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Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables
The notion of -fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.
The Wiener-Itô chaos expansion of a square integrable random variable which was first proved in  plays fundamental role in Malliavin calculus of variations [2, 3] which appeared to be a powerful instrument in the analysis of functionals of Brownian motion. The Malliavin calculus has found extensive applications to stochastic differential equations arising as models of various random phenomena. One of the important sources of such equations is markets modeling in financial mathematics [4, 5].
In the last decades, many researchers’ interest has been drawn to stochastic differential equations in infinite dimensional Hilbert spaces driven by a cylindrical Wiener process or, equivalently, by countable set of Brownian motions [6, 7]. For example, in financial mathematics, such equations are used in modeling interest rates term structure or zero-coupon bond market [8, 9]. The present work was motivated by the need to make the Malliavin calculus applicable to Hilbert space valued stochastic processes. The first step in this direction is to obtain the generalization of the Wiener-Itô chaos expansion for Hilbert space valued random variables.
In Section 3, we introduce iterated Itô integrals and multiple Itô integrals with respect to a cylindrical Wiener process. In the Hilbert space valued case, the integrand of an -fold iterated Itô integral is a function defined on a certain subset of and taking values in a certain space of Hilbert space valued continuous -linear forms defined on the th Cartesian power of the Hilbert space where the Wiener process takes values.
Section 4 contains main results of the paper which are stated in Theorem 9 and Corollary 11. The proof of the theorem follows the scheme of the proof of the Wiener-Itô chaos expansion in the -valued case in .
2. Itô Representation Theorem for Hilbert Space Valued Random Variables
Let be a probability space. For any separable Hilbert space , we denote by , or for short, the space of all Bochner square integrable -valued random variables on . Let be an orthonormal basis in and let be a sequence of independent identically distributed Brownian motions on the probability space. Consider the corresponding cylinder Wiener process, defined by It is easy to see that the series is not convergent in ; however, for any , we have that is a random variable belonging to . Denote by the -algebra, generated by the Wiener process at , that is, the -algebra, generated by the set of random variables , where , . (Note that the series (1) is not convergent in at although for any , .) The family is called the filtration generated by the Wiener process . Note that the Brownian motions are martingales with respect to the filtration .
Let be another separable Hilbert space and let be an orthonormal basis in . Then, the family of operators , defined by the equality forms an orthonormal basis in the space of all Hilbert-Schmidt operators acting from to . Any has the decomposition where and For any -valued random process adapted to the filtration , where , satisfying the property the stochastic Itô integral with respect to the cylindrical Wiener process is well-defined and is an element of the space (see the definition and the properties in ). Note that if the function is -measurable, the equality (5) implies
Theorem 1. For any -measurable random variable , there exists a unique -adapted -valued random process satisfying (5) such that
Proof. For any , we have . Therefore, we have the following decomposition for :
Denote , where is the filtration generated by the -dimensional Wiener process
By the finite dimensional Itô representation theorem (, Theorem .), there exists a unique -valued random process
such that the following conditions hold:(i)all the mappings are -measurable,(ii),(iii)the processes are adapted to the filtration .
Moreover, we have Here, denotes the -dimensional Brownian motion Since for , we have It follows from the uniqueness of the representation (12) that for . Thus, the equality (12) can be rewritten as By the Jensen inequality, we have Therefore, Thus, Consequently, Since and , we have For any , it holds that It follows from (20) that since the series in the right-hand side of the equality are convergent in by the estimate (21). Thus, the equality (8) holds true. It also follows from this estimate by the Levy theorem that we can pass to the limits at in the equality. As a result, we obtain where By the polarization identity, we obtain the following assertion.
Corollary 2. Let and be -valued random processes, adapted to the filtration , such that Then,
3. Multiple Itô Integrals with respect to the Cylindrical Wiener Process
Let be a function defined for , taking values in the space and -measurable. If for any the random process is adapted to the filtration and the condition holds true, then the Itô integral is well defined and is an element of the space for all . As a function of , it is an -valued random process adapted to the filtration . We also have If moreover we have then the following iterated Itô integral is well defined: and it satisfies the equality Note that the operators defined by the equality form an orthonormal basis in the space and any operator has the following decomposition: We can identify the decomposition (34) with the -valued bilinear form on defined by the equality . Thus, the space can be identified with the Hilbert space of all continuous bilinear forms of the form (34) with the norm generated by the scalar product where , . So, the iterated integral (31) with respect to the cylindrical Wiener process is well defined for any -valued -measurable function , defined on , where if it satisfies the condition (30).
One can easily extend the above definition to the case of arbitrary , defining the times iterated integral inductively for any function , where is the space of all continuous -valued -linear forms on having form such that if it satisfies the following conditions: (n-i) is -measurable;(n-ii) is -measurable for any ;(n-iii).
For the defined iterated integral, we have We state without proof the next two lemmas which are straightforward generalizations of the corresponding properties of the -valued iterated Itô integrals.
Lemma 3. Let and satisfy the conditions (n-i), (n-ii) and (n-iii) with and correspondingly. Then,
Lemma 4. Let satisfy the conditions (n-i), (n-ii) and (n-iii). Then, Denote by the space of all symmetric functions satisfying the condition
Definition 5. For any , define the multiple -fold Itô integral by the equality if the right-hand side iterated integral exists.
4. The Decomposition Theorem
To establish the main result, we need a few lemmas.
Lemma 6. The set of random variables is dense in .
Proof. Let be a dense subset in . Let be a fixed ordering of the countable set of random variables . Denote by be the -algebra generated by . We have for all and is the smallest -algebra containing all .
For any , we have where the limit is pointwise a.e. with respect to and in . By the Doob-Dynkin lemma for any , there exists a Borel function such that Let be the probability measure on generated by . Since can be approximated in by functions , the assertion follows.
Consider the following exponential functionals of Brownian motions :
Lemma 7. The linear span of the set is dense in .
Proof. Let be orthogonal to all functions of the family (50). Take , where , . Then, we have
It follows that
for all , , , .
Fix . The corresponding function is real analytic. Consider its analytic extension onto . It is also equal to zero. Taking , we obtain For any denoting by its Fourier transform, we obtain Thus, is orthogonal to a dense subset in .
Lemma 8. For any , , the product is a linear combination of the iterated Itô integrals of the form where and .
Proof. We use induction with respect to . The assertion is evident for . Suppose it is true for some . Let . We introduce the following notation: This means that satisfy the following stochastic differential equations: and we have By the Itô formula, we obtain Since , we come to the equality Since , we can apply the assumption to the products in the brackets, thus completing the proof.
Theorem 9. Any -measurable random variable has the unique decomposition where , . It holds that where .
Proof. By Theorem 1, we have
where is a -valued random variable, -measurable for any . For its norm, we have
Let . For any by Theorem 1, taking for and for , we obtain the following representation:
where , with the following equality for the norm:
Set . Substituting the representation (66) into (64), we come to the equality
From (65) and (67), it follows that
Applying further in similar manner Theorem 1 and setting
at the th step of this process, we obtain
It follows from here that
This means that the series
is convergent in the space and
is an element of this space. By Lemma 3, the integral is orthogonal in to the integrals for all , . It follows that is orthogonal to for all , .
Note that by the well-known connection between iterated Itô integrals with respect to a Brownian motion and the Hermite polynomials, we have for any , .
Let , where . For any and any , we have from where, by Lemma 8 and the equality (77), it follows that, for any , is orthogonal in to the products of the form It follows from here and the connection between the Hermite polynomials and general powers that It follows from the expansion that a product of the form has a decomposition into the series of the products of the form convergent in the space . Consequently, is orthogonal to any product of the form (82). By Lemma 7, this means that for any ; that is, . Passing to the limit in the equality (71), we obtain
Setting the functions to be equal to zero on and defining , where is the symmetrization of , we obtain (62). By the equality passing to the limit in (72), we obtain (63).
Definition 10. For any , define the -fold Itô integral by the equality (45), where the iterated integral is understood as the limit of the integrals at in the space if the limit exists.
Corollary 11. Any can be uniquely decomposed into the series where , . The equality holds true, where .
Proof. Let . For any , it can be uniquely represented in the form
where and is the symmetrization of the function set to be equal to zero at the set . For its norm, we have
Suppose that the functions and are defined everywhere in and are equal to zero on and correspondingly.
It follows from the equality where , from the uniqueness of the representation (89), and from the assertion of Lemma 4 that the equality holds true and therefore for and any . It follows from here that for any there exists a unique function such that for any the equality holds true and there exists a unique function such that for any it holds . By the estimates it follows that in the space . Consequently, exist in the space . Passing to the limits in the equalities (89) and (90) we obtain (87) and (88).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by the Ministry of Education and Science of Russian Federation (Program 1.1016.2011), by RFBR, Project 13-01-00090, and by the Program of State Support of RF Leading Universities (Agreement no. 02.A03.21.0006 of 27.08.2013).
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Copyright © 2014 M. A. Alshanskiy. 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.