Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let $Z$ be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of $Z$. Then, we use the Fock transform to define some fundamental operators on generalized functionals of $Z$, and apply the above mentioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of $Z$, and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of $Z$.


Introduction
One of the important theorems in Privault's discrete-time chaotic calculus [1,2] is its Clark-Ocone formula, which reads where = ( 푘 ) is a discrete-time normal noise, L 2 ( ) is the space of square integrable functionals of , F 푘 is the -field generated by ( 푗 ; 0 ≤ ≤ ), 푘 is the annihilation operator on L 2 ( ), and the series on the right-hand side converges in the norm of L 2 ( ). The Clark-Ocone formula (1) directly gives the predictable representation of functionals of , which implies the predictable representation property of discrete-time martingales associated with . The formula can also be used to establish the corresponding covariant identities [1]. More importantly, as was shown by Gao and Privault [3], this formula plays an important role in proving logarithmic Sobolev inequalities for Bernoulli measures. There are other applications based on the formula [2].
Despite its multiple uses, however, the Clark-Ocone formula (1) still suffers from a main drawback. That is, it holds only for the square integrable functionals of , which excludes many other interesting functionals of .
On the other hand, as is shown in [4], one can use the canonical orthonormal basis of L 2 ( ) to construct a nuclear space S( ) such that S( ) is densely contained in L 2 ( ). Thus, by identifying L 2 ( ) with its dual, one can get a Gel'fand triple where S * ( ) is the dual of S( ), which is endowed with the strong topology, which cannot be induced by any norm [5]. As usual, S( ) is called the testing functional space of , while S * ( ) is called the generalized functional space of . It turns out [6] that the generalized functional space S * ( ) can accommodate many quantities of theoretical interest that cannot be covered by L 2 ( ).
In this paper, we would like to extend the Clark-Ocone formula (1) to the generalized functionals of . More precisely, we would like to establish a Clark-Ocone formula for all elements of S * ( ). Our main work is as follows.
We first prove a result concerning the regularity of generalized functionals in S * ( ) in Section 2. Then, in Section 3, we use the Fock transform [6] to define some fundamental operators on S * ( ) and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish our formula, namely, the Clark-Ocone formula, for generalized functionals in S * ( ) in Section 3 and show its application results in Section 4, which include the covariant identity result and the variant upper bound result for generalized functionals in S * ( ).
Throughout this paper, N designates the set of all nonnegative integers and Γ the finite power set of N; namely, where #( ) means the cardinality of as a set. If ∈ N and ∈ Γ, then we simply write ∪ for ∪ { }. Similarly, we use \ .

Generalized Functionals of Discrete-Time Normal Noises
In all the following sections, we always assume that (Ω, F, ) is a given probability space. We use E to mean the expectation with respect to . As usual, L 2 (Ω, F, ) denotes the Hilbert space of square integrable complex-valued measurable functions on (Ω, F, ). We use ⟨⋅, ⋅⟩ and ‖ ⋅ ‖ to mean the inner product and norm of L 2 (Ω, F, ), respectively. By convention, ⟨⋅, ⋅⟩ is conjugate-linear in its first argument and linear in its second argument.

Discrete-Time Normal Noises.
Example 1. Let = ( 푛 ) 푛∈N be an independent sequence of random variables on (Ω, F, ) with Write G −1 = {0, Ω} and G 푛 = ( 푘 ; 0 ≤ ≤ ) for ∈ N. Then, one can immediately see that Thus, is a discrete-time normal noise. Note that, by letting = ( 푛 ) be the partial sum sequence of , one gets the classical random walk.
In the literature, F ∞ -measurable functions on Ω are also known as functionals of . Thus, elements of L 2 (Ω, F ∞ , ) are naturally called square integrable functionals of .
Thus, if a discrete-time normal noise has the chaotic representation property, then its canonical functional system { 휎 | ∈ Γ} is actually an orthonormal basis of L 2 (Ω, F ∞ , ).

Generalized Functionals.
From now on, we always assume that = ( 푛 ) 푛∈N is a given discrete-time normal noise on (Ω, F, ) that has the chaotic representation property.
It is easy to see that 휎 ≥ 1 for all ∈ Γ. This implies that ‖ ⋅ ‖ 푝 ≤ ‖ ⋅ ‖ 푞 and S 푞 ( ) ⊂ S 푝 ( ) whenever 0 ≤ ≤ . Thus, we actually get a chain of Hilbert spaces of functionals of : We now put and endow it with the topology generated by the norm sequence {‖ ⋅ ‖ 푝 } 푝≥0 . Note that, for each ≥ 0, S 푝 ( ) is just the completion of S( ) with respect to ‖ ⋅ ‖ 푝 . Thus, S( ) is a countably Hilbert space [5,8]. The next lemma, however, shows that S( ) even has a much better property.
Lemma 6 (see [4,6]). The space S( ) is a nuclear space; namely, for any ≥ 0, there exists > such that the inclusion The lemma below is then an immediate consequence of the general theory of countably Hilbert spaces (see, e.g., [8] or [5]).
Lemma 7 (see [4,6]). Let S * ( ) be the dual of S( ) and endow it with the strong topology. Then, and moreover the inductive limit topology over S * ( ) given by space sequence {S * 푝 ( )} 푝≥0 coincides with the strong topology.
We mention that, by identifying L 2 ( ) with its dual, one comes to a Gel'fand triple which we refer to as the Gel'fand triple associated with the discrete-time normal noise .
Theorem 8 (see [6]). The system { 휎 | ∈ Γ} is contained in S( ) and moreover it forms a basis for S( ) in the sense that where ⟨⋅, ⋅⟩ is the inner product of L 2 ( ) and the series converges in the topology of S( ).
Definition 9 (see [4,6]). Elements of S * ( ) are called generalized functionals of , while elements of S( ) are called testing functionals of . Thus, S * ( ) and S( ) can be accordingly called the generalized functional space and the testing functional space of , respectively. It turns out [6] that S * ( ) can accommodate many quantities of theoretical interest that cannot be covered by L 2 ( ).
It is easy to verify that, for Φ, Ψ ∈ S * ( ), Φ = Ψ if and only ifΦ =Ψ. Thus, a generalized functional of is completely determined by its Fock transform. The following theorem characterizes generalized functionals of through their Fock transforms.
Theorem 11 (see [6]). Let be a function on Γ. Then, is the Fock transform of an element Φ of S * ( ) if and only if it satisfies for some constants ≥ 0 and ≥ 0. In that case, for > + 1/2, one has and in particular Φ ∈ S * 푞 ( ).
The theorem below describes the regularity of generalized functionals of via their Fock transforms.
In that case, the norm Proof. The "Only If" Part. By the well-known Riesz representation theorem [9], there exists a unique ∈ S 푝 ( ) such that Thus, which implies (22) and (23).
The "If" Part. For each ∈ S( ), using Theorem 8, we have Thus, Φ is a bounded functional on the space Remark 13. There exists a continuous linear mapping R : where (⋅, ⋅) is the canonical bilinear form on S * ( ) × S( ). We call R the Riesz mapping.

Clark-Ocone Formula for Generalized Functionals
In this section, we first introduce some fundamental operators on the space S * ( ). And then we establish our Clark-Ocone formula for functionals in S * ( ).

Annihilation and Creation Operators
Theorem 15. Let ∈ N. Then, there exists a continuous linear operator a 푘 : S * ( ) → S * ( ) such that Proof. For each Φ ∈ S * ( ), by Theorem 11, there exist constants , ≥ 0 such that which means that the function which, together with Theorem 11, implies that there exists a unique Ψ Φ ∈ S * ( ) such that Now, consider the mapping a 푘 : S * ( ) → S * ( ) defined by It is not hard to verify that a 푘 is a linear operator and satisfies (29). To complete the proof, we still need to show that a 푘 : S * ( ) → S * ( ) is continuous with respect to the strong topology over S * ( ). Let ≥ 0 and denote by j 푘 : S * 푝 ( ) → S * ( ) the inclusion mapping; namely, j 푘 is the mapping defined by Then, the composition mapping a 푘 ∘ j 푘 is a linear operator from S * 푝 ( ) to S * ( ). For each Φ ∈ S * 푝 ( ), we have Journal of Function Spaces 5 which together with Theorem 12 implies that a 푘 ∘ j 푘 (Φ) ∈ S * 푝 ( ) and Thus, a 푘 ∘j 푘 (S * 푝 ( )) ⊂ S * 푝 ( ) and a 푘 ∘j 푘 : S * 푝 ( ) → S * 푝 ( ) is a bounded operator, which implies that a 푘 ∘ j 푘 is continuous as an operator from S * 푝 ( ) to S * ( ). Since the choice of the above ≥ 0 is arbitrary, we actually arrive at a conclusion that the composition mapping a 푘 ∘ j 푘 : S * 푝 ( ) → S * ( ) is continuous for all ≥ 0. Therefore, a 푘 : S * ( ) → S * ( ) is continuous with respect to the inductive limit topology over S * ( ), which together with Lemma 7 implies that a 푘 : S * ( ) → S * ( ) is continuous with respect to the strong topology over S * ( ).
Carefully checking the proof of Theorem 15, one can find the next result already proven.
Theorem 16. Let ∈ N. Then, for each ≥ 0, S * 푝 ( ) keeps invariant under the action of a 푘 , and moreover With the same arguments, we can prove the next two theorems, which are dual forms of Theorems 15 and 16, respectively. Proof. For each Φ ∈ S * ( ), by Theorem 11, there exist constants , ≥ 0 such that which means that the function → 1 휎 ( )Φ( \ ) satisfies which, together with Theorem 11, implies that there exists a unique Θ Φ ∈ S * ( ) such that Now, consider the mapping a † 푘 : S * ( ) → S * ( ) defined by It is not hard to verify that a † 푘 is a linear operator and satisfies (38). To complete the proof, we still need to show that a † 푘 : S * ( ) → S * ( ) is continuous with respect to the strong topology over S * ( ).
From the proof of Theorem 17, we can easily get the next result concerning the operator a † 푘 .
Theorem 18. Let ∈ N. Then, for each ≥ 0, S * 푝 ( ) keeps invariant under the action of a † 푘 , and moreover Remark 19. For ≥ 0, the corresponding annihilation operator 푘 on L 2 ( ) and its dual * 푘 (known as the creation operator) admit the property And moreover, they satisfy the canonical anticommutation relation (CAR) in equal-time * where means the identity operator on L 2 ( ). We refer to [2,6] and for details about these operators. The next theorem shows the link between a 푘 and 푘 , as well as between a † 푘 and * 푘 . 6

Journal of Function Spaces
Theorem 20. Let ≥ 0. Then, the operators a 푘 and a † 푘 satisfy where R is the Riesz mapping as indicated in Remark 13.
Proof. Let ∈ L 2 ( ). Then, for all ∈ Γ, we havê which implies a 푘 R = R 푘 . It then follows by the arbitrariness of ∈ L 2 ( ) that a 푘 R = R 푘 . Similarly, we can prove a † 푘 R = R * 푘 . In view of Theorem 20, we give the following definition to name the operators a 푘 and a † 푘 .
Definition 21. For ≥ 0, the operators a 푘 and a † 푘 are called the annihilation and creation operators on generalized functionals of , respectively.

Expectation and Conditional Expectation Operators.
For the Riesz mapping R, using Theorem 12, we can prove that R ∈ S * 0 ( ) for all ∈ L 2 ( ). In particular, we have R1 ∈ S * 0 ( ).
Theorem 23. The mapping E : S * ( ) → S * ( ) defined by is a continuous linear operator from S * ( ) to itself. And, moreover, Proof. Clearly, E : S * ( ) → S * ( ) is a linear operator and satisfies (54). Next, let us show that E : S * ( ) → S * ( ) is continuous with respect to the strong topology over S * ( ). Let ≥ 0 and denote by j 푘 : S * 푝 ( ) → S * ( ) the inclusion mapping. Then, the composition mapping E ∘ j 푘 is a linear operator from S * 푝 ( ) to S * ( ). For each Φ ∈ S * 푝 ( ), we have which together with Theorem 12 implies that E ∘ j 푘 (Φ) ∈ S * 푝 ( ) and Thus, E ∘ j 푘 (S * 푝 ( )) ⊂ S * 푝 ( ) and E ∘ j 푘 : S * 푝 ( ) → S * 푝 ( ) is a bounded operator, which implies that E ∘ j 푘 is continuous as an operator from S * 푝 ( ) to S * ( ). Since the choice of the above ≥ 0 is arbitrary, we actually arrive at a conclusion that the composition mapping E ∘ j 푘 : S * 푝 ( ) → S * ( ) is continuous for all ≥ 0. Therefore, E : S * ( ) → S * ( ) is continuous with respect to the inductive limit topology over S * ( ), which together with Lemma 7 implies that E : S * ( ) → S * ( ) is continuous with respect to the strong topology over S * ( ).
Definition 24. The operator E is called the expectation operator on generalized functionals of .
Since 1 ∈ L 2 ( ), the expectation E with respect to is actually a bounded operator from L 2 ( ) to itself. The next theorem shows the link between the operators E and E, which justifies the above definition.
Theorem 25. It holds that ER = RE, where R is the Riesz mapping.
Proof. For any ∈ L 2 ( ) and any ∈ Γ, by a direct computation, we havê where Proof. We omit the proof because it is quite similar to that of Theorem 15.
Using Theorems 12 and 26, we can easily prove the next theorem, which shows that the operator E 푘 has a type of contraction property on S * ( ).
Theorem 27. Let ≥ 0. Then, for each ≥ 0, S * 푝 ( ) keeps invariant under the action of E 푘 , and moreover Definition 28. The operators E 푘 , ≥ 0, are called the conditional expectation operators on generalized functionals of .
For ≥ 0, we set 푘 = E[⋅ | F 푘 ], the expectation given F 푘 , where F 푘 is the -field generated by ( 푗 ; 0 ≤ ≤ ) as mentioned above. 푘 is usually known as a conditional expectation operator on square integrable functionals of . The theorem below then justifies Definition 28.

Clark-Ocone Formula for Generalized Functionals.
In this subsection, we establish our Clark-Ocone formula for generalized functionals of .
Theorem 30. For all generalized functionals Φ ∈ S * ( ), it holds that where the series on the right-hand side converges strongly in S * ( ).
Proposition 31. For each ≥ 0, it holds that Proof. Let ≥ 0. Then, for all Φ ∈ S * ( ) and ∈ Γ, by Theorems 17 and 26, we get where equality Combining Theorem 30 with Proposition 31, we arrive at the next interesting result, which we call the Clark-Ocone formula for generalized functionals of .
Theorem 32. For all generalized functionals Φ ∈ S * ( ), it holds that where E −1 = E and the series on the right-hand side converges strongly in S * ( ).
Remark 33. As mentioned above, 푘 and * 푘 are the annihilation and creation operators on L 2 ( ), respectively, and is the conditional expectation operator on L 2 ( ). It can be verified that * where −1 = E and 푘 is the -component of the discretetime normal noise . Thus, the Clark-Ocone formula (1) can be rewritten as the following form: where the series on the right-hand side converges in the norm of L 2 ( ). This observation justifies calling formula (67) the Clark-Ocone formula for generalized functionals of .
Theorem 35. Let Φ, Ψ ∈ S * 푝 ( ) for some ≥ 0. Then, their -covariant cov 푝 (Φ, Ψ) makes sense, and moreover Proof. By Theorem 12, the series on the right-hand side of (73) converges absolutely. On the other hand, by Theorem 30, we have which together with the fact This completes the proof.
Theorem 35 sets up covariant identities for generalized functionals of . The next theorem then gives meaningful upper bounds to variants of generalized functionals of .
Proof. This is an immediate consequence of Theorem 32.
Remark 38. A generalized functional of , or, in other words, a generalized functional in S * ( ), can be interpreted as a generalized random variable on the probability space (Ω, F, ). Accordingly, a sequence of generalized functionals of can be viewed as a generalized stochastic process. Theorem 37 then shows that each generalized random variable on (Ω, F, ) can be represented as the generalized stochastic integral of an (E 푘 )-predictable generalized stochastic process with respect to (a † 푘 ).

Conflicts of Interest
The authors declare that they have no conflicts of interest.