Abstract and Applied Analysis

Volume 2013 (2013), Article ID 152043, 12 pages

http://dx.doi.org/10.1155/2013/152043

Research Article

Interpolation Methods for Stochastic Processes Spaces

E. Nursultanov¹ and T. Aubakirov²

¹Lomonosov Moscow State University (Kazakh Branch) and Gumilyov Eurasian National University, Munatpasova 7, Astana 010010, Kazakhstan

²Autonomous Organization of Education Nazarbayev Intellectual Schools, Astana 010010, Kazakhstan

Received 2 September 2013; Revised 4 November 2013; Accepted 18 November 2013

Academic Editor: Henryk Hudzik

Copyright © 2013 E. Nursultanov and T. Aubakirov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The scales of classes of stochastic processes are introduced. New interpolation theorems and boundedness of some transforms of stochastic processes are proved. Interpolation method for generously monotonous processes is entered. Conditions and statements of interpolation theorems concern the fixed stochastic process, which differs from the classical results.

1. Introduction

Interpolation methods of functional spaces are one of the basic tools to get inequalities in parametrical spaces. These methods are widely applied in the theory of stochastic processes (see [1–5] and other).

In this paper classes of stochastic processes are considered, which, in some sense, are analogues of the net spaces which were investigated in [6–8].

Assume that is a complete probability space. A family of -algebras such that is called a filtration.

Let be a filtration, a sequence of random variables measurable function with respect to the -algebra . Then we say that the set is a stochastic process.

Let be a system of sets satisfying the condition . We say that a stochastic process is defined on a system if . For a stochastic process , which is defined on a system , we define the sequence of numbers , where

We call this sequence a majorant of a process on a system of sets .

Let us give some examples of a choice of a system of sets : , in this case the sequence is a sequence of averages of a process ; for it is a majorant of sequence of conditional averages ; and for it is a majorant of a process . The following cases are interesting: (1) , where is the fixed stopping time, and ; (2) , where , and is the sequence of the stopping times.

We consider the classes of stochastic processes defined on , which characterize the speed of convergence of sequence to zero.

By , ,   we denote the set of all stochastic processes , defined on for which

if and

if .

Let us denote

for , and

for , where

We consider that

where is the integer part of the number . In particular .

A random variable , which takes values in the set , is called the Markov time of the filtration , if for any . The Markov time , for which (a.p.) [9], is called the stopping time.

Let be a stochastic process and be the Markov time. By we denote the stopped process , where and is the characteristic function of the set .

We assume also that

.

The spaces are spaces of converging stochastic processes, where parameters , and characterize the speed and the metrics, in which a given process converges.

In this paper we prove a Marcinkiewicz-type interpolation theorem for the introduced space. An interpolation method, essentially related to the properties of the Markov stopping times, is introduced. In the last paragraph the given interpolation method is applied to Besov type space with variable approximation properties. Part of the results were announced in [10].

We write (or ) if (or ) for some positive constant independent of appropriate quantities involved in the expressions and . Notation means that and .

2. Properties of the Spaces and

We say [9] that stochastic process is a martingale if for every the following conditions hold: (1) ; (2) (a.p.). If instead of property (2) it is assumed that , then we say that a process is a submartingale (supermartingale).

Definition 1. Let be a fixed system of sets, be a stochastic process defined on . We say that a process belongs to the class if there exists a constant such that for every and for every

This inequality implies that for every . The class contains martingales, nonnegative submartingales, and nonpositive supermartingales. The stochastic process from we call generalized monotone.

Lemma 2. Let , , and . If , then there exists a random variable such that (a.p.).

Proof. Let be the Marcinkiewicz-Lorentz space and . Using the equivalent norm of spaces (see [6]) and measurability of function with respect to -algebra , we get the following:

Taking into account that , for , we have .

But , therefore by the Doob theorem ([11]), the process converges almost surely.

Lemma 3. Let ,  ,  , and . Then

where

Proof. The existence of follows from Lemma 2. Further, we have

Therefore, using Lemma 8, we obtain

The reverse inequality follows from the expression:

The lemma is proved.

Lemma 4. Let . Then

(1)for ,

(2)for , ,

where ,   depend only on the indicated parameters.

Proof. Let us prove inequalities (16), (18). The proof of inequalities (17), (19) is similar. Let . By Minkowski’s inequality and by the generalized monotonicity of a process we get the following:

To prove the second statement it is enough to show that and apply the first statement. Since , we have the following:

Remark 5. Properties of the spaces given in Lemma 4 show that the second parameter is weak with respect to the first . These properties of the spaces are important in the interpolation.

Lemma 6. Let ,  . If , then for

and for

Proof. Using the generalized monotonicity of a process , we have the following:

One can prove the reverse estimate in a similar way.

Lemma 7 (Hölder inequality). Let ,   and , . If stochastic processes and , then and

Proof. Since is measurable with respect to an algebra , we have and hence

We will need the following Hardy-type inequalities.

Lemma 8. Let , , , , and ; then for a nonnegative sequence the following inequalities hold:

3. Interpolation Method for Stochastic Processes

Let be a transform that transforms a stochastic process , which is defined on the system , to the stochastic process , which is defined on the system . We say that the transform is quasilinear if there exists a constant such that for any the following inequality holds almost surely:

It is known ([9]) that if a process is a martingale (submartingale), then the process is also a martingale (submartingale).

Denote and .

The transforms and of the stochastic process are examples of quasilinear transforms.

Let be a pair of quasinormed own subspaces of linear Hausdorff stochastic processes spaces , which is defined on a probability space with a filtration . Obviously, this pair is compatible pair and hence the scale of interpolation spaces is defined with respect to the real method ([12]).

Moreover, let for

and for

where

is the Peetre functional.

Let be a sequence of stopping times with respect to a filtration and be a pair of quasinormed own subspaces . Let and . We define the following:

Here the infimum is taken over all stopping times from . Moreover for

and for

Theorem 9. Let be two compatible pairs of stochastic processes and let be some fixed family of Markov times with respect to a filtration . If is a quasilinear map for stochastic processes and

for all stopping times , then

where the constant is from the definition of quasilinearity of the operator .

Proof. Consider the following:

The theorem is proved.

Lemma 10. Let and be stopping times. Then for

The proof is similar to the proof of the Lemma 4.

4. Interpolation Properties of the Spaces

Theorem 11. Let , , , and be the stopping times. Then for any stochastic process ,

where the constant depends only on parameters ,  ,  , .

If , then

where the constant also depends only on parametres ,.

Proof. Let be any representation of a process , where , . To prove the first statement of the theorem, we use the following inequality: .

For any we have the following:

By putting , we get . Therefore, using (22) and Lemma 8, we have the following:

Let us prove the second statement of the theorem. Let . By using Lemmas 8 and 3 we have the following:

Let , then taking into account that ,  , we obtain the following:

Thus, we obtain the following:

Further, we have the following:

By using Lemma 8 for , (46), and (47), we have the following:

By applying Minkowski’s inequality to (45) and using Lemma 4, estimates (48), we obtain

Corollary 12. Let , , , , , , , and be a quasilinear transform. If for any the following conditions hold:

then

where depends only on ,  ,  ,  , and .

Taking into account that the measurable function may be considered as a martingale, by corollary we may receive Marcinkiewicz-Calderon interpolation theorem (see [13]).

Corollary 13 (Marcinkiewicz-Calderon theorem). Let ,  , ,  , and . If   is a quasilinear map and

then

Corollary 14. Let be a quasilinear transform such that for any and for any the following weak inequality holds:

Then

for any and such that ,  .

5. Boundedness of Some Operators in Class

Let be a stochastic sequence and be a predicted sequence . A stochastic sequence such that

where is called the transform of with respect to . If is a martingale then we say that is the martingale transform.

Theorem 15. Let ,   and . Let be a martingale transform of a martingale by predicted sequence . If

then

where a constant depends only on parametres , and .

Proof. Let be a martingale transform of a martingale by predicted sequence ; that is,

where ,  . By Abel’s transform , we get the following:

Taking into account that are measurable functions with respect to the algebra , we have and

Hence the weak inequality is proved as follows:

for .

Let , , and . Let a pair of numbers and satisfy the following condition:

Then from that is proved above it follows that

for .

Taking into account that for any stopping time processes and are martingales, it is possible to apply Theorem 9. Then

where

Note that there exists such that . Then it follows from (63) that .

Theorem 16. Let ,   and and be the Markov time and let be a stopped process. Then

Proof. Denote

Let us show that .

If , then

Now, using Corollary 12 we get the statement of the Theorem 16.

Corollary 17. Let ,   and a process be a nonnegative submartingale. Then the process is also submartingale and

Proof. It follows from Theorem 16 that

The reverse inequality is trivial.

6. Interpolation Properties of the Space , the Embedding Theorems

Theorem 18. Let , and . Then

Proof. Using Lemmas 10 and 4 we have the following:

Putting we get the following:

It follows from the definition that for for for for ; therefore,

Substituting these equalities in (74) and applying Lemmas 4 and 8, we get the following:

For the proof of reverse estimate we use the fact that for any and the following equality holds:

Then we have the following:

Substituting and using Lemma 4, we have the following:

since , the proof is complete.

Theorem 19. Let ,   and and let the filtration be such that for every and for all the following condition holds:

where the constant does not depend on . Then

Proof. Let us show that

According to the condition (80), for we have that . Therefore for we get the following:

Thus, (82) is proved.

Now, let ,  , and such that

Then using interpolation Theorems 18 and 11 we obtain the following:

It follows from (82) that . Hence, , where . The proof is complete.

7. Spaces with Variable Approximation Properties by Haar System

In this paragraph we consider some applications of the introduced interpolation method to Besov type spaces with variable approximation properties.

Let and let be a -algebra of Borel subsets of set , a linear Lebesgue measure on , the Haar filtration, and a sequence of stopping times such that for any the following conditions hold: (a.p.) and

For a function we denote by the Fourier coefficients by Haar functions system ([14]). For the given stopping time we denote

which we call the Fourier-Haar partial sum of a function , corresponding to the Markov time .

Let , , . By we denote the set of functions , for which

for ,

for .

Conceptually, the introduced spaces are close to spaces with variable smoothness. Here we mention works of Leopold [15], Cobos and Fernandez [16], and Besov [17–20].

Lemma 20. Let , and let be the Fourier-Haar partial sum with respect to the Markov time . Then

Proof. Denote

Let be the Marcinkiewicz-Lorentz space. Using the equivalent norm of spaces (see [6]) and martingale properties of Fourier-Haar partial sums we get the following:

Now, applying the interpolation theorem (see [12]), we obtain the statement of the lemma.

Lemma 21. Let , , , and

Then

Proof is similar to the proof of Lemma 3.

Theorem 22. Let , and . Then

Proof. By using Lemma 21 we have the following:

By applying Lemma 8 we get the following:

Let us prove the reverse embedding. Let , be an arbitrary representation of a function, and . Then

Since the representation is arbitrary, we have the following:

Hence, putting we get the following:

The theorem is proved.

Acknowledgment

This research was partially supported by Ministry of Education and Science of the Republic of Kazakhstan (0112RK02176, 0112RK00608).

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