Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space

Cho, Dong Hyun

doi:https://doi.org/10.1155/2014/916423

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 916423 | https://doi.org/10.1155/2014/916423

Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space

Dong Hyun Cho¹

Academic Editor: Allan Peterson

Received04 Apr 2014

Accepted27 May 2014

Published18 Jun 2014

Abstract

Let denote a generalized Wiener space, the space of real-valued continuous functions on the interval and define a stochastic process by , for and , where with a.e. and is a continuous function on . Let and be given by and , where is a partition of . In this paper we derive two simple formulas for generalized conditional Wiener integrals of functions on with the conditioning functions and which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function including the time integral on .

1. Introduction

A time integral is simply the Riemann integral of a function of the continuous random variable with respect to the parameter for which is the Wiener space, the space of continuous real-valued functions on with [1]. This means that the time integral of is a random variable on satisfying where is Riemann integrable on and is the Lebesgue measure on . A study of the Feynman integral provides ready examples of the utility of the time integral. The Feynman-Kac formula represents an important step in the process of providing a rigorous definition of the Feynman integral. A detailed explanation of this formula can be found in [2]. The Feynman-Kac functional is given by including the time integral, where is a standard Brownian motion process on and is a complex-valued potential. Calculations involving the conditional expectations of the functional are important in the study of the Feynman integral. More examples of functionals involving time integrals are found in [2–4]. In [5] Park and Skoug introduced a generalized Brownian motion process defined by where is of bounded variation with a.e. on and the integral denotes the Paley-Wiener-Zygmund stochastic integral of according to [6], and then they generalized various theories related to the conditional Wiener integrals on .

On the other hand let denote the space of continuous real-valued functions on the interval . Im et al. [6–8] introduced a probability measure on , where is a probability measure on the Borel class of . We note that is exactly the Wiener measure on if , the Dirac measure concentrated at . Let be a continuous function on . Define stochastic processes by for and for . Let and be given by where is a partition of . The author [9–11] derived various properties of and then extended his works on to . In fact he [9] established that is a generalized Brownian process and derived a simple formula for a conditional expectation which evaluates conditional expectations in terms of ordinary expectations. Moreover he [10] extended the results on to those on with drift using only a translation theorem [6, Theorem 3.1]. In [11] he investigated the distribution of and proved that is a generalized Brownian motion process if the initial distribution is degenerated. He also established a generalized Wiener integration theorem which extends Lemma 2.1 of [6]. Furthermore he derived a generalized Paley-Wiener theorem which generalizes Theorem 3.5 of [6]. As applications of the theorems he evaluated generalized Wiener integrals of various functions including for , where and is a positive integer.

In this paper, using the results in [11], we derive two simple formulas for generalized conditional Wiener integrals of functions on with the conditioning functions and which contain drift and initial distribution. As applications of these simple formulas we evaluate several generalized conditional Wiener integrals of the functions given by (6). We note that if and on , then as above is exactly the stochastic process given by so that these works generalize the results of [5, 9, 10, 12–15] in which the works are the first results among them.

2. Simple Formulas for Generalized Conditional Wiener Integrals

In this section we derive two simple formulas for generalized conditional Wiener integrals on the space which is introduced in the previous section.

Let denote the Lebesgue measure on the Borel class of and let denote the dot product on . Let be the analogue of Wiener space associated with a probability measure on , where denotes the Borel class of [6–8]. Let be integrable and be a random vector on assuming that the value space of is a normed space equipped with the Borel -algebra. Then we have the conditional expectation of given from a well-known probability theory. Furthermore there exists a -integrable complex valued function on the value space of such that for a.e. , where is the probability distribution of . The function is called the conditional -integral of given and it is also denoted by .

For in and in let denote the Paley-Wiener-Zygmund integral of according to [6] and let denote the inner product over . Let be of bounded variation with a.e. on and let . Define for and define stochastic processes by for and for . Let and let be a partition of . For let for and for . Define random vectors and by for . For any function on define the polygonal function of by for , where and denote the indicator functions. For define the polygonal function of by (11), where is replaced by , respectively, for . If , is interpreted as on . For and for let and for let for . It is not difficult to show that

Now we have the following theorem [11].

Theorem 1. For , and are independent so that and are also independent.

Theorem 2. Let for some . Then is normally distributed with mean and variance .

Proof. By (14) and Theorem 3.4 of [6], is normally distributed with mean and variance . A simple calculation with an aid of (13) shows that which completes the proof.

Remark 3. (1) We can prove Theorem 2 using the Fourier transform of , Lemma 2.4 of [11], and Corollaries 2.11, 2.12 of [11], but the proof is tedious.
(2) We can also prove Theorem 2 using Theorem 2.9 of [9] and Theorem 2.13 of [11].

Theorem 4. The process and are stochastically independent.

Proof. Note that for . and are independent by Theorem 2.10 of [9] for , and and are also independent by Theorem 1 which completes the proof of the theorem.

Since for and , we have the following theorem by Theorem 16 of [10].

Theorem 5. The processes , where , are stochastically independent.

For a function let for . Applying the same method as used in the proof of Theorem 2 in [15] with Problem 4 of [16, page 216], we have the following theorem from Theorem 4.

Theorem 6. Let be a complex valued function on and let be integrable over . Then for a Borel subset of where is the probability distribution of on , so that for a.e. where the expectation is taken over the variable .

Using similar method as used in the proof of Theorem 18 of [10], we can prove the following theorem.

Theorem 7. Let be a complex valued function on and let be integrable over . Let be the probability distribution of on . Then for a.e. where .

Remark 8. (1) The conditioning functions and describe the positions of paths at the times , (the present time). depends on the present position of the path for while does not. Moreover if , , and which is the Dirac measure concentrated at , then we can obtain the space in [12] by Theorem 2.13 of [11]. Furthermore if is replaced by the generalized Brownian motion process on and , then we can also obtain Theorem 3.4 of [12] by Theorem 6. If and , then we can obtain Theorem 3 of [5] by Theorem 6. If and , then we can obtain Remark 2.2 of [17] by Theorem 6. If a.e. and , then we can obtain Theorem 2.9 of [13] and Theorem 2.5 of [14] by Theorems 6 and 7, respectively. Finally if a.e., , and , then we can obtain Theorem 2 of [15] by Theorem 6 which is among the first result expressing the conditional Wiener integrals of functions on as ordinary Wiener integrals.
(2) Theorems 6 and 7 are not generalizations of Theorem 2.12 of [9] and Theorems 17 and 18 of [10]. In Theorems 6 and 7 the conditioning functions depend on the initial distribution while the conditioning functions in [9, 10] do not.

3. A Multivariate Normal Distribution

In this section we investigate the joint distribution of , where for . In fact we prove that the random vector as above has a multivariate normal distribution which plays a key role in the next sections.

Now we begin this section with the introduction of the following lemma.

Lemma 9. Let and be given by (13) for . Then is a linearly independent set in .

Proof. For let for a.e. . For take which satisfies the above equation and . Replacing by we have the following linear equation system with unknowns : The determinant of coefficient matrix of the system is given by Now we have which completes the proof.

A random variable on is said to be degenerated if there exists a constant satisfying [18].

Lemma 10. Let the assumptions and notations be as given in Lemma 9. If and is degenerated, then for all .

Proof. By (14) we can take a real constant satisfying for a.e. . Then we have by Theorem 3.4 of [6] that so that in . Now by Lemma 9 which completes the proof.

Lemma 11. Let the assumptions and notations be as given in Lemma 9. Then for the covariance matrix of the random variables , , exists and is positive definite. Moreover is given by and the determinant is positive so that is nonsingular and the inverse matrix of is also positive definite.

Proof. By Theorem 3.4 of [6] so that the covariance of and is given by . Now we have for which proves (26). We have for Moreover if , then for a.e. ; that is, is degenerated. Thus the covariance matrix is positive definite by Lemma 10. Since is symmetric and positive definite, the eigenvalues of are real and positive. Since , we have so that is invertible. Since and implies ; that is, , is positive definite.

Remark 12. Using the same process as used in the proof of Theorem 3.4 in [9] we can prove (26).

For simplicity let for and , where and . By Lemmas 9, 10, 11, and Theorem 4 of [1] we have the following theorem which is our main result in this section.

Theorem 13. Let the assumptions and notations be as given in Lemma 11. Then for every Borel measurable function where is the positive definite matrix satisfying and means that if either side exists, then both sides exist and they are equal.

4. Conditional Expectations of Functions on Time Integrals

In this section we evaluate generalized conditional Wiener integrals of the function including a time integral. To do this we have the following theorem by Theorems 6 and 13 and Theorem 3.3 of [11].

Theorem 14. Let the assumptions and notations be as given in Theorem 13 and let for . Suppose that . Then for a.e. where .

Example 15. Let the assumptions and notations be as given in Theorem 14. If , then by Theorem 2 we have for a.e. If , thenMoreover for a.e. which is a generalization of (2) in Theorem 23 of [10].

By Theorem 7 we have the following theorem.

Theorem 16. Let the assumptions and notations be as given in Theorem 14 and let . Then for a.e.

Example 17. Let the assumptions and notations be as given in Theorem 16. If , then by Theorem 16 we have for a.e. If , then we have for a.e. which is a generalization of (2) in Theorem 24 of [10].

Theorem 18. Let the assumptions and notations be as given in Theorem 14 and let . For and let for , where . Then for a.e. where denotes the greatest integer less than or equal to .

Proof. For let , where . By Theorems 7 and 14 we have for a.e. Let . Then by the change of variable theorem which completes the proof.

Example 19. Let the assumptions and notations be as given in Theorem 18. If , then by Theorem 18 we have for a.e. If , then we have for a.e. which is a generalization of (5) in Theorem 24 of [10].

Theorem 20. Let the notations be as given in Theorem 14 and let for a.e. . Suppose that and for . Then for a.e. exists and where and is as given in Theorem 14. Moreover if exists for , then can be represented by

Proof. By Theorem 2.1 and Lemma 2.1 of [6] we have for so that exists and which implies the existence of . By the monotone convergence theorem we have so that we have for each positive integer . Furthermore for a.e. we have by Theorem 6 Now by Theorems 5, 6, 14, the Fubini’s theorem, and the dominated convergence theorem we have for a.e. which proves the first equality of the theorem. Furthermore Since (52) holds for , we have by Theorem 13 and the dominated convergence theorem which proves the second equality of the theorem.

Theorem 21. Let the notations and the first part of the assumptions in Theorem 20 be given. Then for a.e. is given by the first equality in Theorem 20 replacing by , where the s are as given in Theorems 16 and 18. Moreover if the second part of the assumptions in Theorem 20 holds, then is given by the second equality in Theorem 20 replacing by where and .

Proof. The first part of the theorem immediately follows from Theorems 7, 16, 18, and 20. Suppose that the second part of the assumptions in Theorem 20 holds. For let , where . Then we have Now the second part of the theorem follows from Theorems 7 and 20.

Example 22. (1) Let . Then so that for which is finite by Theorem 1.4 of [19]. Hence and are given by Theorems 20 and 21, respectively, with and for .
(2) Let for and suppose that , where . Then for and we have by the integration parts formula so that by Hölder’s inequality which is finite by Theorem 1.4 of [19]. Now and are given by Theorems 20 and 21, respectively, with and for .

5. Evaluation Formulas for Other Functions

In this section, using the simple formulas in Section 2, we derive evaluation formulas for generalized conditional Wiener integrals of various functions which are of interest in Feynman integration theories themselves and quantum mechanics.

Since for and we have the following theorems from Theorem 3.2 of [11] and Theorems 21, 22, 23, 24, 25, and 26 of [10].

Theorem 23. Let and for . Suppose that . Then for a.e. where denotes the greatest integer less than or equal to .

Theorem 24. Let the assumptions be as given in Theorem 23 and for let Then for a.e.

Theorem 25. Let and let , with . For let and suppose that . Then for a.e.

Theorem 26. Let the assumptions be as given in Theorem 25.(1)If , , and , then for a.e. (2)If and , then for a.e.

For , let , let be the subspace of generated by , let be the orthogonal complement of and let be the orthogonal projection. Let be the class of all complex valued Borel measures of bounded variation over and let be the space of all functions which for have the form for a.e. . Note that is a Banach algebra [6].

Theorem 27. Let be absolutely continuous on . Let and be related by (67). Then for a.e. , is given by

Theorem 28. Let the assumptions be as given in Theorem 27 and for let Then for a.e. , is given by

Remark 29. Suppose that , if necessary, in each lemma, theorem, and example of this paper. Then by Lemma 2.5 of [11] both and exist. Let and for . Since for and can be replaced by and , respectively, in each result of this paper.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2058991).

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Copyright © 2014 Dong Hyun Cho. 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.

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