A Generalization of the Clark-Ocone Formula

In this paper, we use a white noise approach to Malliavin calculus to prove the generalization of the Clark-Ocone formula ( ) [ ] [ ] ( ) 0 | d , T t t F E F E D F W t t ω = + ◊ ∫  where [ ] E F denotes the generalized expectation, ( ) d d t F D F ω ω = is the (generalized) Malliavin derivative, ◊ is the Wick product and ( ) W t is the 1-dimensional Gaussian white noise.


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 ( ) B t 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 ( ) W t is the one dimensional Gaussian white noise.This formula holds for all * F ∈ , where *  is a space of stochastic distribution.In particular, if ( )


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 ( ) 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 β −  .

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]).

( )
S Ω =  be the space of tempered distribution on the set  of real number and let µ be the Gaussian white noise probability measure on Ω such that ( ) where , ω φ denotes the action of ( ) ., 0, ., , where E E µ = denotes the expectation with respect to µ .This isometry allows us to define a Brownian motion ( ) ( ) then the iterated Itô integral is given by In the following we let ( ) ( ) be the Hermite polynomials and let { } 1 be the basis of ( ) 1 !e 2 , 1, 2, The set of multi-indices , , , n is the set of all natural number and , , z z z =  is a sequence of number or function, we use the multi-induces notation ( ) where ◊ denote the Wick product, and extend linearly.Then if

2) Stochastic distribution
For 0 1 β ≤ < , let ( ) * S β be the space of Kondratiev space of stochastic distribution, consist of all formal expansions

2
, for some , where ( ) Note that ( ) * S β is the dual of ( ) S β and we can define the action of ( ) is the usual inner product in  .( ) if the limit exist in ( ) Then there exist a subsequence { } We want to prove that for some q ∈  , ( ) ( ) ( ) Using the fact that ( ) Therefore, ( ) , 1 for a.a.
With this notation we have, for all multi indices α where ( ) . We say that ( ) and equip β  with the projective topology.

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 ( ) Note that this coincides with usual conditional expectation if In particular Proof.Assume that, without loss of generality, and similarly G.By Corollary 2.2 and Definition 4.1, we have Then there exists a subsequence { } for . .
be the chaos expansion of F and put ( )      Proof.In case of 0 β = a complete proof is given in [4].The proof for general 0 1 β ≤ < is a simple modification.Note that both integral in (4.
Malliavin derivative, ◊ is the Wick product and representation to integral in Itô integral representation theorem in the context of analysis on the Wiener space

2 L
γ ∈  .Then we say that F has directional derivative in the direction γ if

2 )
To prove this part, it suffices to prove that if 0

2 )
It suffices to prove that if 0 as k → ∞ , for a.a.t.By Journal of Applied Mathematics and Physics The last assertion follows from (4.2).Theorem 4.4.Suppose λ denote Lebesque measure on  .Let 6)exist by Lemma 4.7.Hence, by Lemma 4.6 and (4.4), we have F denotes the generalized exsection of F.
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