A GENERALIZATION OF THE ITÔ FORMULA

The classical Ito formula is generalized to some anticipating 
processes. The processes we consider are in a Sobolev space which 
is a subset of the space of square integrable functions over a 
white noise space. The proof of the result uses white noise 
techniques.


Introduction.
The well-known Itô formula relies on the fact that the integrands in the Itô processes are nonanticipating.However, in many cases, this essential condition is not satisfied.Take the example of the stochastic integral 1 0 B( 1)dB(t), t < 1, where B(t) is a Brownian motion.Clearly, B( 1) is anticipating and 1 0 B(1)dB(t) cannot be defined as an Itô integral; necessitating an extension.
We are interested in generalizing the Itô formula to anticipating processes in an infinite-dimensional space.For this purpose, we consider the infinite-dimensional space to be the white noise space ( (R), µ), where (R) is the space of tempered distributions and µ is the standard Gaussian measure on (R).We use the Hitsuda-Skorokhod integral as the underlying extension of the Itô integral and we prove the formula for processes of the form θ(X(t), F ) where X(t) is a Wiener integral, θ ∈ Ꮿ 2 b (R 2 ) and F ∈ ᐃ 1/2 , a Sobolev space in the Hilbert space (L 2 ) ≡ L 2 ( (R), µ).
A number of variations to the formula in this paper exist.Prevault [9] developed a formula which was applied to processes of the form Y (t) = θ(X(t), F ), where F is a smooth random variable depending on the whole trajectory of (B(t)) t∈R + , and (X(t)) t∈R + is an adapted Itô process.His proof relies on the expression of infinitesimal time changes on Brownian functionals using the Gross Laplacian.In [6], Kuo developed a formula which was applied to processes of the form Y (t) = θ(X(t), B(c)), 0 ≤ a ≤ c ≤ b, with X(t) a Wiener integral.The main tool of his proof is the very important white noise function, the S-transform, and the fact that both X(t) and B(t) have Gaussian laws.The proof in this paper is a limiting process of Kuo's proof.

Background.
In this section, the basic background from white noise analysis are introduced and the interested reader is provided with the relevant references.

Concept and notations.
Let E be a real separable Hilbert space with norm |•| 0 .Let A be a densely defined selfadjoint operator on E, whose eigenvalues {λ n } n≥1 satisfy the following conditions: For any p ≥ 0, let Ᏹ p be the completion of E with respect to the norm |f | p := |A p f | 0 .Note that Ᏹ p is a Hilbert space with the norm | • | p and Ᏹ p ⊂ Ᏹ q for any p ≥ q.The second condition on the eigenvalues above implies that the inclusion map i : Ᏹ p+1 → Ᏹ p is a Hilbert-Schmidt operator.Let Ᏹ be the projective limit of {Ᏹ p ; p ≥ 0}, and let Ᏹ be the dual space of Ᏹ.Then the space Ᏹ = p≥0 Ᏹ p equipped with the topology given by the family {| • | p } p≥0 of seminorms is a nuclear space.Hence Ᏹ ⊂ E ⊂ Ᏹ is a Gel'fand triple with the following continuous inclusions: We used the Riesz representation theorem to identify the dual of E with itself.
It can be shown that for all p ≥ 0, the dual space Ᏹ p is isomorphic to Ᏹ −p , which is the completion of the space E with respect to the norm Minlo's theorem allows us to define a unique probability measure µ on the Borel subsets of Ᏹ with the property that for all f ∈ Ᏹ, the random variable •,f is normally distributed with mean 0 and variance |f | 2 0 .We are using •, • to denote the duality between Ᏹ and Ᏹ.This means that the characteristic functional of µ is given by The probability space (Ᏹ ,µ) is called the white noise space.The space L 2 (Ᏹ ,µ) will be denoted by (L 2 ); that is, (L 2 ) is the set of functions ϕ : Ᏹ → C such that ϕ is measurable and Ᏹ |ϕ(x)| 2 dµ(x) < ∞.If we denote by E c the complexification of E, the Wiener-Itô theorem allows us to associate to each ϕ ∈ (L 2 ) a unique sequence {f n } n≥0 , f n ∈ E ⊗n c and express ϕ as ϕ = ∞ n=0 I n (f n ) where I n (f n ) is a multiple Wiener integral of order n (see [3]).This decomposition is similar to what is referred to as the Fock-space decomposition as shown in [8].
The (L 2 )-norm ϕ 0 of ϕ is given by where If 0 < p ≤ q, then (Ᏹ q ) ⊂ (Ᏹ p ) with the property that for any q ≥ 0, there exists p > q such that the inclusion map I p,q : (Ᏹ p ) (Ᏹ q ) is a Hilbert-Schmidt operator and I p,q 2 HS ≤ (1 − i p,q 2 HS ) −1 where i p,q is the inclusion map from Ᏹ p into Ᏹ q as noted earlier.
Analogous to the way Ᏹ was defined, we also define (Ᏹ) as the projective limit of {(Ᏹ p ); p ≥ 0} and (Ᏹ) * as the dual space of (Ᏹ).With the above result, (Ᏹ) = p≥0 (Ᏹ p ) with the topology generated by the family { • p ; p ≥ 0} of norms.It is a nuclear space forming the infinite-dimensional Gel'fand triple (Ᏹ) ⊂ (L 2 ) ⊂ (Ᏹ) * .Moreover, we have the following continuous inclusions: (2.6) We note that by letting E = L 2 (R), we get Ᏹ = (R) and obtain the Gel'fand triple Similarly, an element φ ∈ (Ᏹ) * can be written as φ The bilinear pairing between φ and ϕ is then represented as It is possible to construct wider Gel'fand triples than the one above; for example, by Kondratiev and Streit (see [6,Chapter 4] and the Cochran-Kuo-Sengupta (CKS) space in [1]).

Hermite polynomials, Wick tensors, and multiple Wiener integrals.
The Hermite polynomial of degree n with parameter σ 2 is defined by : (2.10) These polynomials have a generating function given by (2.11) The following formulas are also helpful: : (2.12) where The trace operator is the element τ ∈ (Ᏹ c ) ⊗2 defined by τ, ξ ⊗ η = ξ, η , ξ,η∈ Ᏹ c . (2.14) Let x ∈ Ᏹ .The Wick tensor : x ⊗n : of an element x is defined as where τ is the trace operator.The following formula similar to (2.12) is also important for Wick tensors, that is, (2.17) In order to make mathematical computations concerning multiple Wiener integrals easier, they are expressed in terms of Wick tensors.This is achieved via two statements as follows (see [6,Theorem 5.4]): (1) let h 1 ,h 2 ,... ∈ E be orthogonal and let For an element φ ∈ (Ᏹ p ) * , it follows that 3. The white noise differential operator.Let y ∈ Ᏹ and ϕ = : x ⊗n :,f ∈ (Ᏹ).It can be shown that the directional derivative where y ⊗1 is the unique continuous and linear map from This shows that the function ϕ has Gâteaux derivative D y ϕ.
we define an operator D y on (Ᏹ) as It can be shown that D y is a continuous linear operator on (Ᏹ) (see [6,Section 9.1]).By the duality between (Ᏹ) * and (Ᏹ), the adjoint operator D * y of D y can be defined by Now let Ᏹ be the Schwartz space of all infinitely differentiable functions f : R → R such that for all n, k ∈ N, If we take y = δ t , the Dirac delta function at t, then (1) ∂ t ≡ D δ t is called the white noise differential operator, the Hida differential operator, or the annihilation operator, (2)

The S-transform.
Let Φ ∈ (Ᏹ) * .The S-transform is a function on Ᏹ c defined by (2.28) The S-transform is an injective function because the exponential functions span a dense subset of ().If Φ ∈ (L 2 ), the S-transform of Φ is also called the Segal-Bargmann transform of Φ.

White noise integrals.
From now on, our reference Gel'fand triple is A white noise integral is a type of integral for which the integrand takes values in the space (Ᏹ) * β of generalized functions.As an example, consider the integral t 0 e −c(t−s) : Ḃ(s) 2 : ds where Ḃ is white noise.In this case, the integrand is an β -valued function on a measurable space (M, Ꮾ,m), a white noise integral is an integral of the type Despite the fact that (Ᏹ) * β is not a Banach space, these integrals can be defined in the Pettis or Bochner sense by the use of the S-transform.

White noise integrals in the Pettis sense.
We need to define E Φ(u)dm(u) as the generalized function in (Ᏹ) * β that satisfies the following: In particular, if Φ(u) is replaced by ∂ * u Φ(u), we have The above two equations then call for the following conditions on the function Φ to be satisfied: (This can be verified by using the characterization theorem for generalized functions.)The statement in (c) can be rewritten as Since the linear span of the set {: e In terms of the S-transform, Pettis integrability can be characterized using the following theorem.For a proof, see [6,Section 13.4].
Theorem 3.1.Suppose that the function Φ : M → (Ᏹ) * β satisfies the following conditions: (1) S(Φ(•))(ξ) is measurable for any ξ ∈ Ᏹ c ; (2) there exists nonnegative numbers K, a, and p such that Then Φ is Pettis integrable and for any E ∈ Ꮾ, 3.2.White noise integrals in the Bochner sense.We know that (Ᏹ) * β is not a Banach space but (Ᏹ) * β = ∪ p≥0 (Ᏹ p ) * β and each of the spaces (Ᏹ p ) * β is a separable Hilbert space.With this in mind, the white noise integral E Φ(u)dm(u) can be defined in the Bochner sense in the following way.Let Φ : M → (Ᏹ) * β , then Φ is Bochner integrable if it satisfies the following conditions: (1) Φ is weakly measurable; (2) there exists p ≥ 0 such that Φ(u) (3.8) The following theorem contains the conditions for Bochner integrability in terms of the S-transform and helps estimate the norm Φ(u) −p,−β of Φ. See [6, Section 13.5].
Theorem 3.2.Let Φ : M → (Ᏹ) * β be a function satisfying the following conditions: Then Φ is Bochner integrable and M Φ(u)dm(u) ∈ (Ᏹ q ) * β for any q > p such that where b ∞ is the essential supremum of b.It turns out that for such q, M Φ(u)dm(u) An example worth noting is the following.
In this case, Φ(u) = e iu •,f F(u), u ∈ R and it satisfies the conditions in Theorem 3.1 for Pettis integrability.

An extension of the Itô integral.
A number of extensions of the Itô integral exist.One such extension is by Itô [4], where he extended it to stochastic integrals for integrands which may be anticipating.In particular, he showed that 1 0 B(1)dB(t) = B(1) 2 .In [2], a special type of integral called the Hitsuda-Skorokhod integral (see Definition 4.1 below) was introduced as a motivation to obtain an Itô type formula for such functions as θ(B(t), B(c)), t < c, for a Ꮿ 2 -function θ. (We note here that B(c), t < c, is not Ᏺ t measurable.) Consider the Gel'fand triple Consider a stochastic process ϕ(t) in the space L 2 ([a, b] × (R)) which is nonanticipating.The Itô integral b a ϕ(t)dB(t) for the process ϕ(t) can be expressed as a white noise integral in the Pettis sense.The following theorem due to Kubo and Takenaka [5] (see also [6,Theorem 13.12] for a proof) implies that the Hitsuda-Skorokhod integral is an extension of the Itô integral to ϕ(t) which might be anticipating.A look at Example 4.3 will throw some light on the difference between Itô's extension and the Hitsuda-Skorokhod integral.) .We can apply the results from Section 2.2 concerning Wick tensors to get 1 0 (4.4) The operator N is called the number operator.Moreover, the power N r , r ∈ R, of the number operator is defined in the following way: for ϕ = ∞ n=0 : For any r ∈ R, N r is a continuous linear operator from () β into itself and from () * β into itself.Let ᐃ 1/2 be the Sobolev space of order 1/2 for the Gel'fand triple ).The norm on ᐃ 1/2 will be defined as Now if ϕ is given with the property that N 1/2 ϕ ∈ (L 2 ), and and so For more information on Sobolev spaces and the Number operator, see [7].
The following theorem gives a condition on the function ϕ(t) in order for b a ∂ * t ϕ(t)dt to be a Hitsuda-Skorokhod integral.This condition is determined by the number operator N and the space ᐃ 1/2 plays a major role.The proof for this condition can be found in [6,Theorem 13.16].We first identify the space L   The main tool for the proofs of the formulas obtained for the above two generalizations is the S-transform.The result for the generalization of (a) is stated below as a theorem and as earlier explained, my new formula will be using this particular generalization.(See [6,Theorem 13.21] for a complete proof.) is a Hitsuda-Skorokhod integral and the following equalities hold in (L 2 ): (1) (2) ( As hinted earlier, the proof of the main result in this paper is a limiting process of the above theorem.The following is the main result. is a Hitsuda-Skorokhod integral and the following equality holds in (L 2 ): (5.9 The proof is presented in the following steps.First it will be shown that for F = B(c), 0 ≤ a ≤ c ≤ b the above formula holds and that it coincides with the formula in Theorem 3.1.Secondly a special choice of F will be taken in the following way: we know that the span of the set {: e •,g :; . .
then, as N → ∞, F N → F in (L 2 ) where F = λ 1 : e •,g 1 : +λ 2 : e •,g 2 : +•••+λ k : e •,g k :.In our proof we will assume that F N → F in ᐃ 1/2 .Also, g will be taking on the form g = k j=1 α j 1 [0,c j ) , k ∈ N, α j ,c j ∈ R. The formula will then be generalized to θ(X(t), F N ) with F N chosen as above.An extension to θ(X(t), F ) with general F ∈ ᐃ 1/2 will be achieved via a limiting process.
Proof of Theorem 5.2.In the proof of [6, Theorem 1] which uses the S-transform, there are two components that were treated separately: (1) when a ≤ t ≤ c, and We claim that our new formula in the above theorem is correct when we replace F with B(c).To see that this is correct, it is enough to show that B(c) ∈ ᐃ 1/2 .We proceed by computing the norm for B(c) in the space ᐃ 1/2 .Indeed, since B(c) = •, 1 [0,c) , we have (5.12) (5.13) Thus, the two components ( 1) and ( 2) above in Theorem 3.1 are put together as one piece so as to satisfy the above theorem.
In general, suppose (5.16) with (5.28) Therefore, there exists a subsequence {F N k } k≥1 ⊂ {F N } N≥1 such that F N k → F as k → ∞ almost surely.For such a subsequence, the following is true: (5.29) We claim that the following convergences hold almost surely as k → ∞: Proof of (5.32).Let ω ∈ (R) be fixed.Since F N k converges, it is a bounded sequence, that is, there exists M > 0 such that |F N k (ω)| ≤ M for all k.Therefore, the function given by ∂ 2 θ/∂x 2 : [a, b] × [−M, M] → R is continuous on compact sets and hence uniformly continuous.Thus given > 0, there exists δ > 0 such that whenever .
(5.35) Also, by the convergence of {F N k } k≥1 , there exists a number N( ) depending on such that (5.38) By the same uniform continuity argument used above in proving (5.33), as k → ∞, we see that Therefore a change of integrals is justified and the following is true: (5.41) Then, {H N k (ω)} k≥1 is a sequence of positive functions and by the above result, ) and so (5.33) is true.
Proof of (5.34).We approximate the (L 2 )-norm of the difference of the two integrals.We claim the norm of this difference goes to zero.For convenience, use the following notation.Let ( where ((•, •)) 0 is the inner product on (L 2 ).Since F N k (ω) → F(ω) in (L 2 ), we can choose a subsequence so that the convergence is almost surely.Thus for such a subsequence, there exists Ω 0 such that P {Ω 0 } = 1 and for ω fixed, we have F N k (ω) → F(ω).Then, by the mean value theorem, for each s, As earlier noted, the convergence of the subsequence {F N k } k≥1 of numbers implies that for some fixed constant M > 0, the quantity F N k (ω) − F(ω) 0 ≤ M, for all k.Therefore, using the fact that F N k (ω) → F(ω) in (L 2 ), we have and also (5.50) The number operator, N can also be expressed as N = R ∂ * s ∂ s ds (see [6]).Hence, for any ϕ ∈ L 2 ([a, b]; ᐃ 1/2 ), we have But N 1/2 is a bounded operator from ᐃ 1/2 into (L 2 ).Therefore, by the same mean value theorem argument, since F N k → F also in ᐃ 1/2 by our original assumption, we have Therefore, it has been shown that the convergence in (5.34) is true in the (L 2 )norm.We then pick a further subsequence of the subsequence {F N k } k≥1 also denoted by {F N k } k≥1 such that convergence hold almost surely.This then proves all the convergences necessary and completes the proof of the theorem.
β is Pettis integrable.Then the function t → ∂ * t Φ(t) is also Pettis integrable and (3.3) holds.)dt is called the Hitsuda-Skorokhod integral of ϕ if it is a random variable in (L 2 ).
[6,let ∂ t + ϕ(t) and ∂ t − ϕ(t) be the right-hand white noise derivative and left-hand white noise derivative of ϕ, respectively (see[6, Definition 13.25]).The forward integral of ϕ is defined as y 1 ,...,y p ) be a function defined on R p+1 and of class Ꮿ 2 .Then we have the following formula which is simply a generalization of the one above: θ X(t), B c 1 ,...,B c p = θ X(a), B c 1 ,...,B c p

Proof of (5.30) and (5.31).
Because of continuity of θ, since F N k → F almost surely, those two convergences also hold almost surely.
It is known that in a Hilbert space H, if x n → x and y n → y, then x n ,y n → x, y where •, • is the inner product on H.By taking H = L 2 ([a, t]) with the Lebesgue measure it follows then that t a