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. Let be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of . Then, we use the Fock transform to define some fundamental operators on generalized functionals of and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of .
1. Introduction
One of the important theorems in Privault’s discrete-time chaotic calculus [1, 2] is its Clark-Ocone formula, which readswhere is a discrete-time normal noise, is the space of square integrable functionals of , is the -field generated by , is the annihilation operator on , and the series on the right-hand side converges in the norm of .
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 to construct a nuclear space such that is densely contained in . Thus, by identifying with its dual, one can get a Gel’fand triplewhere is the dual of , which is endowed with the strong topology, which cannot be induced by any norm [5]. As usual, is called the testing functional space of , while is called the generalized functional space of . It turns out [6] that the generalized functional space can accommodate many quantities of theoretical interest that cannot be covered by .
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 . Our main work is as follows.
We first prove a result concerning the regularity of generalized functionals in in Section 2. Then, in Section 3, we use the Fock transform [6] to define some fundamental operators on 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 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 .
Throughout this paper, designates the set of all nonnegative integers and the finite power set of ; namely, where means the cardinality of as a set. If and , then we simply write for . Similarly, we use .
2. Generalized Functionals of Discrete-Time Normal Noises
In all the following sections, we always assume that is a given probability space. We use to mean the expectation with respect to . As usual, denotes the Hilbert space of square integrable complex-valued measurable functions on . We use and to mean the inner product and norm of , respectively. By convention, is conjugate-linear in its first argument and linear in its second argument.
2.1. Discrete-Time Normal Noises
A sequence of integrable random variables on is called a discrete-time normal noise if it satisfies(i) for ;(ii) for .
Here, , for and means the conditional expectation given .
Example 1. Let be an independent sequence of random variables on with Write and for . Then, one can immediately see that (i) for ;(ii) for . Thus, is a discrete-time normal noise. Note that, by letting be the partial sum sequence of , one gets the classical random walk.
For a discrete-time normal noise on , one can construct a corresponding family of random variables on in the following manner:We call the canonical functional system of .
Lemma 2 (see [1, 2, 7]). Let be a discrete-time normal noise on . Then, its canonical functional system forms a countable orthonormal system in .
Let be the -field over generated by a discrete-time normal noise on . Then, the canonical functional system is also a countable orthonormal system in the space of square integrable complex-valued measurable functions on .
In the literature, -measurable functions on are also known as functionals of . Thus, elements of are naturally called square integrable functionals of .
Definition 3. A discrete-time normal noise on is said to have the chaotic representation property if its canonical functional system is total in , where .
Thus, if a discrete-time normal noise has the chaotic representation property, then its canonical functional system is actually an orthonormal basis of .
2.2. Generalized Functionals
From now on, we always assume that is a given discrete-time normal noise on that has the chaotic representation property.
For brevity, we use to denote the space of square integrable functionals of ; namely, where . For , we denote by the -field generated by ; namely, We note that shares the same inner product and norm with , and moreover the canonical functional system of forms a countable orthonormal basis for , which we call the canonical orthonormal basis of .
Lemma 4 (see [4]). Let be the -valued function on given byThen, for , the positive term series converges and moreover
Using the -valued function defined by (8), we can construct a chain of Hilbert spaces consisting of functionals of as follows. For , we putand defineIt is not hard to check that, with as the inner product, becomes a Hilbert space. We write for . Clearly, it holds that
Lemma 5 (see [4, 6]). For , one has and moreover the system forms an orthonormal basis for .
It is easy to see that for all . This implies that and whenever . Thus, we actually get a chain of Hilbert spaces of functionals of :We now putand endow it with the topology generated by the norm sequence . Note that, for each , is just the completion of with respect to . Thus, is a countably Hilbert space [5, 8]. The next lemma, however, shows that even has a much better property.
Lemma 6 (see [4, 6]). The space is a nuclear space; namely, for any , there exists such that the inclusion mapping defined by is a Hilbert-Schmidt operator.
For , we denote by the dual of and the norm of . Then, and whenever . 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 be the dual of and endow it with the strong topology. Then,and moreover the inductive limit topology over given by space sequence coincides with the strong topology.
We mention that, by identifying with its dual, one comes to a Gel’fand triplewhich we refer to as the Gel’fand triple associated with the discrete-time normal noise .
Theorem 8 (see [6]). The system is contained in and moreover it forms a basis for in the sense thatwhere is the inner product of and the series converges in the topology of .
Definition 9 (see [4, 6]). Elements of are called generalized functionals of , while elements of are called testing functionals of .
Thus, and can be accordingly called the generalized functional space and the testing functional space of , respectively. It turns out [6] that can accommodate many quantities of theoretical interest that cannot be covered by .
In the following, we denote by the canonical bilinear form on given byNote that is different from the inner product of .
Definition 10 (see [6]). For , its Fock transform is the function on given bywhere is the canonical bilinear form.
It is easy to verify that, for , , if and only if . 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 . Then, is the Fock transform of an element of if and only if it satisfiesfor some constants and . In that case, for , one hasand in particular .
The theorem below describes the regularity of generalized functionals of via their Fock transforms.
Theorem 12. Let and . Then, if and only ifIn that case, the norm of in satisfies
Proof. The “Only If” Part. By the well-known Riesz representation theorem [9], there exists a unique such that and Thus, which implies (22) and (23).
The “If” Part. For each , using Theorem 8, we have Thus, is a bounded functional on the space , which implies since is dense in .
Remark 13. There exists a continuous linear mapping such thatwhere is the canonical bilinear form on . We call the Riesz mapping.
Theorem 14 (see [10]). Let , , , be generalized functionals of . Then, the sequence converges strongly to in if and only if it satisfies the following:(1) for all .(2)There are constants and such that
3. Clark-Ocone Formula for Generalized Functionals
In this section, we first introduce some fundamental operators on the space . And then we establish our Clark-Ocone formula for functionals in .
3.1. Annihilation and Creation Operators
Theorem 15. Let . Then, there exists a continuous linear operator such that
Proof. For each , by Theorem 11, there exist constants , such that which means that the function satisfies which, together with Theorem 11, implies that there exists a unique such thatNow, consider the mapping defined byIt is not hard to verify that is a linear operator and satisfies (29). To complete the proof, we still need to show that is continuous with respect to the strong topology over .
Let and denote by the inclusion mapping; namely, is the mapping defined byThen, the composition mapping is a linear operator from to . For each , we have which together with Theorem 12 implies that and Thus, and is a bounded operator, which implies that is continuous as an operator from to .
Since the choice of the above is arbitrary, we actually arrive at a conclusion that the composition mapping is continuous for all . Therefore, is continuous with respect to the inductive limit topology over , which together with Lemma 7 implies that is continuous with respect to the strong topology over .
Carefully checking the proof of Theorem 15, one can find the next result already proven.
Theorem 16. Let . Then, for each , keeps invariant under the action of , and moreover
With the same arguments, we can prove the next two theorems, which are dual forms of Theorems 15 and 16, respectively.
Theorem 17. Let . Then, there exists a continuous linear operator such that
Proof. For each , by Theorem 11, there exist constants , such that which means that the function satisfies which, together with Theorem 11, implies that there exists a unique such thatNow, consider the mapping defined byIt is not hard to verify that is a linear operator and satisfies (38). To complete the proof, we still need to show that is continuous with respect to the strong topology over .
Let and denote by the inclusion mapping. Then, the composition mapping is a linear operator from to . For each , we have which together with Theorem 12 implies that and Thus, and is a bounded operator, which implies that is continuous as an operator from to .
Since the choice of the above is arbitrary, we actually arrive at a conclusion that the composition mapping is continuous for all . Therefore, is continuous with respect to the inductive limit topology over , which together with Lemma 7 implies that is continuous with respect to the strong topology over .
From the proof of Theorem 17, we can easily get the next result concerning the operator .
Theorem 18. Let . Then, for each , keeps invariant under the action of , and moreover
Remark 19. For , the corresponding annihilation operator on and its dual (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 . We refer to [2, 6] and for details about these operators.
The next theorem shows the link between and , as well as between and .
Theorem 20. Let . Then, the operators and satisfywhere is the Riesz mapping as indicated in Remark 13.
Proof. Let . Then, for all , we have which implies . It then follows by the arbitrariness of that . Similarly, we can prove .
In view of Theorem 20, we give the following definition to name the operators and .
Definition 21. For , the operators and are called the annihilation and creation operators on generalized functionals of , respectively.
Much like the operators on , the operators also satisfy a canonical anticommutation relation (CAR) in equal-time.
Theorem 22. Let be the identity operator on . Then, for , it holds that
Proof. Let . Then, for any , it follows from (29) and (38) thatand thus which implies that . It then follows from the arbitrariness of that .
3.2. Expectation and Conditional Expectation Operators
For the Riesz mapping , using Theorem 12, we can prove that for all . In particular, we have .
Theorem 23. The mapping defined byis a continuous linear operator from to itself. And, moreover,
Proof. Clearly, is a linear operator and satisfies (54). Next, let us show that is continuous with respect to the strong topology over .
Let and denote by the inclusion mapping. Then, the composition mapping is a linear operator from to . For each , we have which together with Theorem 12 implies that and Thus, and is a bounded operator, which implies that is continuous as an operator from to .
Since the choice of the above is arbitrary, we actually arrive at a conclusion that the composition mapping is continuous for all . Therefore, is continuous with respect to the inductive limit topology over , which together with Lemma 7 implies that is continuous with respect to the strong topology over .
Definition 24. The operator is called the expectation operator on generalized functionals of .
Since , the expectation with respect to is actually a bounded operator from to itself. The next theorem shows the link between the operators and , which justifies the above definition.
Theorem 25. It holds that , where is the Riesz mapping.
Proof. For any and any , by a direct computation, we have Thus, .
Theorem 26. Let . Then, there exists a continuous linear operator such thatwhere and denotes the indicator of .
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 has a type of contraction property on .
Theorem 27. Let . Then, for each , keeps invariant under the action of , and moreover
Definition 28. The operators , , are called the conditional expectation operators on generalized functionals of .
For , we set , the expectation given , where is the -field generated by as mentioned above. is usually known as a conditional expectation operator on square integrable functionals of . The theorem below then justifies Definition 28.
Theorem 29. For each , it holds that , where is the Riesz mapping.
Proof. Let . Then, for any and any , by a direct computation, we have Thus, .
3.3. 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 , it holds thatwhere the series on the right-hand side converges strongly in .
Proof. Let and for . Then, for , by a direct computation, we haveIt then follows that for all as . On the other hand, by Theorem 11, there are constants and such that which together with (62) gives Therefore, by Theorem 14, we know converges strongly to in . This completes the proof.
Proposition 31. For each , it holds thatwhere .
Proof. Let . Then, for all and , by Theorems 17 and 26, we get where equality is used. Thus, holds. Similarly, we can verify .
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 , it holds thatwhere and the series on the right-hand side converges strongly in .
Remark 33. As mentioned above, and are the annihilation and creation operators on , respectively, and is the conditional expectation operator on . It can be verified thatwhere and is the -component of the discrete-time 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 . This observation justifies calling formula (67) the Clark-Ocone formula for generalized functionals of .
4. Applications
In the final section, we show some applications of our Clark-Ocone formula.
For and , , we define asprovided the series on the right-hand side absolutely converges. Note that if , , then by Theorem 12 the series in (70) absolutely converges, and hence makes sense, and in particular
Definition 34. For generalized functionals , , their -covariant , , is defined asprovided the right-hand side makes sense.
By convention, is called the -variant of generalized functional . Clearly, if .
Theorem 35. Let , for some . Then, their -covariant 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 gives 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 .
Theorem 36. Let for some . Then, its -variant makes sense, and moreover
Proof. By Theorems 16, 18, and 27, we know that belongs to and This together with (71) and (73) yields This completes the proof.
A sequence of generalized functionals in is said to be -predictable ifIt is said to be -integrable if the series converges strongly in . In that case, we call the generalized stochastic integral of with respect to and write
Theorem 37. Let . Then, the sequence of generalized functionals in is -predictable and -integrable, and moreover
Proof. This is an immediate consequence of Theorem 32.
Remark 38. A generalized functional of , or, in other words, a generalized functional in , can be interpreted as a generalized random variable on the probability space . 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 can be represented as the generalized stochastic integral of an -predictable generalized stochastic process with respect to .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 11461061).