Journal of Applied Mathematics and Physics
Vol.06 No.07(2018), Article ID:86059,11 pages
10.4236/jamp.2018.67121
A Generalization of the Clark-Ocone Formula
Mahmmoud Salih*, Sulieman Jomah
School of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: June 5, 2018; Accepted: July 16, 2018; Published: July 19, 2018
ABSTRACT
In this paper, we use a white noise approach to Malliavin calculus to prove the generalization of the Clark-Ocone formula
where denotes the generalized expectation, is the (generalized) Malliavin derivative, is the Wick product and is the 1-dimensional Gaussian white noise.
Keywords:
White Noise, Malliavin Calculus, Wick Product, Brownian Motion
1. Introduction
In 1975, Hida introduced the theory of white noise with his lecture note on Brownian functionals [1] . After that H. Holden et al. [2] emphasized this theory with stochastic partial differential equations (SPDEs) driven by Brownian motion.
In 1984, Ocone proved the Clark-Ocone formula [3] , to give an explicit representation to integral in Itô integral representation theorem in the context of analysis on the Wiener space , the space of all real continuous functions on starting at 0. He proved that
(1.1)
where is the Malliavin derivative and is the one dimensional Brownian motion on the Winer space. In [4] the authors proved the generalization of Clark-Ocone formula (see, e.g., [5] [6] ). This theorem has many interesting application, for example, computing the replicating portfolio of call option in Black & Scholes type market. They proved that
(1.2)
where denotes the generalized expectation, is the
(generalized) Malliavin derivative, is the Wick product and is the one dimensional Gaussian white noise. This formula holds for all , where is a space of stochastic distribution. In particular, if then equation (1.2) turns out to be
The purpose of this papper is to generalize the well known Clark-Ocone formula to generalized functions of white noise, i.e., to the space . The generalization has the following form
where denotes the generalized expectation, is the
(generalized) Malliavin derivative, is the Wick product, and is the 1-dimensional Gaussian white noise.
The paper is organized as follows. In Section 2 and 3, we recall necessary definitions and results from white noise and prove a new results that we will need. Finally in Section 4, we generalize the Clark-Ocone formula, i.e., to the space .
2. White Noise
In this section we recall necessary definitions and results from white noise. For more information about white noise analysis (see e.g, [7] - [14] ).
Given be the space of tempered distribution on the set of real number and let be the Gaussian white noise probability measure on such that
(2.1)
where denotes the action of on . It follows from (2.1) that
where denotes the expectation with respect to . This isometry allows us to define a Brownian motion as the continuous version of where
Then, for all . Let be the algebra generated by . If , i.e., is symmetric and
then the iterated Itô integral is given by
(2.2)
In the following we let
(2.3)
be the Hermite polynomials and let be the basis of consiting
(2.4)
The set of multi-indices of nonnegative integers is denoted by . Where is the set of all natural number and . If is a sequence of number or function, we use the multi-induces notation
Theorem 2.1. ( [15] ) Let be are an orthonormal function in . Then for all multi-indices , we have
Corollary 2.2.
where denote the Wick product, and extend linearly. Then if , we have
Proof.
3. Stochastic Test Function and Stochastic Distribution (Konddratiev Spaces)
1) Stochastic test function spaces
Suppose , for , let consist of those
such that
where
(3.1)
2) Stochastic distribution
For , let be the space of Kondratiev space of stochastic distribution, consist of all formal expansions
such that
where is defined in (3.1).
Note that is the dual of and we can define the action of on by
where is the usual inner product in .
Definition 3.1. Let be the random variable and let . Then we say that F has directional derivative in the direction if
(3.2)
if the limit exist in .
Definition 3.2. A function -integrable if
Then the -integrable of , denoted by , is the unique element such that
Definition 3.3. Consider such that
and
then we say that F is (Hida) Malliavin differentiable and we put
is called the Hida-Malliavin derivative or stochastic gradient of F at t.
The set of all differentiable is denoted by .
Definition 3.4. Consider . Then we define the stochastic derivative of F at t by
Lemma 3.5.
1) Let . Then for a.a. .
2) Suppose for all and
Then there exist a subsequence such that
Proof. 1) Suppose . Then
where
We want to prove that for some ,
Note that
Moreover,
where for all. Hence,
Using the fact that for all m, we get
(3.3)
Therefore,
2) To prove this part, it suffices to prove that if in, then there exist a subsequence such that in as, for a.a. t. We have prove that
Therefore,
So, there exists a subsequence such that for a.a. t as. This complete the proof.
Suppose is the Hermite functions, and put
(3.4)
and
(3.5)
and
With this notation we have, for all multi indices where.
Definition 3.6. 1) Let. We say that
belong to the space if
we define
and equip with the projective topology.
2) We say that
belong to the space if
we define
and equip with the inductive topology. Then is the dual of, with action
4. The Generalized Clark-Ocone Formula
Now we are prepared to present the main result of this paper. It generalizes the well know Clark-Ocone formula to generalized functions, i.e., to the space.
Definition 3.1. Suppose. Then the conditional expectation of F with respect to is given by
(4.1)
Note that this coincides with usual conditional expectation if, and
(4.2)
In particular
(4.3)
Lemma 4.2. Suppose. Then
Proof. Assume that, without loss of generality,
and similarly G. By Corollary 2.2 and Definition 4.1, we have
Lemma 4.3.
Let. Then for a.a..
Consider for all and
Then there exists a subsequence such that
Proof. 1) Suppose. Then
where
Choose such that. We will prove that
Note that
So
Hence, using the fact that for all n, we get
(4.4)
Therefore,
and
2) It suffices to prove that if in, then there exists a
subsequence such that in as, for a.a. t. By (4.4) we can see that in. So there exists a subsequence
such that
(4.5)
Therefore,
The last assertion follows from (4.2).
Theorem 4.4. Suppose denote Lebesque measure on. Let be -measurable. Then
and
Proof. Let be the chaos expansion of F and put
where. Then by Lemma 3.8 (see [4] ), we have
By Itô representation theorem there is a unique which is adapted and such that
and such that
since in, we conclude that
Therefore,
on the other hand, by Lemma 4.1, we have
By taking another subsequence, we obtain that
We conclude that
This completes the proof.
Lemma 4.5. Suppose and. Then
where
Proof. Let. Then
Lemma 4.6. Suppose. Then
Proof. By Lemma 4.3 and (4.4), we have
Lemma 4.7. Let and in. Then
(4.6)
Proof. In case of a complete proof is given in [4] . The proof for general is a simple modification. Note that both integral in (4.6) exist by Lemma 4.7. Hence, by Lemma 4.6 and (4.4), we have
This completes the proof.
Theorem 4.8. Let be -measurable. Then is integrable in and
where, denotes the generalized exsection of F.
Proof. Let. Then, by Lemma 3.8 (see [4] ), we have
therefore,
the limit exist in and hence in. The result follows from Lemma 4.7.
Cite this paper
Salih, M. and Jomah, S. (2018) A Generalization of the Clark-Ocone Formula. Journal of Applied Mathematics and Physics, 6, 1443-1453. https://doi.org/10.4236/jamp.2018.67121
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