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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 152043, 12 pages
Interpolation Methods for Stochastic Processes Spaces
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.
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.
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).
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.) , 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 .
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  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 ) 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 (), the process converges almost surely.
Lemma 3. Let , , , and . Then where
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 () 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 ().
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 ).
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:
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
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.