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

Authors: Caishi Wang; Shuai Lin; Ailing Huang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

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. LetZ 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 abovementioned 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.


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## Body

## 1. Introduction

One of the important theorems in Privault’s discrete-time chaotic calculus [1, 2] is its Clark-Ocone formula, which reads(1)ξ=Eξ+∑k=0∞ZkE∂kξ∣Fk-1,ξ∈L2Z,where Z=(Zk) is a discrete-time normal noise, L2(Z) is the space of square integrable functionals of Z, Fk is the σ-field generated by (Zj;0≤j≤k), ∂k is the annihilation operator on L2(Z), and the series on the right-hand side converges in the norm of L2(Z).The Clark-Ocone formula (1) directly gives the predictable representation of functionals of Z, which implies the predictable representation property of discrete-time martingales associated with Z. 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 ξ of Z, which excludes many other interesting functionals of Z.On the other hand, as is shown in [4], one can use the canonical orthonormal basis of L2(Z) to construct a nuclear space S(Z) such that S(Z) is densely contained in L2(Z). Thus, by identifying L2(Z) with its dual, one can get a Gel’fand triple(2)SZ⊂L2Z⊂S∗Z,where S∗(Z) is the dual of S(Z), which is endowed with the strong topology, which cannot be induced by any norm [5]. As usual, S(Z) is called the testing functional space of Z, while S∗(Z) is called the generalized functional space of Z. It turns out [6] that the generalized functional space S∗(Z) can accommodate many quantities of theoretical interest that cannot be covered by L2(Z).In this paper, we would like to extend the Clark-Ocone formula (1) to the generalized functionals of Z. More precisely, we would like to establish a Clark-Ocone formula for all elements of S∗(Z). Our main work is as follows.We first prove a result concerning the regularity of generalized functionals inS∗(Z) in Section 2. Then, in Section 3, we use the Fock transform [6] to define some fundamental operators on S∗(Z) 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∗(Z) 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∗(Z).Throughout this paper,N designates the set of all nonnegative integers and Γ the finite power set of N; namely, (3)Γ=σ∣σ⊂N,#σ<∞,where #(σ) means the cardinality of σ as a set. If k∈N and σ∈Γ, then we simply write σ∪k for σ∪{k}. Similarly, we use σ∖k.


## 2. Generalized Functionals of Discrete-Time Normal Noises

In all the following sections, we always assume that(Ω,F,P) is a given probability space. We use E to mean the expectation with respect to P. As usual, L2(Ω,F,P) denotes the Hilbert space of square integrable complex-valued measurable functions on (Ω,F,P). We use 〈·,·〉 and ∥·∥ to mean the inner product and norm of L2(Ω,F,P), respectively. By convention, 〈·,·〉 is conjugate-linear in its first argument and linear in its second argument.


### 2.1. Discrete-Time Normal Noises

A sequenceZ=(Zn)n∈N of integrable random variables on (Ω,F,P) is called a discrete-time normal noise if it satisfies(i)
EZn∣Fn-1=0 for n≥0;(ii)
EZn2∣Fn-1=1 for n≥0.Here,F-1={∅,Ω}, Fn=σ(Zk;0≤k≤n) for n∈N and E·∣Fn means the conditional expectation given Fn.Example 1.
Letζ=(ζn)n∈N be an independent sequence of random variables on (Ω,F,P) with (4)Pζn=-1=Pζn=1=12,n∈N.Write G-1={∅,Ω} and Gn=σ(ζk;0≤k≤n) for n∈N. Then, one can immediately see that (i)
Eζn∣Gn-1=0 for n≥0;(ii)
Eζn2∣Gn-1=1 for n≥0.
Thus,ζ is a discrete-time normal noise. Note that, by letting X=(Xn) be the partial sum sequence of ζ, one gets the classical random walk.For a discrete-time normal noiseZ=(Zn)n∈N on (Ω,F,P), one can construct a corresponding family Zσ∣σ∈Γ of random variables on (Ω,F,P) in the following manner:(5)Z∅=1,Zσ=∏i∈σZi,σ∈Γ,σ≠∅.We call Zσ∣σ∈Γ the canonical functional system of Z.Lemma 2 (see [1, 2, 7]).
LetZ=(Zn)n∈N be a discrete-time normal noise on (Ω,F,P). Then, its canonical functional system Zσ∣σ∈Γ forms a countable orthonormal system in L2(Ω,F,P).LetF∞=σ(Zn;n∈N) be the σ-field over Ω generated by a discrete-time normal noise Z=(Zn)n∈N on (Ω,F,P). Then, the canonical functional system Zσ∣σ∈Γ is also a countable orthonormal system in the space L2(Ω,F∞,P) of square integrable complex-valued measurable functions on (Ω,F∞,P).In the literature,F∞-measurable functions on Ω are also known as functionals of Z. Thus, elements of L2(Ω,F∞,P) are naturally called square integrable functionals of Z.Definition 3.
A discrete-time normal noiseZ=(Zn)n∈N on (Ω,F,P) is said to have the chaotic representation property if its canonical functional system Zσ∣σ∈Γ is total in L2(Ω,F∞,P), where F∞=σ(Zn;n∈N).
Thus, if a discrete-time normal noiseZ has the chaotic representation property, then its canonical functional system Zσ∣σ∈Γ is actually an orthonormal basis of L2(Ω,F∞,P).


### 2.2. Generalized Functionals

From now on, we always assume thatZ=(Zn)n∈N is a given discrete-time normal noise on (Ω,F,P) that has the chaotic representation property.For brevity, we useL2(Z) to denote the space of square integrable functionals of Z; namely, (6)L2Z=L2Ω,F∞,P,where F∞=σ(Zn;n∈N). For k≥0, we denote by Fk the σ-field generated by (Zj;0≤j≤k); namely, (7)Fk=σZj;0≤j≤k.We note that L2(Z) shares the same inner product 〈·,·〉 and norm ∥·∥ with L2(Ω,F,P), and moreover the canonical functional system Zσ∣σ∈Γ of Z forms a countable orthonormal basis for L2(Z), which we call the canonical orthonormal basis of L2(Z).Lemma 4 (see [4]).
Letσ↦λσ be the N-valued function on Γ given by(8)λσ=∏k∈σk+1,σ≠∅,σ∈Γ;1,σ=∅,σ∈Γ.Then, for p>1, the positive term series ∑σ∈Γλσ-p converges and moreover(9)∑σ∈Γλσ-p≤exp⁡∑k=1∞k-p<∞.Using theN-valued function defined by (8), we can construct a chain of Hilbert spaces consisting of functionals of Z as follows. For p≥0, we put(10)SpZ=ξ∈L2Z∣∑σ∈Γλσ2pZσ,ξ2<∞and define(11)ξ,ηp=∑σ∈Γλσ2pZσ,ξ¯Zσ,η,ξ,η∈SpZ.It is not hard to check that, with 〈·,·〉p as the inner product, Sp(Z) becomes a Hilbert space. We write ξp=〈ξ,ξ〉p for ξ∈Sp(Z). Clearly, it holds that(12)ξp2=∑σ∈Γλσ2pZσ,ξ2,ξ∈SpZ.Lemma 5 (see [4, 6]).
Forp≥0, one has Zσ∣σ∈Γ⊂Sp(Z) and moreover the system λσ-pZσ∣σ∈Γ forms an orthonormal basis for Sp(Z).It is easy to see thatλσ≥1 for all σ∈Γ. This implies that ∥·∥p≤∥·∥q and Sq(Z)⊂Sp(Z) whenever 0≤p≤q. Thus, we actually get a chain of Hilbert spaces of functionals of Z:(13)⋯⊂Sp+1Z⊂SpZ⊂⋯⊂S1Z⊂S0Z=L2Z.We now put(14)SZ=⋂p=0∞SpZand endow it with the topology generated by the norm sequence {∥·∥p}p≥0. Note that, for each p≥0, Sp(Z) is just the completion of S(Z) with respect to ∥·∥p. Thus, S(Z) is a countably Hilbert space [5, 8]. The next lemma, however, shows that S(Z) even has a much better property.Lemma 6 (see [4, 6]).
The spaceS(Z) is a nuclear space; namely, for any p≥0, there exists q>p such that the inclusion mapping ipq:Sq(Z)→Sp(Z) defined by ipq(ξ)=ξ is a Hilbert-Schmidt operator.Forp≥0, we denote by Sp∗(Z) the dual of Sp(Z) and ∥·∥-p the norm of Sp∗(Z). Then, Sp∗(Z)⊂Sq∗(Z) and ∥·∥-p≥∥·∥-q whenever 0≤p≤q. 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]).
LetS∗(Z) be the dual of S(Z) and endow it with the strong topology. Then,(15)S∗Z=⋃p=0∞Sp∗Zand moreover the inductive limit topology over S∗(Z) given by space sequence {Sp∗(Z)}p≥0 coincides with the strong topology.We mention that, by identifyingL2(Z) with its dual, one comes to a Gel’fand triple(16)SZ⊂L2Z⊂S∗Z,which we refer to as the Gel’fand triple associated with the discrete-time normal noise Z.Theorem 8 (see [6]).
The systemZσ∣σ∈Γ is contained in S(Z) and moreover it forms a basis for S(Z) in the sense that(17)ξ=∑σ∈ΓZσ,ξZσ,ξ∈SZ,where 〈·,·〉 is the inner product of L2(Z) and the series converges in the topology of S(Z).Definition 9 (see [4, 6]).
Elements ofS∗(Z) are called generalized functionals of Z, while elements of S(Z) are called testing functionals of Z.
Thus,S∗(Z) and S(Z) can be accordingly called the generalized functional space and the testing functional space of Z, respectively. It turns out [6] that S∗(Z) can accommodate many quantities of theoretical interest that cannot be covered by L2(Z).
In the following, we denote by⟪·,·⟫ the canonical bilinear form on S∗(Z)×S(Z) given by(18)⟪Φ,ξ⟫=Φξ,Φ∈S∗Z,ξ∈SZ.Note that ⟪·,·⟫ is different from the inner product 〈·,·〉 of L2(Z).Definition 10 (see [6]).
ForΦ∈S∗(Z), its Fock transform is the function Φ^ on Γ given by(19)Φ^σ=⟪Φ,Zσ⟫,σ∈Γ,where ⟪·,·⟫ is the canonical bilinear form.
It is easy to verify that, forΦ, Ψ∈S∗(Z), Φ=Ψ if and only if Φ^=Ψ^. Thus, a generalized functional of Z is completely determined by its Fock transform. The following theorem characterizes generalized functionals of Z through their Fock transforms.Theorem 11 (see [6]).
LetF be a function on Γ. Then, F is the Fock transform of an element Φ of S∗(Z) if and only if it satisfies(20)Fσ≤Cλσp,σ∈Γfor some constants C≥0 and p≥0. In that case, for q>p+1/2, one has(21)Φ-q≤C∑σ∈Γλσ-2q-p1/2and in particular Φ∈Sq∗(Z).The theorem below describes the regularity of generalized functionals ofZ via their Fock transforms.Theorem 12.
LetΦ∈S∗(Z) and p≥0. Then, Φ∈Sp∗(Z) if and only if(22)∑σ∈Γλσ-2pΦ^σ2<∞.In that case, the norm Φ-p of Φ in Sp∗(Z) satisfies(23)Φ-p2=∑σ∈Γλσ-2pΦ^σ2.Proof.
The “Only If” Part. By the well-known Riesz representation theorem [9], there exists a unique η∈Sp(Z) such that ηp=Φ-p and (24)Φξ=η,ξp,ξ∈SpZ.Thus, (25)∑σ∈Γλσ-2pΦ^σ2=∑σ∈Γλσ-2pZσ,ηp2=∑σ∈Γλσ2pZσ,η2=ηp2=Φ-p2,which implies (22) and (23).
The “If” Part. For each ξ∈S(Z), using Theorem 8, we have (26)Φξ=∑σ∈ΓZσ,ξΦZσ=∑σ∈ΓZσ,ξΦ^σ≤∑σ∈Γλσ2pZσ,ξ21/2∑σ∈Γλσ-2pΦ^σ21/2=ξp∑σ∈Γλσ-2pΦ^σ21/2.Thus, Φ is a bounded functional on the space (S(Z),∥·∥p), which implies Φ∈Sp∗(Z) since S(Z) is dense in Sp(Z).Remark 13.
There exists a continuous linear mappingR:L2(Z)→S∗(Z) such that(27)⟪Rη,ξ⟫=η,ξ,η∈L2Z,ξ∈SZ,where ⟪·,·⟫ is the canonical bilinear form on S∗(Z)×S(Z). We call R the Riesz mapping.Theorem 14 (see [10]).
LetΦ, Φn∈S∗(Z), n≥1, be generalized functionals of Z. Then, the sequence (Φn) converges strongly to Φ in S∗(Z) if and only if it satisfies the following:(1)
Φn^(σ)→Φ^(σ) for all σ∈Γ.(2)
There are constantsC≥0 and p≥0 such that(28)supn≥1⁡Φn^σ≤Cλσp,σ∈Γ.


## 2.1. Discrete-Time Normal Noises

A sequenceZ=(Zn)n∈N of integrable random variables on (Ω,F,P) is called a discrete-time normal noise if it satisfies(i)
EZn∣Fn-1=0 for n≥0;(ii)
EZn2∣Fn-1=1 for n≥0.Here,F-1={∅,Ω}, Fn=σ(Zk;0≤k≤n) for n∈N and E·∣Fn means the conditional expectation given Fn.Example 1.
Letζ=(ζn)n∈N be an independent sequence of random variables on (Ω,F,P) with (4)Pζn=-1=Pζn=1=12,n∈N.Write G-1={∅,Ω} and Gn=σ(ζk;0≤k≤n) for n∈N. Then, one can immediately see that (i)
Eζn∣Gn-1=0 for n≥0;(ii)
Eζn2∣Gn-1=1 for n≥0.
Thus,ζ is a discrete-time normal noise. Note that, by letting X=(Xn) be the partial sum sequence of ζ, one gets the classical random walk.For a discrete-time normal noiseZ=(Zn)n∈N on (Ω,F,P), one can construct a corresponding family Zσ∣σ∈Γ of random variables on (Ω,F,P) in the following manner:(5)Z∅=1,Zσ=∏i∈σZi,σ∈Γ,σ≠∅.We call Zσ∣σ∈Γ the canonical functional system of Z.Lemma 2 (see [1, 2, 7]).
LetZ=(Zn)n∈N be a discrete-time normal noise on (Ω,F,P). Then, its canonical functional system Zσ∣σ∈Γ forms a countable orthonormal system in L2(Ω,F,P).LetF∞=σ(Zn;n∈N) be the σ-field over Ω generated by a discrete-time normal noise Z=(Zn)n∈N on (Ω,F,P). Then, the canonical functional system Zσ∣σ∈Γ is also a countable orthonormal system in the space L2(Ω,F∞,P) of square integrable complex-valued measurable functions on (Ω,F∞,P).In the literature,F∞-measurable functions on Ω are also known as functionals of Z. Thus, elements of L2(Ω,F∞,P) are naturally called square integrable functionals of Z.Definition 3.
A discrete-time normal noiseZ=(Zn)n∈N on (Ω,F,P) is said to have the chaotic representation property if its canonical functional system Zσ∣σ∈Γ is total in L2(Ω,F∞,P), where F∞=σ(Zn;n∈N).
Thus, if a discrete-time normal noiseZ has the chaotic representation property, then its canonical functional system Zσ∣σ∈Γ is actually an orthonormal basis of L2(Ω,F∞,P).


## 2.2. Generalized Functionals

From now on, we always assume thatZ=(Zn)n∈N is a given discrete-time normal noise on (Ω,F,P) that has the chaotic representation property.For brevity, we useL2(Z) to denote the space of square integrable functionals of Z; namely, (6)L2Z=L2Ω,F∞,P,where F∞=σ(Zn;n∈N). For k≥0, we denote by Fk the σ-field generated by (Zj;0≤j≤k); namely, (7)Fk=σZj;0≤j≤k.We note that L2(Z) shares the same inner product 〈·,·〉 and norm ∥·∥ with L2(Ω,F,P), and moreover the canonical functional system Zσ∣σ∈Γ of Z forms a countable orthonormal basis for L2(Z), which we call the canonical orthonormal basis of L2(Z).Lemma 4 (see [4]).
Letσ↦λσ be the N-valued function on Γ given by(8)λσ=∏k∈σk+1,σ≠∅,σ∈Γ;1,σ=∅,σ∈Γ.Then, for p>1, the positive term series ∑σ∈Γλσ-p converges and moreover(9)∑σ∈Γλσ-p≤exp⁡∑k=1∞k-p<∞.Using theN-valued function defined by (8), we can construct a chain of Hilbert spaces consisting of functionals of Z as follows. For p≥0, we put(10)SpZ=ξ∈L2Z∣∑σ∈Γλσ2pZσ,ξ2<∞and define(11)ξ,ηp=∑σ∈Γλσ2pZσ,ξ¯Zσ,η,ξ,η∈SpZ.It is not hard to check that, with 〈·,·〉p as the inner product, Sp(Z) becomes a Hilbert space. We write ξp=〈ξ,ξ〉p for ξ∈Sp(Z). Clearly, it holds that(12)ξp2=∑σ∈Γλσ2pZσ,ξ2,ξ∈SpZ.Lemma 5 (see [4, 6]).
Forp≥0, one has Zσ∣σ∈Γ⊂Sp(Z) and moreover the system λσ-pZσ∣σ∈Γ forms an orthonormal basis for Sp(Z).It is easy to see thatλσ≥1 for all σ∈Γ. This implies that ∥·∥p≤∥·∥q and Sq(Z)⊂Sp(Z) whenever 0≤p≤q. Thus, we actually get a chain of Hilbert spaces of functionals of Z:(13)⋯⊂Sp+1Z⊂SpZ⊂⋯⊂S1Z⊂S0Z=L2Z.We now put(14)SZ=⋂p=0∞SpZand endow it with the topology generated by the norm sequence {∥·∥p}p≥0. Note that, for each p≥0, Sp(Z) is just the completion of S(Z) with respect to ∥·∥p. Thus, S(Z) is a countably Hilbert space [5, 8]. The next lemma, however, shows that S(Z) even has a much better property.Lemma 6 (see [4, 6]).
The spaceS(Z) is a nuclear space; namely, for any p≥0, there exists q>p such that the inclusion mapping ipq:Sq(Z)→Sp(Z) defined by ipq(ξ)=ξ is a Hilbert-Schmidt operator.Forp≥0, we denote by Sp∗(Z) the dual of Sp(Z) and ∥·∥-p the norm of Sp∗(Z). Then, Sp∗(Z)⊂Sq∗(Z) and ∥·∥-p≥∥·∥-q whenever 0≤p≤q. 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]).
LetS∗(Z) be the dual of S(Z) and endow it with the strong topology. Then,(15)S∗Z=⋃p=0∞Sp∗Zand moreover the inductive limit topology over S∗(Z) given by space sequence {Sp∗(Z)}p≥0 coincides with the strong topology.We mention that, by identifyingL2(Z) with its dual, one comes to a Gel’fand triple(16)SZ⊂L2Z⊂S∗Z,which we refer to as the Gel’fand triple associated with the discrete-time normal noise Z.Theorem 8 (see [6]).
The systemZσ∣σ∈Γ is contained in S(Z) and moreover it forms a basis for S(Z) in the sense that(17)ξ=∑σ∈ΓZσ,ξZσ,ξ∈SZ,where 〈·,·〉 is the inner product of L2(Z) and the series converges in the topology of S(Z).Definition 9 (see [4, 6]).
Elements ofS∗(Z) are called generalized functionals of Z, while elements of S(Z) are called testing functionals of Z.
Thus,S∗(Z) and S(Z) can be accordingly called the generalized functional space and the testing functional space of Z, respectively. It turns out [6] that S∗(Z) can accommodate many quantities of theoretical interest that cannot be covered by L2(Z).
In the following, we denote by⟪·,·⟫ the canonical bilinear form on S∗(Z)×S(Z) given by(18)⟪Φ,ξ⟫=Φξ,Φ∈S∗Z,ξ∈SZ.Note that ⟪·,·⟫ is different from the inner product 〈·,·〉 of L2(Z).Definition 10 (see [6]).
ForΦ∈S∗(Z), its Fock transform is the function Φ^ on Γ given by(19)Φ^σ=⟪Φ,Zσ⟫,σ∈Γ,where ⟪·,·⟫ is the canonical bilinear form.
It is easy to verify that, forΦ, Ψ∈S∗(Z), Φ=Ψ if and only if Φ^=Ψ^. Thus, a generalized functional of Z is completely determined by its Fock transform. The following theorem characterizes generalized functionals of Z through their Fock transforms.Theorem 11 (see [6]).
LetF be a function on Γ. Then, F is the Fock transform of an element Φ of S∗(Z) if and only if it satisfies(20)Fσ≤Cλσp,σ∈Γfor some constants C≥0 and p≥0. In that case, for q>p+1/2, one has(21)Φ-q≤C∑σ∈Γλσ-2q-p1/2and in particular Φ∈Sq∗(Z).The theorem below describes the regularity of generalized functionals ofZ via their Fock transforms.Theorem 12.
LetΦ∈S∗(Z) and p≥0. Then, Φ∈Sp∗(Z) if and only if(22)∑σ∈Γλσ-2pΦ^σ2<∞.In that case, the norm Φ-p of Φ in Sp∗(Z) satisfies(23)Φ-p2=∑σ∈Γλσ-2pΦ^σ2.Proof.
The “Only If” Part. By the well-known Riesz representation theorem [9], there exists a unique η∈Sp(Z) such that ηp=Φ-p and (24)Φξ=η,ξp,ξ∈SpZ.Thus, (25)∑σ∈Γλσ-2pΦ^σ2=∑σ∈Γλσ-2pZσ,ηp2=∑σ∈Γλσ2pZσ,η2=ηp2=Φ-p2,which implies (22) and (23).
The “If” Part. For each ξ∈S(Z), using Theorem 8, we have (26)Φξ=∑σ∈ΓZσ,ξΦZσ=∑σ∈ΓZσ,ξΦ^σ≤∑σ∈Γλσ2pZσ,ξ21/2∑σ∈Γλσ-2pΦ^σ21/2=ξp∑σ∈Γλσ-2pΦ^σ21/2.Thus, Φ is a bounded functional on the space (S(Z),∥·∥p), which implies Φ∈Sp∗(Z) since S(Z) is dense in Sp(Z).Remark 13.
There exists a continuous linear mappingR:L2(Z)→S∗(Z) such that(27)⟪Rη,ξ⟫=η,ξ,η∈L2Z,ξ∈SZ,where ⟪·,·⟫ is the canonical bilinear form on S∗(Z)×S(Z). We call R the Riesz mapping.Theorem 14 (see [10]).
LetΦ, Φn∈S∗(Z), n≥1, be generalized functionals of Z. Then, the sequence (Φn) converges strongly to Φ in S∗(Z) if and only if it satisfies the following:(1)
Φn^(σ)→Φ^(σ) for all σ∈Γ.(2)
There are constantsC≥0 and p≥0 such that(28)supn≥1⁡Φn^σ≤Cλσp,σ∈Γ.


## 3. Clark-Ocone Formula for Generalized Functionals

In this section, we first introduce some fundamental operators on the spaceS∗(Z). And then we establish our Clark-Ocone formula for functionals in S∗(Z).


### 3.1. Annihilation and Creation Operators

Theorem 15.
Letk∈N. Then, there exists a continuous linear operator ak:S∗(Z)→S∗(Z) such that(29)akΦ^σ=1-1σkΦ^σ∪k,σ∈Γ,Φ∈S∗Z.Proof.
For eachΦ∈S∗(Z), by Theorem 11, there exist constants C, p≥0 such that (30)Φ^σ≤Cλσp,σ∈Γ,which means that the function σ↦1-1σkΦ^(σ∪k) satisfies (31)1-1σkΦ^σ∪k≤1-1σkCλσ∪kp=1-1σkC1+kpλσp≤C1+kpλσp,σ∈Γ,which, together with Theorem 11, implies that there exists a unique ΨΦ∈S∗(Z) such that(32)ΨΦ^σ=1-1σkΦ^σ∪k,σ∈Γ.Now, consider the mapping ak:S∗(Z)→S∗(Z) defined by(33)akΦ=ΨΦ,Φ∈S∗Z.It is not hard to verify that ak is a linear operator and satisfies (29). To complete the proof, we still need to show that ak:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).
Letp≥0 and denote by jk:Sp∗(Z)→S∗(Z) the inclusion mapping; namely, jk is the mapping defined by(34)jkΦ=Φ,Φ∈Sp∗Z.Then, the composition mapping ak∘jk is a linear operator from Sp∗(Z) to S∗(Z). For each Φ∈Sp∗(Z), we have (35)∑σ∈Γλσ-2pak∘jkΦ^σ2=∑σ∈Γλσ-2pakΦ^σ2=∑σ∈Γλσ-2p1-1σkΦ^σ∪k2=∑k∉σ∈Γ1+k2pλσ∪k-2pΦ^σ∪k2≤1+k2p∑τ∈Γλτ-2pΦ^τ2,which together with Theorem 12 implies that ak∘jk(Φ)∈Sp∗(Z) and (36)ak∘jkΦ-p≤1+kpΦ-p.Thus, ak∘jk(Sp∗Z)⊂Sp∗(Z) and ak∘jk:Sp∗(Z)→Sp∗(Z) is a bounded operator, which implies that ak∘jk is continuous as an operator from Sp∗(Z) to S∗(Z).
Since the choice of the abovep≥0 is arbitrary, we actually arrive at a conclusion that the composition mapping ak∘jk:Sp∗(Z)→S∗(Z) is continuous for all p≥0. Therefore, ak:S∗(Z)→S∗(Z) is continuous with respect to the inductive limit topology over S∗(Z), which together with Lemma 7 implies that ak:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).Carefully checking the proof of Theorem15, one can find the next result already proven.Theorem 16.
Letk∈N. Then, for each p≥0, Sp∗(Z) keeps invariant under the action of ak, and moreover(37)akΦ-p≤1+kpΦ-p,Φ∈Sp∗Z.With the same arguments, we can prove the next two theorems, which are dual forms of Theorems15 and 16, respectively.Theorem 17.
Letk∈N. Then, there exists a continuous linear operator ak†:S∗(Z)→S∗(Z) such that(38)ak†Φ^σ=1σkΦ^σ∖k,σ∈Γ,Φ∈S∗Z.Proof.
For eachΦ∈S∗(Z), by Theorem 11, there exist constants C, p≥0 such that (39)Φ^σ≤Cλσp,σ∈Γ,which means that the function σ↦1σkΦ^(σ∖k) satisfies (40)1σkΦ^σ∖k≤1σkCλσ∖kp=1σkC1+k-pλσp≤C1+k-pλσp,σ∈Γ,which, together with Theorem 11, implies that there exists a unique ΘΦ∈S∗(Z) such that(41)ΘΦ^σ=1σkΦ^σ∖k,σ∈Γ.Now, consider the mapping ak†:S∗(Z)→S∗(Z) defined by(42)ak†Φ=ΘΦ,Φ∈S∗Z.It is not hard to verify that ak† is a linear operator and satisfies (38). To complete the proof, we still need to show that ak†:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).
Letp≥0 and denote by jk:Sp∗(Z)→S∗(Z) the inclusion mapping. Then, the composition mapping ak†∘jk is a linear operator from Sp∗(Z) to S∗(Z). For each Φ∈Sp∗(Z), we have (43)∑σ∈Γλσ-2pak†∘jkΦ^σ2=∑σ∈Γλσ-2pak†Φ^σ2=∑σ∈Γλσ-2p1σkΦ^σ∖k2=∑k∈σ∈Γ1+k-2pλσ∖k-2pΦ^σ∖k2≤1+k-2p∑τ∈Γλτ-2pΦ^τ2,which together with Theorem 12 implies that ak†∘jk(Φ)∈Sp∗(Z) and (44)ak†∘jkΦ-p≤1+k-pΦ-p.Thus, ak†∘jk(Sp∗(Z))⊂Sp∗(Z) and ak†∘jk:Sp∗(Z)→Sp∗(Z) is a bounded operator, which implies that ak†∘jk is continuous as an operator from Sp∗(Z) to S∗(Z).
Since the choice of the abovep≥0 is arbitrary, we actually arrive at a conclusion that the composition mapping ak†∘jk:Sp∗(Z)→S∗(Z) is continuous for all p≥0. Therefore, ak†:S∗(Z)→S∗(Z) is continuous with respect to the inductive limit topology over S∗(Z), which together with Lemma 7 implies that ak†:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).From the proof of Theorem17, we can easily get the next result concerning the operator ak†.Theorem 18.
Letk∈N. Then, for each p≥0, Sp∗(Z) keeps invariant under the action of ak†, and moreover(45)ak†Φ-p≤1+k-pΦ-p,Φ∈Sp∗Z.Remark 19.
Fork≥0, the corresponding annihilation operator ∂k on L2(Z) and its dual ∂k∗ (known as the creation operator) admit the property (46)∂kZσ=1σkZσ∖k,∂k∗Zσ=1-1σkZσ∪k,σ∈Γ.And moreover, they satisfy the canonical anticommutation relation (CAR) in equal-time (47)∂k∗∂k+∂k∂k∗=I,where I means the identity operator on L2(Z). We refer to [2, 6] and for details about these operators.
The next theorem shows the link betweenak and ∂k, as well as between ak† and ∂k∗.Theorem 20.
Letk≥0. Then, the operators ak and ak† satisfy(48)akR=R∂k,ak†R=R∂k∗,where R is the Riesz mapping as indicated in Remark 13.Proof.
Letη∈L2(Z). Then, for all σ∈Γ, we have (49)akRη^σ=1-1σkη,Zσ∪k=η,∂k∗Zσ=∂kη,Zσ=R∂kη^σ,which implies akRη=R∂kη. It then follows by the arbitrariness of η∈L2(Z) that akR=R∂k. Similarly, we can prove ak†R=R∂k∗.In view of Theorem20, we give the following definition to name the operators ak and ak†.Definition 21.
Fork≥0, the operators ak and ak† are called the annihilation and creation operators on generalized functionals of Z, respectively.Much like the operators{∂k,∂k∗} on L2(Z), the operators {ak,ak†} also satisfy a canonical anticommutation relation (CAR) in equal-time.Theorem 22.
LetI be the identity operator on S∗(Z). Then, for k≥0, it holds that(50)ak†ak+akak†=I.Proof.
LetΦ∈S∗(Z). Then, for any σ∈Γ, it follows from (29) and (38) that(51)ak†akΦ^σ=1σkakΦ^σ∖k=1σkΦ^σ,akak†Φ^σ=1-1σkak†Φ^σ∪k=1-1σkΦ^σ,and thus (52)ak†ak+akak†^Φσ=akak†Φ^σ+akak†Φ^σ=Φ^σ,which implies that (ak†ak+akak†)Φ=Φ. It then follows from the arbitrariness of Φ∈S∗(Z) that ak†ak+akak†=I.


### 3.2. Expectation and Conditional Expectation Operators

For the Riesz mappingR, using Theorem 12, we can prove that Rη∈S0∗(Z) for all η∈L2(Z). In particular, we have R1∈S0∗(Z).Theorem 23.
The mappingE:S∗(Z)→S∗(Z) defined by(53)EΦ=Φ^∅R1,Φ∈S∗Z,is a continuous linear operator from S∗(Z) to itself. And, moreover,(54)EΦ^σ=Φ^∅1,Zσ,σ∈Γ,Φ∈S∗Z.Proof.
Clearly,E:S∗(Z)→S∗(Z) is a linear operator and satisfies (54). Next, let us show that E:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).
Letp≥0 and denote by jk:Sp∗(Z)→S∗(Z) the inclusion mapping. Then, the composition mapping E∘jk is a linear operator from Sp∗(Z) to S∗(Z). For each Φ∈Sp∗(Z), we have (55)∑σ∈Γλσ-2pE∘jkΦ^σ2=∑σ∈Γλσ-2pEΦ^σ2=∑σ∈Γλσ-2pΦ^∅1,Zσ2≤∑σ∈Γλσ-2pΦ^σ2,which together with Theorem 12 implies that E∘jk(Φ)∈Sp∗(Z) and (56)E∘jkΦ-p≤Φ-p.Thus, E∘jk(Sp∗(Z))⊂Sp∗(Z) and E∘jk:Sp∗(Z)→Sp∗(Z) is a bounded operator, which implies that E∘jk is continuous as an operator from Sp∗(Z) to S∗(Z).
Since the choice of the abovep≥0 is arbitrary, we actually arrive at a conclusion that the composition mapping E∘jk:Sp∗(Z)→S∗(Z) is continuous for all p≥0. Therefore, E:S∗(Z)→S∗(Z) is continuous with respect to the inductive limit topology over S∗(Z), which together with Lemma 7 implies that E:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).Definition 24.
The operatorE is called the expectation operator on generalized functionals of Z.Since1∈L2(Z), the expectation E with respect to P is actually a bounded operator from L2(Z) to itself. The next theorem shows the link between the operators E and E, which justifies the above definition.Theorem 25.
It holds thatER=RE, where R is the Riesz mapping.Proof.
For anyξ∈L2(Z) and any σ∈Γ, by a direct computation, we have (57)REξ^σ=Eξ,Zσ=ξ,Z∅1,Zσ=Rξ^∅1,Zσ=ERξ^σ.Thus, ER=RE.Theorem 26.
Letk≥0. Then, there exists a continuous linear operator Ek:S∗(Z)→S∗(Z) such that(58)EkΦ^σ=1ΓkσΦ^σ,σ∈Γ,where Γk=σ∈Γ∣max⁡σ≤k and 1Γk(·) denotes the indicator of Γk.Proof.
We omit the proof because it is quite similar to that of Theorem15.Using Theorems12 and 26, we can easily prove the next theorem, which shows that the operator Ek has a type of contraction property on S∗(Z).Theorem 27.
Letk≥0. Then, for each p≥0, Sp∗(Z) keeps invariant under the action of Ek, and moreover(59)EkΦ-p≤Φ-p,∀Φ∈Sp∗Z.Definition 28.
The operatorsEk, k≥0, are called the conditional expectation operators on generalized functionals of Z.Fork≥0, we set Pk=E·∣Fk, the expectation given Fk, where Fk is the σ-field generated by (Zj;0≤j≤k) as mentioned above. Pk is usually known as a conditional expectation operator on square integrable functionals of Z. The theorem below then justifies Definition 28.Theorem 29.
For eachk≥0, it holds that EkR=RPk, where R is the Riesz mapping.Proof.
Letk≥0. Then, for any ξ∈L2(Z) and any σ∈Γ, by a direct computation, we have (60)RPkξ^σ=Pkξ,Zσ=ξ,PkZσ=1Γkσξ,Zσ=1ΓkσRξ^σ=EkRξ^σ.Thus, EkR=RPk.


### 3.3. Clark-Ocone Formula for Generalized Functionals

In this subsection, we establish our Clark-Ocone formula for generalized functionals ofZ.Theorem 30.
For all generalized functionalsΦ∈S∗(Z), it holds that(61)Φ=EΦ+∑k=0∞Ekak†akΦ,where the series on the right-hand side converges strongly in S∗(Z).Proof.
LetΦ∈S∗(Z) and Ψn=∑k=0nEkak†akΦ for n≥0. Then, for σ∈Γ, by a direct computation, we have(62)Ψn^σ=∑k=0n1Γkσ1σkΦ^σ=0,σ=∅;0,σ≠∅,n<max⁡σ;Φ^σ,σ≠∅,n≥max⁡σ.It then follows that Ψn^(σ)→Φ-EΦ^(σ) for all σ∈Γ as n→∞. On the other hand, by Theorem 11, there are constants C≥0 and p≥0 such that (63)Φ^σ≤Cλσp,σ∈Γ,which together with (62) gives (64)supn≥0⁡Ψn^σ≤Φ^σ≤Cλσp,σ∈Γ.Therefore, by Theorem 14, we know (Ψn) converges strongly to Φ-EΦ in S∗(Z). This completes the proof.Proposition 31.
For eachk≥0, it holds that(65)Ekak†=ak†Ek,Ekak=Ek-1ak,where E-1=E.Proof.
Letk≥0. Then, for all Φ∈S∗(Z) and σ∈Γ, by Theorems 17 and 26, we get (66)Ekak†Φ^σ=1Γkσ1σkΦ^σ∖k=1Γkσ∖k1σkΦ^σ∖k=ak†EkΦ^σ,where equality 1Γk(σ)1σ(k)=1Γk(σ∖k)1σ(k) is used. Thus, Ekak†=ak†Ek holds. Similarly, we can verify Ekak=Ek-1ak.Combining Theorem30 with Proposition 31, we arrive at the next interesting result, which we call the Clark-Ocone formula for generalized functionals of Z.Theorem 32.
For all generalized functionalsΦ∈S∗(Z), it holds that(67)Φ=EΦ+∑k=0∞ak†Ek-1akΦ,where E-1=E and the series on the right-hand side converges strongly in S∗(Z).Remark 33.
As mentioned above,∂k and ∂k∗ are the annihilation and creation operators on L2(Z), respectively, and Pk=E·∣Fk is the conditional expectation operator on L2(Z). It can be verified that(68)∂k∗Pk-1η=ZkPk-1η,∀k≥0,∀η∈L2Z,where P-1=E and Zk is the k-component of the discrete-time normal noise Z. Thus, the Clark-Ocone formula (1) can be rewritten as the following form:(69)ξ=Eξ+∑k=0∞∂k∗Pk-1∂kξ,ξ∈L2Z,where the series on the right-hand side converges in the norm of L2(Z). This observation justifies calling formula (67) the Clark-Ocone formula for generalized functionals of Z.


## 3.1. Annihilation and Creation Operators

Theorem 15.
Letk∈N. Then, there exists a continuous linear operator ak:S∗(Z)→S∗(Z) such that(29)akΦ^σ=1-1σkΦ^σ∪k,σ∈Γ,Φ∈S∗Z.Proof.
For eachΦ∈S∗(Z), by Theorem 11, there exist constants C, p≥0 such that (30)Φ^σ≤Cλσp,σ∈Γ,which means that the function σ↦1-1σkΦ^(σ∪k) satisfies (31)1-1σkΦ^σ∪k≤1-1σkCλσ∪kp=1-1σkC1+kpλσp≤C1+kpλσp,σ∈Γ,which, together with Theorem 11, implies that there exists a unique ΨΦ∈S∗(Z) such that(32)ΨΦ^σ=1-1σkΦ^σ∪k,σ∈Γ.Now, consider the mapping ak:S∗(Z)→S∗(Z) defined by(33)akΦ=ΨΦ,Φ∈S∗Z.It is not hard to verify that ak is a linear operator and satisfies (29). To complete the proof, we still need to show that ak:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).
Letp≥0 and denote by jk:Sp∗(Z)→S∗(Z) the inclusion mapping; namely, jk is the mapping defined by(34)jkΦ=Φ,Φ∈Sp∗Z.Then, the composition mapping ak∘jk is a linear operator from Sp∗(Z) to S∗(Z). For each Φ∈Sp∗(Z), we have (35)∑σ∈Γλσ-2pak∘jkΦ^σ2=∑σ∈Γλσ-2pakΦ^σ2=∑σ∈Γλσ-2p1-1σkΦ^σ∪k2=∑k∉σ∈Γ1+k2pλσ∪k-2pΦ^σ∪k2≤1+k2p∑τ∈Γλτ-2pΦ^τ2,which together with Theorem 12 implies that ak∘jk(Φ)∈Sp∗(Z) and (36)ak∘jkΦ-p≤1+kpΦ-p.Thus, ak∘jk(Sp∗Z)⊂Sp∗(Z) and ak∘jk:Sp∗(Z)→Sp∗(Z) is a bounded operator, which implies that ak∘jk is continuous as an operator from Sp∗(Z) to S∗(Z).
Since the choice of the abovep≥0 is arbitrary, we actually arrive at a conclusion that the composition mapping ak∘jk:Sp∗(Z)→S∗(Z) is continuous for all p≥0. Therefore, ak:S∗(Z)→S∗(Z) is continuous with respect to the inductive limit topology over S∗(Z), which together with Lemma 7 implies that ak:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).Carefully checking the proof of Theorem15, one can find the next result already proven.Theorem 16.
Letk∈N. Then, for each p≥0, Sp∗(Z) keeps invariant under the action of ak, and moreover(37)akΦ-p≤1+kpΦ-p,Φ∈Sp∗Z.With the same arguments, we can prove the next two theorems, which are dual forms of Theorems15 and 16, respectively.Theorem 17.
Letk∈N. Then, there exists a continuous linear operator ak†:S∗(Z)→S∗(Z) such that(38)ak†Φ^σ=1σkΦ^σ∖k,σ∈Γ,Φ∈S∗Z.Proof.
For eachΦ∈S∗(Z), by Theorem 11, there exist constants C, p≥0 such that (39)Φ^σ≤Cλσp,σ∈Γ,which means that the function σ↦1σkΦ^(σ∖k) satisfies (40)1σkΦ^σ∖k≤1σkCλσ∖kp=1σkC1+k-pλσp≤C1+k-pλσp,σ∈Γ,which, together with Theorem 11, implies that there exists a unique ΘΦ∈S∗(Z) such that(41)ΘΦ^σ=1σkΦ^σ∖k,σ∈Γ.Now, consider the mapping ak†:S∗(Z)→S∗(Z) defined by(42)ak†Φ=ΘΦ,Φ∈S∗Z.It is not hard to verify that ak† is a linear operator and satisfies (38). To complete the proof, we still need to show that ak†:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).
Letp≥0 and denote by jk:Sp∗(Z)→S∗(Z) the inclusion mapping. Then, the composition mapping ak†∘jk is a linear operator from Sp∗(Z) to S∗(Z). For each Φ∈Sp∗(Z), we have (43)∑σ∈Γλσ-2pak†∘jkΦ^σ2=∑σ∈Γλσ-2pak†Φ^σ2=∑σ∈Γλσ-2p1σkΦ^σ∖k2=∑k∈σ∈Γ1+k-2pλσ∖k-2pΦ^σ∖k2≤1+k-2p∑τ∈Γλτ-2pΦ^τ2,which together with Theorem 12 implies that ak†∘jk(Φ)∈Sp∗(Z) and (44)ak†∘jkΦ-p≤1+k-pΦ-p.Thus, ak†∘jk(Sp∗(Z))⊂Sp∗(Z) and ak†∘jk:Sp∗(Z)→Sp∗(Z) is a bounded operator, which implies that ak†∘jk is continuous as an operator from Sp∗(Z) to S∗(Z).
Since the choice of the abovep≥0 is arbitrary, we actually arrive at a conclusion that the composition mapping ak†∘jk:Sp∗(Z)→S∗(Z) is continuous for all p≥0. Therefore, ak†:S∗(Z)→S∗(Z) is continuous with respect to the inductive limit topology over S∗(Z), which together with Lemma 7 implies that ak†:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).From the proof of Theorem17, we can easily get the next result concerning the operator ak†.Theorem 18.
Letk∈N. Then, for each p≥0, Sp∗(Z) keeps invariant under the action of ak†, and moreover(45)ak†Φ-p≤1+k-pΦ-p,Φ∈Sp∗Z.Remark 19.
Fork≥0, the corresponding annihilation operator ∂k on L2(Z) and its dual ∂k∗ (known as the creation operator) admit the property (46)∂kZσ=1σkZσ∖k,∂k∗Zσ=1-1σkZσ∪k,σ∈Γ.And moreover, they satisfy the canonical anticommutation relation (CAR) in equal-time (47)∂k∗∂k+∂k∂k∗=I,where I means the identity operator on L2(Z). We refer to [2, 6] and for details about these operators.
The next theorem shows the link betweenak and ∂k, as well as between ak† and ∂k∗.Theorem 20.
Letk≥0. Then, the operators ak and ak† satisfy(48)akR=R∂k,ak†R=R∂k∗,where R is the Riesz mapping as indicated in Remark 13.Proof.
Letη∈L2(Z). Then, for all σ∈Γ, we have (49)akRη^σ=1-1σkη,Zσ∪k=η,∂k∗Zσ=∂kη,Zσ=R∂kη^σ,which implies akRη=R∂kη. It then follows by the arbitrariness of η∈L2(Z) that akR=R∂k. Similarly, we can prove ak†R=R∂k∗.In view of Theorem20, we give the following definition to name the operators ak and ak†.Definition 21.
Fork≥0, the operators ak and ak† are called the annihilation and creation operators on generalized functionals of Z, respectively.Much like the operators{∂k,∂k∗} on L2(Z), the operators {ak,ak†} also satisfy a canonical anticommutation relation (CAR) in equal-time.Theorem 22.
LetI be the identity operator on S∗(Z). Then, for k≥0, it holds that(50)ak†ak+akak†=I.Proof.
LetΦ∈S∗(Z). Then, for any σ∈Γ, it follows from (29) and (38) that(51)ak†akΦ^σ=1σkakΦ^σ∖k=1σkΦ^σ,akak†Φ^σ=1-1σkak†Φ^σ∪k=1-1σkΦ^σ,and thus (52)ak†ak+akak†^Φσ=akak†Φ^σ+akak†Φ^σ=Φ^σ,which implies that (ak†ak+akak†)Φ=Φ. It then follows from the arbitrariness of Φ∈S∗(Z) that ak†ak+akak†=I.


## 3.2. Expectation and Conditional Expectation Operators

For the Riesz mappingR, using Theorem 12, we can prove that Rη∈S0∗(Z) for all η∈L2(Z). In particular, we have R1∈S0∗(Z).Theorem 23.
The mappingE:S∗(Z)→S∗(Z) defined by(53)EΦ=Φ^∅R1,Φ∈S∗Z,is a continuous linear operator from S∗(Z) to itself. And, moreover,(54)EΦ^σ=Φ^∅1,Zσ,σ∈Γ,Φ∈S∗Z.Proof.
Clearly,E:S∗(Z)→S∗(Z) is a linear operator and satisfies (54). Next, let us show that E:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).
Letp≥0 and denote by jk:Sp∗(Z)→S∗(Z) the inclusion mapping. Then, the composition mapping E∘jk is a linear operator from Sp∗(Z) to S∗(Z). For each Φ∈Sp∗(Z), we have (55)∑σ∈Γλσ-2pE∘jkΦ^σ2=∑σ∈Γλσ-2pEΦ^σ2=∑σ∈Γλσ-2pΦ^∅1,Zσ2≤∑σ∈Γλσ-2pΦ^σ2,which together with Theorem 12 implies that E∘jk(Φ)∈Sp∗(Z) and (56)E∘jkΦ-p≤Φ-p.Thus, E∘jk(Sp∗(Z))⊂Sp∗(Z) and E∘jk:Sp∗(Z)→Sp∗(Z) is a bounded operator, which implies that E∘jk is continuous as an operator from Sp∗(Z) to S∗(Z).
Since the choice of the abovep≥0 is arbitrary, we actually arrive at a conclusion that the composition mapping E∘jk:Sp∗(Z)→S∗(Z) is continuous for all p≥0. Therefore, E:S∗(Z)→S∗(Z) is continuous with respect to the inductive limit topology over S∗(Z), which together with Lemma 7 implies that E:S∗(Z)→S∗(Z) is continuous with respect to the strong topology over S∗(Z).Definition 24.
The operatorE is called the expectation operator on generalized functionals of Z.Since1∈L2(Z), the expectation E with respect to P is actually a bounded operator from L2(Z) to itself. The next theorem shows the link between the operators E and E, which justifies the above definition.Theorem 25.
It holds thatER=RE, where R is the Riesz mapping.Proof.
For anyξ∈L2(Z) and any σ∈Γ, by a direct computation, we have (57)REξ^σ=Eξ,Zσ=ξ,Z∅1,Zσ=Rξ^∅1,Zσ=ERξ^σ.Thus, ER=RE.Theorem 26.
Letk≥0. Then, there exists a continuous linear operator Ek:S∗(Z)→S∗(Z) such that(58)EkΦ^σ=1ΓkσΦ^σ,σ∈Γ,where Γk=σ∈Γ∣max⁡σ≤k and 1Γk(·) denotes the indicator of Γk.Proof.
We omit the proof because it is quite similar to that of Theorem15.Using Theorems12 and 26, we can easily prove the next theorem, which shows that the operator Ek has a type of contraction property on S∗(Z).Theorem 27.
Letk≥0. Then, for each p≥0, Sp∗(Z) keeps invariant under the action of Ek, and moreover(59)EkΦ-p≤Φ-p,∀Φ∈Sp∗Z.Definition 28.
The operatorsEk, k≥0, are called the conditional expectation operators on generalized functionals of Z.Fork≥0, we set Pk=E·∣Fk, the expectation given Fk, where Fk is the σ-field generated by (Zj;0≤j≤k) as mentioned above. Pk is usually known as a conditional expectation operator on square integrable functionals of Z. The theorem below then justifies Definition 28.Theorem 29.
For eachk≥0, it holds that EkR=RPk, where R is the Riesz mapping.Proof.
Letk≥0. Then, for any ξ∈L2(Z) and any σ∈Γ, by a direct computation, we have (60)RPkξ^σ=Pkξ,Zσ=ξ,PkZσ=1Γkσξ,Zσ=1ΓkσRξ^σ=EkRξ^σ.Thus, EkR=RPk.


## 3.3. Clark-Ocone Formula for Generalized Functionals

In this subsection, we establish our Clark-Ocone formula for generalized functionals ofZ.Theorem 30.
For all generalized functionalsΦ∈S∗(Z), it holds that(61)Φ=EΦ+∑k=0∞Ekak†akΦ,where the series on the right-hand side converges strongly in S∗(Z).Proof.
LetΦ∈S∗(Z) and Ψn=∑k=0nEkak†akΦ for n≥0. Then, for σ∈Γ, by a direct computation, we have(62)Ψn^σ=∑k=0n1Γkσ1σkΦ^σ=0,σ=∅;0,σ≠∅,n<max⁡σ;Φ^σ,σ≠∅,n≥max⁡σ.It then follows that Ψn^(σ)→Φ-EΦ^(σ) for all σ∈Γ as n→∞. On the other hand, by Theorem 11, there are constants C≥0 and p≥0 such that (63)Φ^σ≤Cλσp,σ∈Γ,which together with (62) gives (64)supn≥0⁡Ψn^σ≤Φ^σ≤Cλσp,σ∈Γ.Therefore, by Theorem 14, we know (Ψn) converges strongly to Φ-EΦ in S∗(Z). This completes the proof.Proposition 31.
For eachk≥0, it holds that(65)Ekak†=ak†Ek,Ekak=Ek-1ak,where E-1=E.Proof.
Letk≥0. Then, for all Φ∈S∗(Z) and σ∈Γ, by Theorems 17 and 26, we get (66)Ekak†Φ^σ=1Γkσ1σkΦ^σ∖k=1Γkσ∖k1σkΦ^σ∖k=ak†EkΦ^σ,where equality 1Γk(σ)1σ(k)=1Γk(σ∖k)1σ(k) is used. Thus, Ekak†=ak†Ek holds. Similarly, we can verify Ekak=Ek-1ak.Combining Theorem30 with Proposition 31, we arrive at the next interesting result, which we call the Clark-Ocone formula for generalized functionals of Z.Theorem 32.
For all generalized functionalsΦ∈S∗(Z), it holds that(67)Φ=EΦ+∑k=0∞ak†Ek-1akΦ,where E-1=E and the series on the right-hand side converges strongly in S∗(Z).Remark 33.
As mentioned above,∂k and ∂k∗ are the annihilation and creation operators on L2(Z), respectively, and Pk=E·∣Fk is the conditional expectation operator on L2(Z). It can be verified that(68)∂k∗Pk-1η=ZkPk-1η,∀k≥0,∀η∈L2Z,where P-1=E and Zk is the k-component of the discrete-time normal noise Z. Thus, the Clark-Ocone formula (1) can be rewritten as the following form:(69)ξ=Eξ+∑k=0∞∂k∗Pk-1∂kξ,ξ∈L2Z,where the series on the right-hand side converges in the norm of L2(Z). This observation justifies calling formula (67) the Clark-Ocone formula for generalized functionals of Z.


## 4. Applications

In the final section, we show some applications of our Clark-Ocone formula.Forp≥0 and Φ, Ψ∈S∗(Z), we define 〈Φ,Ψ〉-p as(70)Φ,Ψ-p=∑σ∈Γλσ-2pΦ^σΨ^σ¯provided the series on the right-hand side absolutely converges. Note that if Φ, Ψ∈Sp∗(Z), then by Theorem 12 the series in (70) absolutely converges, and hence 〈Φ,Ψ〉-p makes sense, and in particular(71)Φ,Φ-p=Φ-p2.Definition 34.
For generalized functionalsΦ, Ψ∈S∗(Z), their p-covariant covp⁡(Φ,Ψ), p≥0, is defined as(72)covp⁡Φ,Ψ=Φ-EΦ,Ψ-EΨ-pprovided the right-hand side makes sense.By convention,varp⁡(Φ)≡covp⁡(Φ,Φ) is called the p-variant of generalized functional Φ. Clearly, varp⁡(Φ)=Φ-EΦ-p2 if Φ∈Sp∗(Z).Theorem 35.
LetΦ, Ψ∈Sp∗(Z) for some p≥0. Then, their p-covariant covp⁡(Φ,Ψ) makes sense, and moreover(73)covp⁡Φ,Ψ=∑k=0∞Ekak†akΦ,Ekak†akΨ-p.Proof.
By Theorem12, the series on the right-hand side of (73) converges absolutely. On the other hand, by Theorem 30, we have (74)covp⁡Φ,Ψ=∑σ∈Γλσ-2pΦ-EΦ^σΨ-EΨ^σ¯=∑σ∈Γλσ-2p∑k=0∞1Γkσ1σkΦ^σ∑k=0∞1Γkσ1σkΨ^σ¯,which together with the fact (75)1Γjσ1σj1Γkσ1σk=0,j≠k,j,k≥0,σ∈Γ,gives (76)covp⁡Φ,Ψ=∑σ∈Γλσ-2p∑k=0∞1Γkσ1σkΦ^σΨ^σ¯=∑k=0∞∑σ∈Γλσ-2p1Γkσ1σkΦ^σ1Γkσ1σkΨ^σ¯=∑k=0∞∑σ∈Γλσ-2pEkak†akΦ^σEkak†akΨ^σ¯=∑k=0∞Ekak†akΦ,Ekak†akΨ-p.This completes the proof.Theorem35 sets up covariant identities for generalized functionals of Z. The next theorem then gives meaningful upper bounds to variants of generalized functionals of Z.Theorem 36.
LetΦ∈Sp∗(Z) for some p≥0. Then, its p-variant varp⁡(Φ) makes sense, and moreover(77)varp⁡Φ≤∑k=0∞ak†akΦ-p2.Proof.
By Theorems16, 18, and 27, we know that Ekak†akΦ belongs to Sp∗(Z) and (78)Ekak†akΦ-p≤ak†akΦ-p,k≥0.This together with (71) and (73) yields (79)varp⁡Φ=∑k=0∞Ekak†akΦ-p2≤∑k=0∞ak†akΦ-p2.This completes the proof.A sequenceu=(uk) of generalized functionals in S∗(Z) is said to be (Ek)-predictable if(80)uk=Ek-1uk,k≥0.It is said to be (ak†)-integrable if the series ∑k=0∞ak†uk converges strongly in S∗(Z). In that case, we call ∑k=0∞ak†uk the generalized stochastic integral of u with respect to (ak†) and write(81)Iu=∑k=0∞ak†uk.Theorem 37.
LetΦ∈S∗(Z). Then, the sequence u=(Ek-1akΦ)k≥0 of generalized functionals in S∗(Z) is (Ek)-predictable and (ak†)-integrable, and moreover(82)Φ=EΦ+Iu.Proof.
This is an immediate consequence of Theorem32.Remark 38.
A generalized functional ofZ, or, in other words, a generalized functional in S∗(Z), can be interpreted as a generalized random variable on the probability space (Ω,F,P). Accordingly, a sequence of generalized functionals of Z can be viewed as a generalized stochastic process. Theorem 37 then shows that each generalized random variable on (Ω,F,P) can be represented as the generalized stochastic integral of an (Ek)-predictable generalized stochastic process with respect to (ak†).



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