛=1
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 a separable Hilbert space, and
let 𝑇 > 0 be a finite time horizon. Furthermore, let 𝑀 ∈
M2𝑇 (R) be a square-integrable martingale. Recall that the
quadratic variation ⟨𝑀, 𝑀⟩ is the (up to indistinguishability)
unique real-valued, nondecreasing, predictable process with
⟨𝑀, 𝑀⟩0 = 0 such that 𝑀2 − ⟨𝑀, 𝑀⟩ is a martingale.
(12)
Using Doob’s optional stopping theorem again, we conclude
𝑝
that 𝑀 ∈ 𝑀𝑇 (𝐸), proving the first statement.
𝑝
Now, let (𝑋𝑛 )𝑛∈N ⊂ 𝐴 𝑇 (𝐸) be a sequence, and let 𝑋 ∈
𝑝
𝑝
𝐿 𝑇 (𝐸) be such that 𝑋𝑛 → 𝑋 in 𝐿 𝑇 (𝐸). Then, for each 𝑡 ∈
[0, 𝑇], we have
󵄩𝑝
󵄩
󵄩𝑝
󵄩
E [󵄩󵄩󵄩𝑋𝑡𝑛 − 𝑋𝑡 󵄩󵄩󵄩𝐸 ] ≤ E [ sup 󵄩󵄩󵄩𝑋𝑠𝑛 − 𝑋𝑠 󵄩󵄩󵄩𝐸 ] 󳨀→ 0,
3. The Itô Integral with respect to a RealValued Square-Integrable Martingale
󵄨
󵄨2
= E [∫ ∑ 󵄨󵄨󵄨⟨𝑋𝑠 , 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 𝑑⟨𝑀, 𝑀⟩𝑠 ]
0
𝑘=1
𝑇
󵄩 󵄩2
= E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐻𝑑⟨𝑀, 𝑀⟩𝑠 ] .
0
(20)
4
International Journal of Stochastic Analysis
Therefore, by (17) and Lemma 4, the series (18) converges
unconditionally in 𝑀𝑇2 (𝐻).
Now, let (𝑔𝑘 )𝑘∈N be another orthonormal basis of 𝐻. We
define J𝑓 , J𝑔 ∈ 𝑀𝑇2 (𝐻) by
∞
J𝑓 := ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 ,
𝑘=1
(21)
∞
J𝑔 := ∑ (⟨𝑋, 𝑔𝑘 ⟩𝐻 ⋅ 𝑀) 𝑔𝑘 .
󵄨󵄨 𝑇 𝑛
󵄨󵄨2
󵄨󵄨
󵄨󵄨
󵄨
[
= lim E 󵄨󵄨∫ ( ∑ ⟨ℎ, 𝑓𝑘 ⟩𝐻⟨𝑓𝑘 , 𝑋𝑠 ⟩𝐻 − ⟨ℎ, 𝑋𝑠 ⟩𝐻) 𝑑𝑀𝑠 󵄨󵄨󵄨 ]
𝑛→∞
󵄨󵄨 0 𝑘=1
󵄨󵄨
󵄨]
[󵄨
󵄨󵄨2
𝑇 󵄨󵄨󵄨 𝑛
󵄨󵄨
󵄨
= lim E [∫ 󵄨󵄨󵄨 ∑ ⟨ℎ, 𝑓𝑘 ⟩𝐻⟨𝑓𝑘 , 𝑋⟩𝐻 − ⟨ℎ, 𝑋⟩𝐻󵄨󵄨󵄨 𝑑⟨𝑀, 𝑀⟩𝑠 ]
𝑛→∞
󵄨󵄨
0 󵄨󵄨󵄨𝑘=1
󵄨
𝑘=1
Let ℎ ∈ 𝐻 be arbitrary. Then, we have
⟨ℎ, J𝑓 ⟩𝐻, ⟨ℎ, J𝑔 ⟩𝐻 ∈ 𝑀𝑇2 (R)
Therefore, by the Itô isometry for the real-valued Itô integral
and Lebesgue’s dominated convergence theorem together
with (17), we obtain
󵄩2
󵄩󵄩
󵄩󵄩⟨ℎ, J𝑓 ⟩ − ⟨ℎ, 𝑋⟩𝐻 ⋅ 𝑀󵄩󵄩󵄩 2
󵄩𝑀𝑇 (R)
󵄩
𝐻
= 0.
(22)
(26)
Analogously, we prove that
󵄩󵄩
󵄩2
𝑔
󵄩󵄩⟨ℎ, J ⟩𝐻 − ⟨ℎ, 𝑋⟩𝐻 ⋅ 𝑀󵄩󵄩󵄩𝑀2 (R) = 0.
𝑇
and the identity
󵄩2
󵄩󵄩
󵄩󵄩⟨ℎ, J𝑓 ⟩ − ⟨ℎ, 𝑋⟩𝐻 ⋅ 𝑀󵄩󵄩󵄩 2
󵄩𝑀𝑇 (R)
󵄩
𝐻
󵄩󵄩
󵄩󵄩2
∞
󵄩󵄩
󵄩󵄩
󵄩
= 󵄩󵄩⟨ℎ, ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀)𝑓𝑘 ⟩ − ⟨ℎ, 𝑋⟩𝐻 ⋅ 𝑀󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 2
𝑘=1
󵄩
󵄩𝑀𝑇 (R)
𝐻
󵄩󵄩 ∞
󵄩󵄩2
󵄩󵄩
󵄩󵄩
󵄩
= 󵄩󵄩 ∑ (⟨ℎ, 𝑓𝑘 ⟩𝐻⟨𝑓𝑘 , 𝑋⟩𝐻 ⋅ 𝑀) − ⟨ℎ, 𝑋⟩𝐻 ⋅ 𝑀󵄩󵄩󵄩
󵄩󵄩𝑘=1
󵄩󵄩 2
󵄩
󵄩𝑀𝑇 (R)
󵄨󵄨2
󵄨󵄨 𝑛
󵄨
󵄨󵄨
󵄨󵄨 ∑ ⟨𝑥, 𝑓𝑘 ⟩ ⟨𝑓𝑘 , ℎ⟩ − ⟨𝑥, ℎ⟩𝐻󵄨󵄨󵄨 󳨀→ 0 as 𝑛 󳨀→ ∞, (24)
󵄨󵄨
𝐻
𝐻
󵄨󵄨󵄨𝑘=1
󵄨󵄨
󵄨
and, by the Cauchy-Schwarz inequality,
∞
𝑘=1
𝑘=1
󵄨
󵄨2
󵄨
󵄨2
≤ ( ∑ 󵄨󵄨󵄨⟨𝑥, f𝑘 ⟩𝐻󵄨󵄨󵄨 ) ( ∑ 󵄨󵄨󵄨⟨𝑓𝑘 , ℎ⟩𝐻󵄨󵄨󵄨 )
= ‖𝑥‖2𝐻‖ℎ‖2𝐻
for each 𝑛 ∈ N.
∀ℎ ∈ 𝐻, P-almost surely.
𝐻
(28)
By separability of 𝐻, we deduce that
𝑔
𝐻
For all 𝑥 ∈ 𝐻, we have
∞
̃ 𝑓 ⟩ = ⟨ℎ, J
̃𝑔 ⟩
⟨ℎ, J
𝑇
𝑇 𝐻
𝑓
󵄩󵄩 𝑛
󵄩󵄩2
󵄩󵄩
󵄩󵄩
󵄩
= lim 󵄩󵄩( ∑ ⟨ℎ, 𝑓𝑘 ⟩𝐻⟨𝑓𝑘 , 𝑋⟩𝐻 − ⟨ℎ, 𝑋⟩𝐻) ⋅ 𝑀󵄩󵄩󵄩
.
𝑛 → ∞󵄩
󵄩󵄩 2
󵄩󵄩 𝑘=1
󵄩𝑀𝑇 (R)
(23)
󵄨󵄨 𝑛
󵄨󵄨2
󵄨󵄨
󵄨
󵄨󵄨 ∑ ⟨𝑥, 𝑓𝑘 ⟩ ⟨𝑓𝑘 , ℎ⟩ − ⟨𝑥, ℎ⟩𝐻󵄨󵄨󵄨
󵄨󵄨
𝐻
𝐻
󵄨󵄨󵄨𝑘=1
󵄨󵄨
󵄨
󵄨󵄨2
󵄨󵄨 ∞
󵄨󵄨
󵄨󵄨
󵄨
= 󵄨󵄨 ∑ ⟨𝑥, 𝑓𝑘 ⟩𝐻⟨𝑓𝑘 , ℎ⟩𝐻󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨𝑘=𝑛+1
󵄨
󵄨
̃ 𝑔 ∈ M2 (𝐻) representatives of
̃𝑓, J
Therefore, denoting by J
𝑇
J𝑓 , J𝑔 , we obtain
̃ ⟩
̃ ⟩ = ⟨ℎ, J
⟨ℎ, J
𝑇
𝑇 𝐻
󵄩󵄩 𝑛
󵄩󵄩2
󵄩󵄩
󵄩󵄩
= lim 󵄩󵄩󵄩 ∑ (⟨ℎ, 𝑓𝑘 ⟩𝐻⟨𝑓𝑘 , 𝑋⟩𝐻 ⋅ 𝑀) − ⟨ℎ, 𝑋⟩𝐻 ⋅ 𝑀󵄩󵄩󵄩
𝑛 → ∞󵄩
󵄩󵄩 2
󵄩󵄩𝑘=1
󵄩𝑀𝑇 (R)
(27)
P-almost surely,
∀ℎ ∈ 𝐻. (29)
P-almost surely,
(30)
Consequently, we have
̃𝑓 = J
̃𝑔
J
𝑇
𝑇
Implying that J𝑓 = J𝑔 . This proves that the value of the series
(18) does not depend on the choice of the orthonormal basis.
Now, Proposition 5 gives rise to the following definition.
Definition 6. For every 𝐻-valued, predictable process 𝑋 satis𝑡
fying (17), we define the Itô integral 𝑋⋅𝑀 = (∫0 𝑋𝑠 𝑑𝑀𝑠 )𝑡∈[0,𝑇]
as
𝑋 ⋅ 𝑀 := ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 ,
𝑘∈N
(31)
where (𝑓𝑘 )𝑘∈N denotes an orthonormal basis of 𝐻.
According to Proposition 5, definition (31) of the Itô
integral is independent of the choice of the orthonormal basis
(𝑓𝑘 )𝑘∈N , and the integral process 𝑋 ⋅ 𝑀 belongs to 𝑀𝑇2 (𝐻).
(25)
Remark 7. As the proof of Proposition 5 shows, the components of the Itô integral 𝑋 ⋅ 𝑀 are pairwise orthogonal
elements of the Hilbert space 𝑀𝑇2 (𝐻).
Proposition 8. For every 𝐻-valued, predictable process 𝑋
satisfying (17), one has the Itô isometry
󵄩󵄩2
󵄩󵄩 𝑇
𝑇
󵄩
󵄩
󵄩 󵄩2
E [󵄩󵄩󵄩󵄩∫ 𝑋𝑠 𝑑𝑀𝑠 󵄩󵄩󵄩󵄩 ] = E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐻 𝑑⟨𝑀, 𝑀⟩𝑠 ] .
󵄩󵄩𝐻
󵄩󵄩 0
0
(32)
International Journal of Stochastic Analysis
5
Proof. Let (𝑓𝑘 )𝑘∈N be an orthonormal basis of 𝐻. According
to (19), we have
⟨(⟨𝑋, 𝑓𝑗 ⟩𝐻 ⋅𝑀) 𝑓𝑗 , (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅𝑀) 𝑓𝑘 ⟩
𝑀𝑇2 (𝐻)
= 0 for 𝑗 ≠ 𝑘.
(33)
Lemma 10. Let 𝑋 be a 𝐻-valued, predictable process satisfying
(17). Then, for every orthonormal basis (𝑓𝑘 )𝑘∈N of 𝐻, one has
∞
󵄨
󵄨2
∑ 󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 ⋅ ⟨𝑀, 𝑀⟩ = ‖𝑋‖2𝐻 ⋅ ⟨𝑀, 𝑀⟩,
where the convergence takes place in 𝐴1𝑇 (R).
Thus, by Lemma 4 and (20), we obtain
󵄩󵄩 𝑇
󵄩󵄩2
󵄩
󵄩
E [󵄩󵄩󵄩󵄩∫ 𝑋𝑠 𝑑𝑀𝑠 󵄩󵄩󵄩󵄩 ] = ‖𝑋 ⋅ 𝑀‖2𝑀2 (𝐻)
𝑇
󵄩󵄩 0
󵄩󵄩𝐻
Proof. We define the integral process
󵄩󵄩2
󵄩󵄩 ∞
󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩 ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀)𝑓𝑘 󵄩󵄩󵄩
󵄩󵄩 2
󵄩󵄩𝑘=1
󵄩𝑀𝑇 (𝐻)
󵄩
I := ‖𝑋‖2𝐻 ⋅ ⟨𝑀, 𝑀⟩
(34)
󵄩
󵄩2
= ∑ 󵄩󵄩󵄩(⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀)𝑓𝑘 󵄩󵄩󵄩𝑀2 (𝐻)
𝑇
𝑛
󵄨
󵄨2
I𝑛 := ∑ 󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 ⋅ ⟨𝑀, 𝑀⟩.
By (17) we have I ∈ 𝐴1𝑇 (R) and (I𝑛 )𝑛∈N ⊂ 𝐴1𝑇 (R). Furthermore, by Lebesgue’s dominated convergence theorem, we
have
𝑇
󵄩 󵄩2
= E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐻𝑑⟨𝑀, 𝑀⟩𝑠 ] ,
0
finishing the proof.
Proposition 9. Let 𝑋 be a 𝐻-valued simple process of the form
𝑛
(35)
𝑖=1
with 0 = 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛+1 = 𝑇 and F𝑡𝑖 -measurable random
variables 𝑋𝑖 : Ω → 𝐻 for 𝑖 = 0, . . . , 𝑛. Then, one has
󵄨
󵄩󵄩
𝑛󵄩
𝑛󵄨
󵄩󵄩I − I 󵄩󵄩󵄩𝐿1 (R) = E [ sup 󵄨󵄨󵄨I𝑡 − I𝑡 󵄨󵄨󵄨]
𝑇
𝑡∈[0,𝑇]
󵄨󵄨 𝑡 ∞
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨2
= E [ sup 󵄨󵄨󵄨∫ ∑ 󵄨󵄨󵄨⟨X𝑠 , 𝑓𝑘 ⟩󵄨󵄨󵄨 𝑑⟨𝑀, 𝑀⟩𝑠 󵄨󵄨󵄨]
󵄨󵄨
𝑡∈[0,𝑇] 󵄨󵄨󵄨 0 𝑘=𝑛+1
󵄨
𝑇
= E [∫
∞
󵄨
󵄨2
∑ 󵄨󵄨󵄨⟨𝑋𝑠 , 𝑓𝑘 ⟩󵄨󵄨󵄨 𝑑⟨𝑀, 𝑀⟩𝑠 ] 󳨀→ 0
0 𝑘=𝑛+1
for 𝑛 󳨀→ ∞,
(42)
𝑛
𝑋 ⋅ 𝑀 = ∑𝑋𝑖 (𝑀𝑡𝑖+1 − 𝑀𝑡𝑖 ) .
(36)
𝑖=1
Proof. Let (𝑓𝑘 )𝑘∈N be an orthonormal basis of 𝐻. Then, for
each 𝑘 ∈ N, the process ⟨𝑋, 𝑓𝑘 ⟩ is a real-valued simple process
with representation
𝑛
⟨𝑋, 𝑓𝑘 ⟩𝐻 = ⟨𝑋0 , 𝑓𝑘 ⟩𝐻1{0} + ∑⟨𝑋𝑖 , 𝑓𝑘 ⟩𝐻1(𝑡𝑖 ,𝑡𝑖+1 ] .
(37)
𝑖=1
Thus, by the definition of the real-valued Itô integral for
simple processes, we obtain
𝑋 ⋅ 𝑀 = ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘
𝑛
= ∑ (∑⟨𝑋𝑖 , 𝑓𝑘 ⟩𝐻 (𝑀t𝑖+1 − 𝑀𝑡𝑖 )) 𝑓𝑘
𝑖=1
(38)
𝑡𝑖+1
= ∑ ( ∑ ⟨𝑋𝑖 , 𝑓𝑘 ⟩𝐻𝑓𝑘 ) (𝑀
𝑖=1
𝑛
𝑘∈N
= ∑𝑋𝑖 (𝑀𝑡𝑖+1 − 𝑀𝑡𝑖 ) ,
𝑖=1
finishing the proof.
which concludes the proof.
Remark 11. As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales 𝑋, 𝑌 ∈
M2𝑇 (𝐻), there exists (up to indistinguishability) a unique
real-valued, predictable process ⟨𝑋, 𝑌⟩ with finite variation
paths and ⟨𝑋, 𝑌⟩0 = 0 such that ⟨𝑋, 𝑌⟩𝐻 − ⟨𝑋, 𝑌⟩ is a
martingale.
Proposition 12. For every 𝐻-valued, predictable process 𝑋
satisfying (17), one has
⟨𝑋 ⋅ 𝑀, 𝑋 ⋅ 𝑀⟩ = ‖𝑋‖2𝐻 ⋅ ⟨𝑀, 𝑀⟩.
𝑘∈N
𝑛
𝑡𝑖
−𝑀 )
(41)
𝑘=1
𝑘=1
𝑋 = 𝑋0 1{0} + ∑𝑋𝑖 1(𝑡𝑖 ,𝑡𝑖+1 ]
(40)
and the sequence (I𝑛 )𝑛∈N of partial sums by
∞
𝑘∈N
(39)
𝑘=1
(43)
Proof. Let (𝑓𝑘 )𝑘∈N be an orthonormal basis of 𝐻. We define
the process J := 𝑋 ⋅ 𝑀 and the sequence (J𝑛 )𝑛∈N of partial
sums by
𝑛
J𝑛 := ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 .
(44)
𝑘=1
By Proposition 5, we have
J𝑛 󳨀→ J
in 𝑀T2 (𝐻) .
(45)
6
International Journal of Stochastic Analysis
Defining the integral process I by (40) and the sequence
(I𝑛 )𝑛∈N of partial sums by (41), using Lemma 10, we have
I𝑛 󳨀→ I
in 𝐴1T (R) .
(46)
Furthermore, we define the process 𝑀 ∈ 𝐴1𝑇 (R) and the
sequence (𝑀𝑛 )𝑛∈N ⊂ 𝐴1𝑇 (R) as
𝑀 := ‖J‖2𝐻 − I,
󵄩 󵄩2
𝑀𝑛 := 󵄩󵄩󵄩J𝑛 󵄩󵄩󵄩𝐻 − I𝑛 ,
(47)
𝑛 ∈ N.
Then, we have (𝑀𝑛 )𝑛∈N ⊂ 𝑀𝑇1 (R). Indeed, for each 𝑛 ∈ N, we
have
󵄩󵄩2
𝑛
󵄩󵄩󵄩 𝑛
󵄩󵄩
󵄩
󵄨
󵄨2
𝑀𝑛 = 󵄩󵄩󵄩 ∑ (⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 󵄩󵄩󵄩 − ∑ 󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 ⋅ ⟨𝑀, 𝑀⟩
󵄩󵄩
󵄩󵄩𝑘=1
󵄩𝐻 𝑘=1
󵄩
𝑛
𝑛
𝑘=1
𝑘=1
󵄩
󵄩2
󵄨
󵄨2
= ∑ 󵄩󵄩󵄩(⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀) 𝑓𝑘 󵄩󵄩󵄩𝐻 − ∑ 󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 ⋅ ⟨𝑀, 𝑀⟩
󵄨2
󵄨
󵄨2 󵄨
= ∑ (󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀󵄨󵄨󵄨 − 󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 ⋅ ⟨𝑀, 𝑀⟩) .
𝑘=1
(48)
For every 𝑘 ∈ N, the quadratic variation of the real-valued
process ⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀 is given by
󵄨2
󵄨
⟨⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀, ⟨𝑋, 𝑓𝑘 ⟩𝐻 ⋅ 𝑀⟩ = 󵄨󵄨󵄨⟨𝑋, 𝑓𝑘 ⟩𝐻󵄨󵄨󵄨 ⋅ ⟨𝑀, 𝑀⟩
(49)
see, for example, [12, Theorem I.4.40.d], which shows that 𝑀𝑛
is a martingale. Since 𝑀𝑛 ∈ 𝐴1𝑇 (R), we deduce that 𝑀𝑛 ∈
𝑀𝑇1 (R).
Next, we prove that 𝑀𝑛 → 𝑀 in 𝐴1𝑇 (R). Indeed, since
󵄨󵄨 󵄩󵄩
󵄨󵄨 ≤ 󵄩󵄩J − J𝑛 󵄩󵄩󵄩󵄩2𝐻 + 2‖J‖𝐻󵄩󵄩󵄩󵄩J − J𝑛 󵄩󵄩󵄩󵄩𝐻,
󵄨
(50)
by the Cauchy-Schwarz inequality and (45) we obtain
𝑡∈[0,𝑇]
󵄩
󵄩2
≤ E [ sup 󵄩󵄩󵄩J𝑡 − J𝑛𝑡 󵄩󵄩󵄩𝐻]
𝑡∈[0,𝑇]
⟨𝑋 ⋅ 𝑀, 𝑌 ⋅ 𝑁⟩ = ⟨𝑋, 𝑌⟩𝐻 ⋅ ⟨𝑀, 𝑁⟩.
(54)
Then, one has
Proof. Using Proposition 12 and the identities
1 󵄩󵄩
󵄩2
󵄩2 󵄩
(󵄩𝑥 + 𝑦󵄩󵄩󵄩𝐻 − 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩𝐻) ,
4 󵄩
𝑥, 𝑦 ∈ 𝐻,
1
⟨𝑀, 𝑁⟩ = (⟨𝑀 + 𝑁, 𝑀 + 𝑁⟩ − ⟨𝑀 − 𝑁, 𝑀 − 𝑁⟩) ,
4
identity (54) follows from a straightforward calculation.
(55)
Proposition 14. Let 𝑁 ∈ M2𝑇 (R) be another squareintegrable martingale such that ⟨𝑀, 𝑁⟩ = 0, and let 𝑋, 𝑌 be
two 𝐻-valued, predictable processes satisfying (17) and (53).
Then, one has
⟨𝑋 ⋅ 𝑀, 𝑌 ⋅ 𝑁⟩𝑀𝑇2 (𝐻) = 0.
(56)
Proof. Using Remark 11, Theorem 13, and the hypothesis
⟨𝑀, 𝑁⟩ = 0, we obtain
⟨𝑋 ⋅ 𝑀, 𝑌 ⋅ 𝑁⟩𝑀𝑇2 (𝐻)
𝑇
𝑇
0
0
𝑇
𝑇
0
0
𝐻
(57)
= E [∫ ⟨𝑋𝑠 , 𝑌𝑠 ⟩𝐻𝑑⟨𝑀, 𝑁⟩𝑠 ] = 0,
0
completing the proof.
4. The Itô Integral with respect to a Sequence
of Standard Lévy Processes
𝑡∈[0,𝑇]
1/2
(53)
𝑇
󵄩
󵄩2
󵄩 󵄩 󵄩
󵄩
≤ E [ sup 󵄩󵄩󵄩J𝑡 − J𝑛𝑡 󵄩󵄩󵄩𝐻] + 2E [ sup 󵄩󵄩󵄩J𝑡 󵄩󵄩󵄩𝐻󵄩󵄩󵄩J𝑡 − J𝑛𝑡 󵄩󵄩󵄩𝐻]
󵄩 󵄩2
+ 2E[ sup 󵄩󵄩󵄩J𝑡 󵄩󵄩󵄩𝐻]
𝑇
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑌𝑠 󵄩󵄩󵄩𝐻𝑑⟨𝑁, 𝑁⟩𝑠 ] < ∞.
0
= E [⟨∫ 𝑋𝑠 𝑑𝑀𝑠 , ∫ 𝑌𝑠 𝑑𝑁𝑠 ⟩]
󵄩󵄩 󵄩󵄩2 󵄩󵄩 𝑛 󵄩󵄩2 󵄨󵄨󵄨
󵄩󵄩J𝑡 󵄩󵄩𝐻 − 󵄩󵄩J𝑡 󵄩󵄩𝐻󵄨󵄨 ]
𝑡∈[0,𝑇]
Theorem 13. Let 𝑁 ∈ M2𝑇 (R) be another square-integrable
martingale, and let 𝑋, 𝑌 be two 𝐻-valued, predictable processes
satisfying (17) and
= E [⟨∫ 𝑋𝑠 𝑑𝑀𝑠 , ∫ 𝑌𝑠 𝑑𝑁𝑠 ⟩ ]
󵄩󵄩 2 󵄩󵄩 𝑛 󵄩󵄩2 󵄩󵄩
󵄩󵄩‖J‖𝐻 − 󵄩󵄩J 󵄩󵄩𝐻󵄩󵄩 1
󵄩
󵄩𝐿 𝑇 (R)
󵄨
= E [ sup 󵄨󵄨󵄨󵄨
𝑡∈[0,𝑇]
(52)
showing that 𝑀𝑛 → 𝑀 in 𝐴1𝑇 (R). Now, Lemma 3 yields that
𝑀 ∈ 𝑀𝑇1 (R), which concludes the proof.
⟨𝑥, 𝑦⟩𝐻 =
𝑛
󵄨󵄨
󵄨󵄨 ‖J‖2𝐻 − 󵄩󵄩󵄩󵄩J𝑛 󵄩󵄩󵄩󵄩2𝐻
󵄨
Therefore, together with (46), we get
󵄩 2 󵄩 𝑛 󵄩2 󵄩
󵄩󵄩
𝑛󵄩
󵄩󵄩𝑀 − 𝑀 󵄩󵄩󵄩𝐿1 (R) ≤ 󵄩󵄩󵄩󵄩‖J‖ − 󵄩󵄩󵄩J 󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝐿1 (R)
𝑇
𝑇
󵄩󵄩
𝑛󵄩
󵄩
+ 󵄩󵄩I − I 󵄩󵄩𝐿1 (R) 󳨀→ 0,
𝑇
1/2
In this section, we introduce the Itô integral for ℓ2 (𝐻)valued processes with respect to a sequence of standard Lévy
processes, which is based on the Itô integral (31) from the
previous section. We define the space of sequences
󵄩
󵄩2
E[ sup 󵄩󵄩󵄩J𝑡 − J𝑛𝑡 󵄩󵄩󵄩𝐻]
𝑡∈[0,𝑇]
󵄩2
󵄩
= 󵄩󵄩󵄩J − J𝑛 󵄩󵄩󵄩𝐿2 (𝐻) + 2‖J‖𝐿2𝑇 (𝐻)
𝑇
󵄩
󵄩
× 󵄩󵄩󵄩J − J𝑛 󵄩󵄩󵄩𝐿2 (𝐻) 󳨀→ 0.
𝑇
(51)
∞
}
{
󵄩 󵄩2
ℓ2 (𝐻) := {(ℎ𝑗 )𝑗∈N ⊂ 𝐻 : ∑ 󵄩󵄩󵄩󵄩ℎ𝑗 󵄩󵄩󵄩󵄩𝐻 < ∞} ,
𝑗=1
}
{
(58)
International Journal of Stochastic Analysis
7
which, equipped with the inner product
∞
⟨ℎ, 𝑔⟩ℓ2 (𝐻) = ∑⟨ℎ𝑗 , 𝑔𝑗 ⟩𝐻,
𝑗=1
(59)
Proposition 18. For each ℓ2 (𝐻)-valued, predictable process 𝑋
satisfying (61), one has the Itô isometry
is a separable Hilbert space.
Definition 15. A sequence (𝑀𝑗 )𝑗∈N of real-valued Lévy processes is called a sequence of standard Lévy processes if it
consists of square-integrable martingales with ⟨𝑀𝑗 , 𝑀𝑘 ⟩𝑡 =
𝛿𝑗𝑘 ⋅ 𝑡 for all 𝑗, 𝑘 ∈ N. Here, 𝛿𝑗𝑘 denotes the Kronecker delta
𝛿𝑗𝑘 = {
Remark 17. As the proof of Proposition 16 shows, the components of the Itô integral ∑𝑗∈N 𝑋𝑗 ⋅ 𝑀𝑗 are pairwise orthogonal
elements of the Hilbert space 𝑀𝑇2 (𝐻).
1, if 𝑗 = 𝑘,
0, if 𝑗 ≠ 𝑘.
(60)
For the rest of this section, let (𝑀𝑗 )𝑗∈N be a sequence of
standard Lévy processes.
󵄩󵄩
󵄩󵄩2
𝑇
𝑇
󵄩󵄩
󵄩󵄩
󵄩 󵄩2
E [󵄩󵄩󵄩󵄩 ∑ ∫ 𝑋𝑠𝑗 𝑑𝑀𝑠𝑗 󵄩󵄩󵄩󵄩 ] = E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] .
󵄩󵄩𝑗∈N 0
󵄩󵄩
0
󵄩𝐻 ]
[󵄩
Proof. Using (63), Lemma 4, and identity (64), we obtain
󵄩󵄩 ∞ 𝑇
󵄩󵄩2
󵄩󵄩2
󵄩󵄩 ∞
󵄩󵄩
󵄩
󵄩
󵄩󵄩
𝑗
𝑗󵄩
𝑗
𝑗󵄩
󵄩
󵄩
󵄩
[
]
E 󵄩󵄩󵄩∑ ∫ 𝑋𝑠 𝑑𝑀𝑠 󵄩󵄩󵄩
= 󵄩󵄩󵄩 ∑𝑋 ⋅ 𝑀 󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 2
󵄩
0
󵄩𝐻] 󵄩󵄩𝑗=1
󵄩𝑀 (𝐻)
[󵄩󵄩𝑗=1
𝑇
∞
󵄩
󵄩2
= ∑󵄩󵄩󵄩󵄩𝑋𝑗 ⋅ 𝑀𝑗 󵄩󵄩󵄩󵄩𝑀2 (𝐻)
𝑇
2
Proposition 16. For every ℓ (𝐻)-valued, predictable process
𝑋 with
𝑇
󵄩 󵄩2
= E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] ,
0
(61)
completing the proof.
the series
𝑗
𝑗
∑𝑋 ⋅ 𝑀
(62)
𝑗∈N
𝑛
⟨𝑋𝑗 ⋅ 𝑀𝑗 , 𝑋𝑘 ⋅ 𝑀𝑘 ⟩𝑀2 (𝐻) = 0.
𝑇
(63)
Moreover, by the Itô isometry (Proposition 8) and the monotone convergence theorem, we have
𝑗=1
𝑇
𝑛
𝑗
𝑋 ⋅ 𝑀 = ∑ ∑ 𝑋𝑖 ((𝑀𝑗 )
𝑡𝑖+1
𝑡𝑖
− (𝑀𝑗 ) ) .
𝑖=1 𝑗∈N
𝑛
𝑗
𝑗
𝑋𝑗 = 𝑋0 1{0} + ∑𝑋𝑖 1(𝑡𝑖 ,𝑡𝑖+1 ] .
𝑇
󵄩 󵄩2
= ∑ E [∫ 󵄩󵄩󵄩󵄩𝑋𝑠𝑗 󵄩󵄩󵄩󵄩𝐻𝑑𝑠]
0
𝑗=1
with 0 = 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛+1 = 𝑇 and F𝑡𝑖 -measurable random
variables 𝑋𝑖 : Ω → ℓ2 (𝐻) for 𝑖 = 0, . . . , 𝑛. Then, one has
(68)
Proof. For each 𝑗 ∈ N, the process 𝑋𝑗 is a 𝐻-valued simple
process having the representation
󵄩󵄩2
󵄩󵄩 𝑇
󵄩
󵄩
= ∑ E [󵄩󵄩󵄩󵄩∫ 𝑋𝑠𝑗 𝑑𝑀𝑠𝑗 󵄩󵄩󵄩󵄩 ]
󵄩󵄩𝐻
󵄩
0
󵄩
𝑗=1
∞
∞
(67)
𝑖=1
Proof. For 𝑗, 𝑘 ∈ N with 𝑗 ≠ 𝑘, we have ⟨𝑀𝑗 , 𝑀𝑘 ⟩ = 0, and,
hence, by Proposition 14, we obtain
󵄩
󵄩2
∑󵄩󵄩󵄩󵄩𝑋𝑗 ⋅ 𝑀𝑗 󵄩󵄩󵄩󵄩𝑀2 (𝐻)
Proposition 19. Let 𝑋 be a ℓ2 (𝐻)-valued simple process of the
form
𝑋 = 𝑋0 1{0} + ∑𝑋𝑖 1(𝑡𝑖 ,𝑡𝑖+1 ]
converges unconditionally in 𝑀𝑇2 (𝐻).
∞
(66)
𝑗=1
𝑇
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] < ∞,
0
(65)
(69)
𝑖=1
(64)
𝑇 ∞
󵄩 󵄩2
= E [∫ ∑ 󵄩󵄩󵄩󵄩𝑋𝑠𝑗 󵄩󵄩󵄩󵄩𝐻𝑑𝑠]
0
[ 𝑗=1
]
𝑇
󵄩 󵄩2
= E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] .
0
Thus, by (61) and Lemma 4, the series (62) converges unconditionally in 𝑀𝑇2 (𝐻).
Hence, by Proposition 9, we obtain
𝑋 ⋅ 𝑀 = ∑ 𝑋𝑗 ⋅ 𝑀𝑗
j∈N
𝑛
𝑡𝑖+1
𝑡𝑖
− (𝑀𝑗 ) )
𝑗∈N 𝑖=1
𝑛
𝑗
𝑡𝑖+1
= ∑ ∑ 𝑋𝑖 ((𝑀𝑗 )
𝑖=1 𝑗∈N
2
Therefore, for a ℓ (𝐻)-valued, predictable process 𝑋
satisfying (61) we can define the Itô integral as the series (62).
𝑗
= ∑ ∑𝑋𝑖 ((𝑀𝑗 )
which finishes the proof.
𝑡𝑖
− (𝑀𝑗 ) ) ,
(70)
8
International Journal of Stochastic Analysis
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.
Definition 20. A 𝑈-valued càdlàg, adapted process 𝐿 is called
a Lévy process if the following conditions are satisfied.
(1) We have 𝐿 0 = 0.
(3) We have 𝐿 𝑡 − 𝐿 𝑠 = 𝐿 𝑡−𝑠 for all 𝑠 ≤ 𝑡.
Definition 21. A 𝑈-valued Lévy process 𝐿 with E[‖𝐿 𝑡 ‖2𝑈] < ∞
and E[𝐿 𝑡 ] = 0 for all 𝑡 ≥ 0 is called a square-integrable Lévy
martingale.
Note that any square-integrable Lévy martingale 𝐿 is
indeed a martingale; that is,
E [⟨𝐿 𝑡 , 𝑢1 ⟩𝑈⟨𝐿 𝑠 , 𝑢2 ⟩𝑈] = (𝑡 ∧ 𝑠) ⟨𝑄𝑢1 , 𝑢2 ⟩𝑈.
(72)
Moreover, for all 𝑢1 , 𝑢2 ∈ 𝑈, the angle bracket process is given
by
𝑡 ≥ 0;
(73)
see [5, Theorem 4.49].
Lemma 22. Let 𝐿 be a 𝑈-valued square-integrable Lévy
martingale with covariance operator 𝑄, let 𝑉 be another
separable Hilbert space, and let Φ : 𝑈 → 𝑉 be an isometric
isomorphism. Then, the process Φ(𝐿) is a 𝑉-valued squareintegrable Lévy martingale with covariance operator 𝑄Φ :=
Φ𝑄Φ−1 .
Proof. The process Φ(𝐿) is a 𝑉-valued càdlàg, adapted process with Φ(𝐿 0 ) = Φ(0) = 0. Let 𝑠 ≤ 𝑡 be arbitrary. Then, the
random variable Φ(𝐿 𝑡 ) − Φ(𝐿 𝑠 ) = Φ(𝐿 𝑡 − 𝐿 𝑠 ) is independent
of F𝑠 , and we have
d
Φ (𝐿 𝑡 ) − Φ (𝐿 𝑠 ) = Φ (𝐿 𝑡 − 𝐿 𝑠 ) = Φ (𝐿 𝑡−𝑠 ) .
(74)
Moreover, for each 𝑡 ∈ R+ , we have
󵄩
󵄩2
󵄩 󵄩2
E [󵄩󵄩󵄩Φ (𝐿 𝑡 )󵄩󵄩󵄩𝑉] = E [󵄩󵄩󵄩𝐿 𝑡 󵄩󵄩󵄩𝑈] < ∞,
E [Φ (𝐿 𝑡 )] = ΦE (𝐿 𝑡 ) = 0,
= E [⟨𝐿 𝑡 , 𝑢1 ⟩𝑈⟨𝐿 𝑠 , 𝑢2 ⟩𝑈] = (𝑡 ∧ 𝑠) ⟨𝑄𝑢1 , u2 ⟩𝑈
(76)
showing that the Lévy martingale Φ(𝐿) has the covariance
operator 𝑄Φ .
Now, let 𝑄 ∈ 𝐿(𝑈) be a self-adjoint, positive definite trace
class operator. Then, there exists a sequence (𝜆 𝑗 )𝑗∈N ⊂ (0, ∞)
(𝜆)
with ∑∞
𝑗=1 𝜆 𝑗 < ∞ and an orthonormal basis (𝑒𝑗 )𝑗∈N of 𝑈
such that
(71)
see [5, Proposition 3.25]. According to [5, Theorem 4.44],
for each square-integrable Lévy martingale 𝐿, there exists a
unique self-adjoint, nonnegative definite trace class operator
𝑄 ∈ 𝐿(𝑈), called the covariance operator of 𝐿, such that for all
𝑡, 𝑠 ∈ R+ and 𝑢1 , 𝑢2 ∈ 𝑈, we have
⟨⟨𝐿, 𝑢1 ⟩𝑈, ⟨𝐿, 𝑢2 ⟩𝑈⟩𝑡 = 𝑡⟨𝑄𝑢1 , 𝑢2 ⟩𝑈,
= E [⟨Φ(𝐿 𝑡 ), Φ(𝑢1 )⟩𝑉⟨Φ(𝐿 𝑠 ), Φ(𝑢2 )⟩𝑉]
= (𝑡 ∧ 𝑠) ⟨Φ𝑄Φ−1 V1 , V2 ⟩𝑉 = (𝑡 ∧ 𝑠) ⟨𝑄Φ V1 , V2 ⟩𝑉,
d
∀𝑠 ≤ t
E [⟨Φ(𝐿 𝑡 ), V1 ⟩𝑉⟨Φ(𝐿 𝑠 ), V2 ⟩𝑉]
= (𝑡 ∧ 𝑠) ⟨𝑄Φ−1 V1 , Φ−1 V2 ⟩𝑈
(2) 𝐿 𝑡 − 𝐿 𝑠 is independent of F𝑠 for all 𝑠 ≤ 𝑡.
E [𝑋𝑡 | F𝑠 ] = 𝑋𝑠
Let 𝑡, 𝑠 ∈ R+ and V𝑖 ∈ 𝑉, 𝑖 = 1, 2 be arbitrary, and set
𝑢𝑖 := Φ−1 V𝑖 ∈ 𝑈, 𝑖 = 1, 2. Then, we have
(75)
showing that Φ(𝐿) is a 𝑉-valued square-integrable Lévy
martingale.
𝑄𝑒𝑗(𝜆) = 𝜆 𝑗 𝑒𝑗(𝜆)
∀𝑗 ∈ N.
(77)
We define the sequence of pairwise orthogonal vectors (𝑒𝑗 )𝑗∈N
as
𝑒𝑗 := √𝜆 𝑗 𝑒𝑗(𝜆) ,
𝑗 ∈ N.
(78)
Proposition 23. Let 𝐿 be a 𝑈-valued square-integrable Lévy
martingale with covariance operator 𝑄. Then, the sequence
(𝑀𝑗 )𝑗∈N given by
𝑀𝑗 :=
1
√𝜆 𝑗
⟨𝐿, 𝑒𝑗(𝜆) ⟩ ,
𝑗 ∈ N,
𝑈
(79)
is a sequence of standard Lévy processes.
Proof. For each 𝑗 ∈ N, the process 𝑀𝑗 is a real-valued squareintegrable Lévy martingale. By (73), for all 𝑗, 𝑘 ∈ N, we obtain
⟨𝑀𝑗 , 𝑀𝑘 ⟩𝑡 =
=
1
√𝜆 𝑗 𝜆 𝑘
⟨⟨𝐿, 𝑒𝑗(𝜆) ⟩ , ⟨𝐿, 𝑒𝑘(𝜆) ⟩𝑈⟩
𝑈
𝑡⟨𝑄𝑒𝑗(𝜆) , 𝑒𝑘(𝜆) ⟩
𝑈
√𝜆 𝑗 𝜆 𝑘
=
𝑡
𝑡𝜆 𝑗 ⟨𝑒𝑗(𝜆) , 𝑒𝑘(𝜆) ⟩
𝑈
√𝜆 𝑗 𝜆 𝑘
= 𝛿𝑗𝑘 ⋅ 𝑡,
(80)
showing that (𝑀𝑗 )𝑗∈N is a sequence of standard Lévy processes.
6. The Itô Integral with respect to an ℓ𝜆2 Valued Lévy Process
In this section, we introduce the Itô integral for ℓ2 (𝐻)-valued
processes with respect to an ℓ𝜆2 -valued Lévy process, which is
based on the Itô integral (62) from Section 4.
International Journal of Stochastic Analysis
9
Let (𝜆 𝑗 )𝑗∈N ⊂ (0, ∞) be a sequence with ∑∞
𝑗=1 𝜆 𝑗 < ∞ and
denote by ℓ𝜆2 the weighted space of sequences
∞
{
}
󵄨 󵄨2
ℓ𝜆2 := {(V𝑗 )𝑗∈N ⊂ R : ∑ 𝜆 𝑗 󵄨󵄨󵄨󵄨V𝑗 󵄨󵄨󵄨󵄨 < ∞} ,
𝑗=1
{
}
∞
𝑗=1
(𝜇)
(82)
𝑔2 = (0, 1, 0, . . .) , . . . .
(84)
Then, the system (𝑔𝑗(𝜆) )𝑗∈N defined as
𝑔𝑗(𝜆) :=
𝑔𝑗
√𝜆 𝑗
,
𝑗 ∈ N,
𝑁𝑘 :=
∀𝑗 ∈ N.
1
√𝜆 𝑗
⟨𝐿, 𝑔𝑗(𝜆) ⟩ 2 ,
ℓ𝜆
𝑗 ∈ N,
(87)
is a sequence of standard Lévy processes.
Definition 24. For every ℓ (𝐻)-valued, predictable process
𝑋 satisfying (61), we define the Itô integral 𝑋 ⋅ 𝐿 :=
𝑡
(∫0 𝑋𝑠 𝑑𝐿 𝑠 )𝑡∈[0,𝑇] as
𝑗∈N
𝑘 ∈ N,
(90)
Remark 25. Note that 𝐿02 (𝐻) ≅ ℓ2 (𝐻), where 𝐿02 (𝐻) denotes
the space of Hilbert-Schmidt operators from ℓ2 to 𝐻. In [21],
the Itô integral for 𝐿02 (𝐻)-valued processes with respect to
an ℓ𝜆2 -valued Wiener process has been constructed in the
usual fashion (first for elementary and afterwards for general
processes), and then the series representation (88) has been
proven; see [21, Proposition 2.2.1].
(91)
𝑇
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩Ψ(𝑋𝑠 )󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] < ∞
0
(92)
𝑋 ⋅ 𝐿 = Ψ (𝑋) ⋅ Φ (𝐿) .
(93)
and the identity
Proof. Since Ψ is an isometry, by (61), we have
𝑇
𝑇
󵄩
󵄩 󵄩2
󵄩2
E [∫ 󵄩󵄩󵄩Ψ(𝑋𝑠 )󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] = E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] < ∞, (94)
0
0
showing (92). Moreover, by (89), we have
(𝜇)
(𝜇)
Φ𝑄Φ−1 𝑔𝑘 = 𝑄Φ 𝑔𝑘 = 𝜇𝑘 𝑔𝑘
∀𝑘 ∈ N,
(95)
and, hence, we get
(𝜇)
(88)
∀ℎ ∈ 𝐻, 𝑤 ∈ ℓ2 (H) .
Then, for every ℓ2 (𝐻)-valued, predictable process 𝑋 satisfying
(61), one has
(𝜇)
2
𝑋 ⋅ 𝐿 := ∑ 𝑋𝑗 ⋅ 𝑀𝑗 .
⟨ℎ, Ψ(𝑤)⟩𝐻 = Φ (⟨ℎ, 𝑤⟩𝐻)
(86)
Then, 𝑄 is a nuclear, self-adjoint, positive definite operator.
Let 𝐿 be an ℓ𝜆2 -valued, square-integrable Lévy martingale
with covariance operator 𝑄. According to Proposition 23, the
sequence (𝑀j )𝑗∈N given by
𝑀𝑗 :=
1
(𝜇)
⟨Φ(𝐿), 𝑔𝑘 ⟩ℓ2 ,
𝜇
√𝜇𝑘
Theorem 26. Let Ψ ∈ 𝐿(ℓ2 (𝐻)) be an isometric isomorphism
such that
(85)
is an orthonormal basis of ℓ𝜆2 . Let 𝑄 ∈ 𝐿(ℓ𝜆2 ) be a linear
operator such that
𝑄𝑔𝑗(𝜆) = 𝜆 𝑗 𝑔𝑗(𝜆) ,
(89)
is a sequence of standard Lévy processes.
(83)
We denote by (𝑔𝑗 )𝑗∈N the standard orthonormal basis of ℓ2 ,
which is given by
𝑔1 = (1, 0, . . .) ,
∀𝑘 ∈ N.
By Lemma 22, the process Φ(𝐿) is a ℓ𝜇2 -valued, square
integrable Lévy martingale with covariance operator 𝑄Φ , and
by Proposition 23, the sequence (𝑁𝑘 )𝑘∈N given by
is a separable Hilbert space. Note that we have the strict
inclusion ℓ2 ⫋ ℓ𝜆2 , where ℓ2 denotes the space of sequences
∞
{
}
󵄨 󵄨2
ℓ2 = {(V𝑗 )𝑗∈N ⊂ R : ∑ 󵄨󵄨󵄨󵄨V𝑗 󵄨󵄨󵄨󵄨 < ∞} .
𝑗=1
{
}
(𝜇)
𝑄Φ 𝑔𝑘 = 𝜇𝑘 𝑔𝑘 ,
(81)
which, equipped with the inner product
⟨V, 𝑤⟩ℓ𝜆2 = ∑ 𝜆 𝑗 V𝑗 𝑤𝑗 ,
Now, let (𝜇𝑘 )𝑘∈N be another sequence with ∑∞
𝑘=1 𝜇𝑘 < ∞,
and let Φ : ℓ𝜆2 → ℓ𝜇2 be an isometric isomorphism such that
(𝜇)
𝑄 (Φ−1 𝑔𝑘 ) = 𝜇𝑘 (Φ−1 𝑔𝑘 )
∀𝑘 ∈ N.
(96)
(𝜇)
By (86) and (96), the vectors (𝑔𝑗(𝜆) )𝑗∈N and (Φ−1 𝑔𝑘 )𝑘∈N are
eigenvectors of 𝑄 with corresponding eigenvalues (𝜆 𝑗 )𝑗∈N and
(𝜇𝑘 )𝑘∈N . Therefore, and since Φ is an isometry, for 𝑗, 𝑘 ∈ N
with 𝜆 𝑗 ≠ 𝜇𝑘 , we obtain
(𝜇)
(𝜇)
⟨Φ𝑔𝑗(𝜆) , 𝑔𝑘 ⟩ 2 = ⟨𝑔𝑗(𝜆) , Φ−1 𝑔𝑘 ⟩ 2 = 0.
ℓ𝜇
ℓ𝜆
(97)
10
International Journal of Stochastic Analysis
Let ℎ ∈ 𝐻 be arbitrary. Then, we have
Thus, taking into account (91) gives us
∞
⟨ℎ, 𝑋 ⋅ 𝐿⟩𝐻 = ∑ ⟨ℎ, Ψ(𝑋)𝑘 ⟩𝐻 ⋅ 𝑁𝑘
∞
⟨ℎ, 𝑋 ⋅ 𝐿⟩𝐻 = ⟨ℎ, ∑𝑋𝑗 ⋅ 𝑀𝑗 ⟩
𝑗=1
𝑘=1
∞
𝐻
= ∑ ⟨ℎ, Ψ(𝑋)𝑘 ⋅ 𝑁𝑘 ⟩𝐻
∞
= ∑⟨ℎ, 𝑋𝑗 ⋅ 𝑀𝑗 ⟩𝐻
𝑘=1
𝑗=1
∞
= ⟨ℎ, ∑ Ψ(𝑋)𝑘 ⋅ 𝑁𝑘 ⟩
∞
= ∑⟨ℎ, 𝑋𝑗 ⟩𝐻 ⋅ 𝑀𝑗
𝑘=1
𝑗=1
∞
1
=∑
𝑗=1 √𝜆 𝑗
∞
1
=∑
𝑗=1 √𝜆 𝑗
⋅
1
√𝜆 𝑗
∞
⋅
1
√𝜆 𝑗
⟨⟨ℎ, 𝑋⟩𝐻, 𝑔𝑗(𝜆) ⟩ 2 ⋅
ℓ𝜆
1
√𝜆 𝑗
⟨𝐿, 𝑔𝑗(𝜆) ⟩
ℓ𝜆2
ℓ𝜇2
7. The Itô Integral with respect to a General
Lévy Process
⟨Φ(⟨ℎ, 𝑋⟩𝐻), Φ𝑔𝑗(𝜆) ⟩
ℓ𝜇2
∞
(𝜇)
(𝜇)
( ∑ ⟨Φ(𝐿), 𝑔𝑘 ⟩ℓ2 ⟨𝑔𝑘 , Φ𝑔𝑗(𝜆) ⟩ 2 ) .
𝑘=1
𝜇
ℓ𝜇
(98)
Since (𝜆 𝑗 )𝑗∈N and (𝜇𝑘 )𝑘∈N are eigenvalues of 𝑄, for each 𝑗 ∈ N,
there are only finitely many 𝑘 ∈ N such that 𝜆 𝑗 = 𝜇𝑘 .
Therefore, by (97), and since (Φ(𝑔𝑗(𝜆) ))𝑗∈N is an orthonormal
basis of ℓ𝜇2 , we obtain
⟨ℎ, 𝑋 ⋅ 𝐿⟩𝐻
∞
Since ℎ ∈ 𝐻 was arbitrary, using the separability of 𝐻 as in
the proof of Proposition 5, we arrive at (93).
Remark 27. From a geometric point of view, Theorem 26 says
that the “angle” measured by the Itô integral is preserved
under isometries.
⟨Φ(⟨ℎ, 𝑋⟩𝐻), Φ𝑔𝑗(𝜆) ⟩
ℓ𝜇2
𝑗=1 √𝜆 𝑗
∞
1
(𝜇)
⟨Φ(⟨ℎ, 𝑋⟩𝐻), Φ𝑔𝑗(𝜆) ⟩ 2 ⟨Φ𝑔𝑗(𝜆) , 𝑔𝑘 ⟩ 2 )
ℓ𝜇
ℓ𝜇
𝑗=1 𝜆 𝑗
In this section, we define the Itô integral with respect to a
general Lévy process, which is based on the Itô integral (88)
from the previous section.
Let 𝑈 be a separable Hilbert space, and let 𝑄 ∈ 𝐿(𝑈) be
a nuclear, self-adjoint, positive definite linear operator. Then,
there exist a sequence (𝜆 𝑗 )𝑗∈N ⊂ (0, ∞) with ∑∞
𝑗=1 𝜆 𝑗 < ∞
and an orthonormal basis (𝑒𝑗(𝜆) )𝑗∈N of 𝑈 such that
𝑄𝑒𝑗(𝜆) = 𝜆 𝑗 𝑒𝑗(𝜆) ,
(𝜇)
⋅ ⟨Φ (𝐿) , 𝑔𝑘 ⟩ℓ𝜇2
∞
1 ∞
(𝜇)
( ∑⟨Φ(⟨ℎ, 𝑋⟩𝐻), Φ𝑔𝑗(𝜆) ⟩ 2 ⟨Φ𝑔𝑗(𝜆) , 𝑔𝑘 ⟩ 2 )
ℓ𝜇
ℓ𝜇
𝜇
𝑗=1
𝑘=1 𝑘
⋅
∞
1
1
(𝜇)
(𝜇)
⟨Φ(⟨ℎ, 𝑋⟩𝐻), 𝑔𝑘 ⟩ℓ2 ⋅
⟨Φ(𝐿), 𝑔𝑘 ⟩ℓ2
𝜇
𝜇
𝜇
𝜇
√
√
𝑘
𝑘
𝑘=1
=∑
∞
⟨𝑢, V⟩𝑈0 := ⟨𝑄−1/2 𝑢, 𝑄−1/2 V⟩𝑈,
𝑘=1
(99)
(102)
is another separable Hilbert space, and the sequence (𝑒𝑗 )𝑗∈N
given by
𝑒𝑗 = √𝜆 𝑗 𝑒𝑗(𝜆) ,
𝑗 ∈ N,
(103)
is an orthonormal basis of 𝑈0 . We denote by 𝐿02 (𝐻) :=
𝐿 2 (𝑈0 , 𝐻) the space of Hilbert-Schmidt operators from 𝑈0
into 𝐻, which, endowed with the Hilbert-Schmidt norm
‖𝑆‖𝐿02 (𝐻)
∞
󵄩 󵄩2
:= ( ∑󵄩󵄩󵄩󵄩𝑆𝑒𝑗 󵄩󵄩󵄩󵄩𝐻)
1/2
,
𝑗=1
𝑆 ∈ 𝐿02 (𝐻) ,
(104)
itself is a separable Hilbert space. We define the isometric
isomorphisms
Φ𝜆 : 𝑈 󳨀→ ℓ𝜆2 ,
Φ𝜆 𝑒𝑗(𝜆) := 𝑔𝑗(𝜆)
for 𝑗 ∈ N,
Ψ𝜆 : 𝐿02 (𝐻) 󳨀→ ℓ2 (𝐻) ,
𝑘
= ∑ Φ(⟨ℎ, 𝑋⟩𝐻) ⋅ 𝑁𝑘 .
(101)
eigenvector corresponding to 𝜆 𝑗 . The space 𝑈0 := 𝑄1/2 (𝑈),
equipped with the inner product
=∑
(𝜇)
⟨Φ(𝐿), 𝑔𝑘 ⟩ℓ2
𝜇
∀𝑗 ∈ N;
namely, 𝜆 𝑗 are the eigenvalues of 𝑄, and each 𝑒𝑗(𝜆) is an
= ∑ (∑
𝑘=1
𝐻
= ⟨ℎ, Ψ (𝑋) ⋅ Φ (𝐿) ⟩𝐻.
⟨Φ(𝐿), Φ𝑔𝑗(𝜆) ⟩
1
=∑
(100)
Ψ𝜆 (𝑆) := (𝑆𝑒𝑗 )𝑗∈N
for 𝑆 ∈ 𝐿02 (𝐻) .
(105)
(106)
International Journal of Stochastic Analysis
11
Recall that (𝑔𝑗(𝜆) )𝑗∈N denotes the orthonormal basis of ℓ𝜆2 ,
which we have defined in (85). Let 𝐿 be a 𝑈-valued squareintegrable Lévy martingale with covariance operator Q.
The following diagram illustrates the situation:
(Φ, Ψ)
(𝓁𝜆2 , 𝓁2(𝐻))
Lemma 28. The following statements are true.
(𝓁𝜇2 , 𝓁2(𝐻))
(Φ𝜆 , Ψ𝜆)
(1) The process Φ𝜆 (𝐿) is an ℓ𝜆2 -valued square-integrable
Lévy martingale with covariance operator 𝑄Φ𝜆 .
(Φ𝜇 , Ψ𝜇 )
(115)
(𝑈, 𝐿02 (𝐻))
(2) One has
𝑄Φ𝜆 𝑔𝑗(𝜆) = 𝜆 𝑗 𝑔𝑗(𝜆) ,
∀𝑗 ∈ N.
(107)
Proof. By Lemma 22, the process Φ𝜆 (𝐿) is an ℓ𝜆2 -valued
square-integrable Lévy martingale with covariance operator
𝑄Φ𝜆 . Furthermore, by (105) and (101), for all 𝑗 ∈ N, we obtain
(𝜆)
(𝜆)
𝑄Φ𝜆 𝑔𝑗(𝜆) = Φ𝜆 𝑄Φ−1
𝜆 𝑔𝑗 = Φ𝜆 𝑄𝑒𝑗
= Φ𝜆 (𝜆 𝑗 𝑒𝑗(𝜆) ) = 𝜆 𝑗 Φ𝜆 𝑒𝑗(𝜆) = 𝜆 𝑗 𝑔𝑗(𝜆) ,
(108)
Ψ𝜆 (𝑋) ⋅ Φ𝜆 (𝐿) = Ψ𝜇 (𝑋) ⋅ Φ𝜇 (𝐿) .
(116)
For this, we prepare the following auxiliary result.
Lemma 29. For all ℎ ∈ 𝐻 and 𝑤 ∈ ℓ2 (𝐻), one has
⟨ℎ, Ψ(𝑤)⟩𝐻 = Φ (⟨ℎ, 𝑤⟩𝐻) .
showing (107).
Now, our idea is to the define the Itô integral for an 𝐿02 (𝐻)-
valued, predictable process 𝑋 with
(117)
(𝜇)
Proof. By (101) and (111), the vectors (𝑒𝑗(𝜆) )𝑗∈N and (𝑓𝑘 )𝑘∈N
are eigenvectors of 𝑄 with corresponding eigenvalues (𝜆 𝑗 )𝑗∈N
and (𝜇𝑘 )𝑘∈N . Therefore, for 𝑗, 𝑘 ∈ N with 𝜆 𝑗 ≠ 𝜇𝑘 , we have
(𝜇)
𝑇
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] < ∞
2
0
In order to show that the Itô integral (110) is well defined,
we have to show that
(109)
⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩𝑈 = 0. For each V ∈ ℓ𝜆2 , we obtain
Φ (V) = (Φ𝜇 ∘ Φ−1
𝜆 ) (V)
by setting
𝑋 ⋅ 𝐿 := Ψ𝜆 (𝑋) ⋅ Φ𝜆 (𝐿) ,
(110)
where the right-hand side of (110) denotes the Itô integral (88)
from Definition 24. One might suspect that this definition
depends on the choice of the eigenvalues (𝜆 𝑗 )𝑗∈N and eigenvectors (𝑒𝑗(𝜆) )𝑗∈N . In order to prove that this is not the case, let
(𝜇𝑘 )𝑘∈N ⊂ (0, ∞) be another sequence with ∑∞
𝑘=1 𝜇𝑘 < ∞, and
(𝜇)
let (𝑓𝑘 )𝑘∈N be another orthonormal basis of 𝑈 such that
(𝜇)
(𝜇)
𝑄𝑓𝑘 = 𝜇𝑘 𝑓𝑘 ,
∀𝑘 ∈ N.
(111)
Then, the sequence (𝑓𝑘 )𝑘∈N given by
(𝜇)
𝑓𝑘 = √𝜇𝑘 𝑓𝑘 ,
𝑘 ∈ N,
(112)
is an orthonormal basis of 𝑈0 . Analogous to (105) and (106),
we define the isometric isomorphisms
Φ𝜇 : 𝑈 󳨀→
ℓ𝜇2 ,
(𝜇)
Φ𝜇 𝑓𝑘
:=
(𝜇)
𝑔𝑘
Ψ𝜇 (𝑆) := (𝑆𝑓𝑘 )𝑘∈N
(𝜇)
∞
(𝜇)
(𝜇)
= ∑ ⟨Φ−1
𝜆 V, 𝑓𝑘 ⟩𝑈 𝑔𝑘
𝑘=1
=(
1
(𝜇)
⟨Φ−1
𝜆 V, 𝑓𝑘 ⟩𝑈 )
√𝜇𝑘
𝑘∈N
=(
1 ∞ −1 (𝜆)
(𝜇)
∑ ⟨Φ𝜆 V, 𝑒𝑗 ⟩ ⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩ )
𝑈
𝑈
√𝜇𝑘 𝑗=1
=(
1 ∞
(𝜇)
∑ ⟨V, 𝑔𝑗(𝜆) ⟩ 2 ⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩ )
ℓ
𝑈
𝜇
𝜆
√ 𝑘 𝑗=1
∞
√𝜆 𝑗
𝑗=1 √𝜇𝑘
(113)
(𝜇)
𝑘=1
= (∑
for 𝑘 ∈ N,
Ψ𝜇 : 𝐿02 (𝐻) 󳨀→ ℓ2 (𝐻) ,
∞
= Φ𝜇 ( ∑ ⟨Φ−1
𝜆 V, 𝑓𝑘 ⟩𝑈 𝑓𝑘 )
∞
𝑗=1
𝑘∈N
𝑘∈N
(𝜇)
⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩ V𝑗 )
𝑈
𝑘∈N
(𝜇)
= ( ∑⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩ V𝑗 )
for 𝑆 ∈ 𝐿02 (𝐻) .
(118)
𝑈
.
𝑘∈N
Furthermore, we define the isometric isomorphisms
2
2
Φ := Φ𝜇 ∘ Φ−1
𝜆 : ℓ𝜆 󳨀→ ℓ𝜇 ,
Ψ := Ψ𝜇 ∘ Ψ𝜆−1 : ℓ2 (𝐻) 󳨀→ ℓ2 (𝐻) .
Let 𝑤 ∈ ℓ2 (𝐻) be arbitrary. By (106), we have
(114)
𝑤 = Ψ𝜆 (Ψ𝜆−1 (𝑤)) = (Ψ𝜆−1 (𝑤)𝑒𝑗 )𝑗∈N ,
(119)
12
International Journal of Stochastic Analysis
and, hence,
Ψ (𝑤) = (Ψ𝜇 ∘
which, together with (109), yields (124). Now, Theorem 26
applies by virtue of Lemma 29 and yields
Ψ𝜆−1 ) (𝑤)
Ψ𝜆 (𝑋) ⋅ Φ𝜆 (𝐿) = Ψ (Ψ𝜆 (𝑋)) ⋅ Φ (Φ𝜆 (𝐿))
= (Ψ𝜆−1 (𝑤)𝑓𝑘 )𝑘∈N
∞
= (Ψ𝜆−1 (𝑤) ( ∑⟨𝑓𝑘 , 𝑒𝑗 ⟩𝑈 𝑒𝑗 ))
0
𝑗=1
∞
0
∞
proving (116).
𝑘∈N
= ( ∑⟨𝑓𝑘 , 𝑒𝑗 ⟩𝑈 Ψ𝜆−1 (𝑤)𝑒𝑗 )
𝑗=1
(120)
𝑘∈N
= ( ∑⟨𝑓𝑘 , 𝑒𝑗 ⟩𝑈 𝑤 )
0
∞
𝑘∈N
(𝜇)
= ( ∑⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩ 𝑤𝑗 )
𝑈
𝑗=1
Definition 31. For every 𝐿02 (𝐻)-valued process 𝑋 satisfying
𝑡
(109), we define the Itô-Integral 𝑋 ⋅ 𝐿 = (∫0 𝑋𝑠 𝑑𝐿 𝑠 )𝑡∈[0,𝑇] by
(110).
By virtue of Proposition 30, Definition (110) of the Itô
integral neither depends on the choice of the eigenvalues
(𝜆 𝑗 )𝑗∈N nor on the eigenvectors (𝑒𝑗(𝜆) )𝑗∈N .
Now, we will the prove the announced series representation of the Itô integral. According to Proposition 23,
the sequences (𝑀𝑗 )𝑗∈N and (𝑁𝑗 )𝑗∈N of real-valued processes
given by
𝑗
𝑗=1
.
𝑘∈N
2
Therefore, for all ℎ ∈ 𝐻 and 𝑤 ∈ ℓ (𝐻), we obtain
⟨ℎ, Ψ(𝑤)⟩𝐻 =
𝑀𝑗 :=
∞
(𝜇)
( ∑⟨𝑒𝑗(𝜆) , 𝑓𝑘 ⟩ ⟨ℎ, 𝑤𝑗 ⟩𝐻)
𝑈
𝑗=1
𝑘∈N
(1) Φ𝜆 (𝐿) is an ℓ𝜆2 -valued Lévy process with covariance
operator 𝑄Φ𝜆 , and one has
𝜆 𝑗 𝑔𝑗(𝜆) ,
∀𝑗 ∈ N.
Lévy process with covariance
(2) Φ𝜇 (𝐿) is an
operator 𝑄Φ𝜇 , and one has
(𝜇)
𝑄Φ𝜇 𝑔𝑘 = 𝜇𝑘 𝑔𝑘
(3) For every
(109), one has
∀k ∈ N.
(123)
predictable process 𝑋 with
𝑇
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩Ψ𝜆 (𝑋𝑠 )󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] < ∞,
0
𝑇
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩󵄩Ψ𝜇 (𝑋𝑠 )󵄩󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] < ∞
0
(124)
√𝜆 𝑗
⟨Φ𝜆 (𝐿), 𝑔𝑗(𝜆) ⟩
ℓ𝜆2
𝜉𝑗 := 𝑋𝑒𝑗 ,
𝑗 ∈ N,
(128)
is a ℓ2 (𝐻)-valued, predictable process, and, one has
𝑋 ⋅ 𝐿 = ∑ 𝜉𝑗 ⋅ 𝑀𝑗 ,
(129)
𝑗∈N
where the right-hand side of (129) converges unconditionally in
𝑀𝑇2 (𝐻).
Proof. Since Φ𝜆 is an isometry, for each 𝑗 ∈ N, we obtain
𝑀𝑗 =
1
√𝜆 𝑗
⟨𝐿, 𝑒𝑗(𝜆) ⟩ =
𝑈
=
1
√𝜆 𝑗
1
√𝜆 𝑗
⟨Φ𝜆 (𝐿), Φ𝜆 𝑒𝑗(𝜆) ⟩
ℓ𝜆2
(130)
⟨Φ𝜆 (𝐿), 𝑔𝑗(𝜆) ⟩ 2
ℓ
𝜆
𝑗
=𝑁.
Thus, by Definitions 31 and 24, we obtain
and the identity (116).
𝑋 ⋅ 𝐿 = Ψ𝜆 (𝑋) ⋅ Φ𝜆 (𝐿)
Proof. The first two statements follow from Lemma 28. Since
Ψ𝜆 and Ψ𝜇 are isometries, we obtain
𝑇
1
(127)
(122)
ℓ𝜇2 -valued
𝐿02 (𝐻)-valued,
𝑁𝑗 :=
𝑈
Proposition 32. For every 𝐿02 (𝐻)-valued, predictable process
𝑋 satisfying (109), the process (𝜉𝑗 )𝑗∈N given by
Proposition 30. The following statements are true.
(𝜇)
√𝜆 𝑗
⟨𝐿, 𝑒𝑗(𝜆) ⟩ ,
are sequences of standard Lévy processes.
finishing the proof.
=
1
(121)
= Φ (⟨ℎ, 𝑤⟩𝐻) ,
𝑄Φ𝜆 𝑔𝑗(𝜆)
(126)
= Ψ𝜇 (𝑋) ⋅ Φ𝜇 (𝐿) ,
𝑇
󵄩
󵄩 󵄩2
󵄩2
E [∫ 󵄩󵄩󵄩Ψ𝜆 (𝑋𝑠 )󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] = E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
0
0
𝑇
󵄩
󵄩2
= E [∫ 󵄩󵄩󵄩󵄩Ψ𝜇 (𝑋𝑠 )󵄩󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠] ,
0
= ∑ Ψ𝜆 (𝑋)𝑗 ⋅ 𝑁𝑗
𝑗∈N
(131)
= ∑ 𝜉𝑗 ⋅ 𝑀𝑗 ,
(125)
𝑗∈N
and, by Proposition 16, the series converges unconditionally
in 𝑀𝑇2 (𝐻).
International Journal of Stochastic Analysis
13
Remark 33. By Remark 17 and the proof of Proposition 32,
the components of the Itô integral ∑𝑗∈N 𝜉𝑗 ⋅ 𝑀𝑗 are pairwise
orthogonal elements of the Hilbert space 𝑀𝑇2 (𝐻).
𝐿02 (𝐻)-valued
Proposition 34. For every
(109), one has the Itô isometry
process 𝑋 satisfying
󵄩󵄩2
󵄩󵄩 𝑇
𝑇
󵄩
󵄩
󵄩 󵄩2
E [󵄩󵄩󵄩󵄩∫ 𝑋𝑠 𝑑𝐿 𝑠 󵄩󵄩󵄩󵄩 ] = E [∫ 󵄩󵄩󵄩𝐿 𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] .
2
󵄩󵄩𝐻
󵄩󵄩 0
0
𝑇
= E [∫
0
𝑗
𝑛
= ∑ ∑ 𝑋𝑖 𝑒𝑗
𝑡𝑖+1
𝑡𝑖
− (𝑁𝑗 ) )
⟨Φ𝜆 (𝐿𝑡𝑖+1 − 𝐿𝑡𝑖 ), 𝑔𝑗(𝜆) ⟩
𝑖=1 𝑗∈N
ℓ𝜆2
√𝜆 𝑗
𝑛
(𝜆)
= ∑ ∑ 𝑋𝑖 𝑒𝑗(𝜆) ⟨𝐿𝑡𝑖+1 − 𝐿𝑡𝑖 , Φ−1
𝜆 𝑔𝑗 ⟩
𝑈
𝑖=1 𝑗∈N
(138)
𝑛
(133)
= ∑ ∑ 𝑋𝑖 𝑒𝑗(𝜆) ⟨𝐿𝑡𝑖+1 − 𝐿𝑡𝑖 , 𝑒𝑗(𝜆) ⟩
𝑈
𝑖=1 𝑗∈N
𝑛
= ∑ 𝑋𝑖 (∑ ⟨𝐿𝑡𝑖+1 − 𝐿𝑡𝑖 , 𝑒𝑗(𝜆) ⟩ 𝑒𝑗(𝜆) )
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑋𝑠 󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] ,
2
𝑖=1
𝑗∈N
𝑈
𝑛
= ∑𝑋𝑖 (𝐿𝑡𝑖+1 − 𝐿𝑡𝑖 ) ,
completing the proof.
We shall now prove that the stochastic integral, which we
have defined so far, coincides with the Itô integral developed
in [5]. For this purpose, it suffices to consider elementary
processes. Note that for each operator 𝑆 ∈ 𝐿(𝑈, 𝐻), the
restriction 𝑆|𝑈0 belongs to 𝐿02 (𝐻), because
∞
∞
󵄩 󵄩2
󵄩 󵄩2
∑ 󵄩󵄩󵄩󵄩𝑆𝑒𝑗 󵄩󵄩󵄩󵄩𝐻 ≤ ∑‖𝑆‖2𝐿(𝑈,𝐻) 󵄩󵄩󵄩󵄩𝑒𝑗 󵄩󵄩󵄩󵄩𝑈
𝑗=1
𝑛
= ∑ ∑ Ψ𝜆 (𝑋𝑖 |𝑈0 ) ((𝑁𝑗 )
(132)
󵄩󵄩 𝑇
󵄩󵄩2
󵄩󵄩2
󵄩󵄩 𝑇
󵄩
󵄩
󵄩
󵄩
E [󵄩󵄩󵄩󵄩∫ 𝑋𝑠 𝑑𝐿 𝑠 󵄩󵄩󵄩󵄩 ] = E [󵄩󵄩󵄩󵄩∫ Ψ𝜆 (𝑋𝑠 )𝑑Φ𝜆 (𝐿)𝑠 󵄩󵄩󵄩󵄩 ]
󵄩󵄩 0
󵄩󵄩𝐻
󵄩󵄩𝐻
󵄩󵄩 0
𝑇
𝑋|𝑈0 ⋅ 𝐿 = Ψ𝜆 (𝑋|𝑈0 ) ⋅ Φ𝜆 (𝐿)
𝑖=1 𝑗∈N
Proof. By the Itô isometry (Proposition 18), and since Ψ𝜆 is
an isometry, we obtain
󵄩
󵄩2
= E [∫ 󵄩󵄩󵄩Ψ𝜆 (𝑋𝑠 )󵄩󵄩󵄩ℓ2 (𝐻) 𝑑𝑠]
0
Thus, by Proposition 19, and since Φ𝜆 is an isometry, we
obtain
𝑗=1
∞󵄩
󵄩󵄩2
󵄩
= ‖𝑆‖2𝐿(𝑈,𝐻) ∑󵄩󵄩󵄩√𝜆 𝑗 𝑒𝑗(𝜆) 󵄩󵄩󵄩
󵄩
󵄩𝑈0
𝑗=1
(134)
∞
Proposition 35. Let 𝑋 be a 𝐿(𝑈, 𝐻)-valued simple process of
the form
𝑛
(135)
𝑖=1
with 0 = 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛+1 = 𝑇 and F𝑡𝑖 -measurable random
variables 𝑋𝑖 : Ω → 𝐿(𝑈, 𝐻) for 𝑖 = 0, . . . , 𝑛. Then, one has
𝑡𝑖+1
𝑋|𝑈0 ⋅ 𝐿 = ∑𝑋𝑖 (𝐿
𝑡𝑖
− 𝐿 ).
(136)
𝑖=1
Proof. The process Ψ𝜆 (𝑋|𝑈0 ) is an ℓ2 (𝐻)-valued simple process having the representation
𝑛
Ψ𝜆 (𝑋|𝑈0 ) = Ψ𝜆 (𝑋0 |𝑈0 ) 1{0} + ∑Ψ𝜆 (𝑋𝑖 |𝑈0 ) 1(𝑡𝑖 ,𝑡𝑖+1 ] . (137)
𝑖=1
Therefore, and since the space of simple processes is
dense in the space of all predictable processes satisfying (109);
see, for example, [5, Corollary 8.17], the Itô integral (110)
coincides with that in [5] for every 𝐿02 (𝐻)-valued, predictable
process 𝑋 satisfying (109). In particular, for a driving Wiener
process, it coincides with the Itô integral from [1–3].
By a standard localization argument, we can extend the
definition of the Itô integral to all predictable processes 𝑋
satisfying
󵄩 󵄩2
P (∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) ds < ∞) = 1,
2
0
𝑗=1
𝑛
completing the proof.
𝑇
= ‖𝑆‖2𝐿(𝑈,𝐻) ∑𝜆 𝑗 < ∞.
𝑋 = 𝑋0 1{0} + ∑𝑋𝑖 1(𝑡𝑖 ,𝑡𝑖+1 ]
𝑖=1
∀𝑇 > 0.
(139)
Since the respective spaces of predictable and adapted,
measurable processes are isomorphic (see [24]), proceeding
as in [24, Section 3.2], we can further extend the definition
of the Itô integral to all adapted, measurable processes 𝑋
satisfying (139).
Acknowledgment
The author is grateful to an anonymous referee for valuable
comments and suggestions.
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