International Journal of Stochastic AnalysisVolume 2013 (2013), Article ID 537023, 17 pageshttp://dx.doi.org/10.1155/2013/537023

Research Article

Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

Akihiro Kitagawa¹ and Atsushi Takeuchi²

¹Aikou Educational Institute, Kinuyama 5-1610-1, Ehime Matsuyama, 791-8501, Japan²Department of Mathematics, Osaka City University, Sugimoto 3-3-138, Sumiyoshi-ku, Osaka 558-8585, Japan

Received 30 September 2012; Accepted 10 December 2012

Academic Editor: S. Mohammed

Copyright © 2013 Akihiro Kitagawa and Atsushi Takeuchi. 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.

Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument.

1. Introduction

Stochastic functional differential equations, or stochastic delay differential equations, determine non-Markovian processes, because the current states of the process in the equation depend on the past histories of the process. Such kind of equations was initiated by Itô and Nisio [1] in their pioneering work about 50 years ago. As stated in [2], there are some difficulties to study such equations, because we cannot use any methods in analysis, partial differential equations, and potential theory at all. On the other hand, it seems to be more natural to consider the models determined by the solutions to the stochastic functional differential equations in finance, physics, biology, and so forth, because such processes include their past histories and can be recognized to reflect real phenomena in various fields much more exactly.

The Malliavin calculus is well known as a powerful tool to study some properties on the density function by a probabilistic approach. There are a lot of works on the densities for diffusion processes by many authors, from the viewpoint of the Malliavin calculus (cf. [3]). Moreover it is also applicable to the case of solutions to stochastic functional differential equations, regarding as one of the examples of the Wiener functionals. Kusuoka and Stroock in [4] studied the application of the Malliavin calculus to the solutions to stochastic functional differential equations and obtained the result on the existence of the smooth density for the solution with respect to the Lebesgue measure. On the other hand, it is well known that the Malliavin calculus is very fruitful to study the asymptotic behavior of the density function related to the large deviations theory (cf. Léandre [5–8] and Nualart [9]). In fact, the Varadhan-type estimate of the density function for the diffusion processes can be also obtained from this viewpoint. Ferrante et al. in [10] discussed such problem in the case of stochastic delay differential equations, where the drift term depends on the whole past histories on the finite time interval, while the diffusion terms depend on the state only for the edges of the finite time interval. Mohammed and Zhang in [11] studied the large deviations for the solution process under a similar situation to [10]. But, the special forms on the diffusion terms play a crucial role throughout their arguments in [10, 11].

In the present paper, we will study the large deviations on the solution process to the stochastic functional differential equations. Our stochastic functional differential equations are much more general, because they are time inhomogeneous, and they are not only the drift terms, but also the diffusion terms in the equation depend on the whole past histories of the process over a finite interval. Furthermore, as a typical application of the large deviation theory and the Malliavin calculus, we will study the asymptotic behavior, so-called the Varadhan-type estimate, of the density function for the solution process, which is quite similar to the case of diffusion processes. The effect of the time delay plays a crucial role in the behavior of the density function, and the obtained result can be also regarded as the natural extension of the estimate for diffusion processes, which are the most interesting points in the present paper.

The paper is organized as follows. In Section 2, we will prepare some notations and introduce our stochastic functional differential equations. Section 3 will be devoted to the brief summary on the Malliavin calculus and its application to our equations. We will consider some estimates which guarantee the smoothness of the solution process and the non degeneracy in the Malliavin sense. The existence of the smooth density will be also discussed in Section 3. The negative-order moments of the Malliavin covariance matrix will be studied there which is important in order to give the estimate of the density function. Sections 4 and 5 are our main goals in the present paper. In Section 4, we will focus on the large deviation principles on the solution processes. As an application of the result obtained in Section 4, we will study the asymptotic behavior on the density for the solution process. Moreover, we can also derive the short time asymptotics on the density function, which can be interpreted as the generalization of the Varadhan-type estimate on diffusion processes (cf. [5–9]).

2. Preliminaries

Let and be positive constants, and denote an -dimensional Brownian motion by . Let be -valued functions on such that, for each , the mapping is smooth in the Frechét sense and all Frechét derivatives of any orders greater than are bounded. Under the conditions stated above, the functions satisfy the linear growth condition and the Lipschitz condition in the functional sense of the form: for , where . Denote by .

Let be sufficiently small. For a deterministic path , we will consider the -valued process given by the stochastic functional differential equation of the form: where is the segment. Since the current state of the solution depends on its past histories, the process is non-Markovian clearly. Since the coefficients of (2) satisfy the Lipschitz and the linear growth condition in the functional sense, there exists a unique solution to (2), via the successive approximation of the solution process to (2) as follows: for (cf. Itô and Nisio [1], Mohammed [2, 12]).

Proposition 1. For any , it holds that

Proof. Let and . The Hölder inequality and the Burkholder inequality tell us to see that from the linear growth condition on the coefficients . Hence, the Gronwall inequality enables us to obtain the assertion for .As for , the Jensen inequality yields us to see that which implies the assertion by using the consequence stated above. The proof is complete.

3. Applications of the Malliavin Calculus

At the beginning, we will introduce the outline of the Malliavin calculus on the Wiener space , briefly, where is the set of -valued continuous functions on starting from the origin. See Di Nunno et al. [13] and Nualart [9, 14] for details. Let be the Cameron-Martin subspace of with the inner product Denote by the set of -valued random variables such that a random variable is represented as the following form: for , where , for , and . Here, we will denote by the set of smooth functions on such that all derivatives of any orders have polynomial growth. For , the -times Malliavin-Shigekawa derivative for is defined by We will consider , which helps us to define the operator for . For and , let be the completion of with respect to the norm Let be the set of -valued random variables with the components of which belong to , and set . For , the -valued random variable given by is well defined, which is called the Malliavin covariance matrix for .

Before studying the application of the Malliavin calculus to the solution process to (2), we will prepare two basic and well-known facts.

Lemma 2 (cf. Kusuoka and Stroock [4], Lemma 2.1). Let be a real separable Hilbert space, and be a progressively measurable process such that for all . Then, for any and , it holds that

Lemma 3 (cf. Nualart [9], Proposition ). Let be a -adapted, -valued process such that for almost all , and that Then, for each , it holds that , and that

Now, we will return our position to study the application of the Malliavin calculus to the solution process to (2).

Proposition 4. Let and . Then, for each , the -valued random variable is in . Moreover, for each , it holds that

Proof. At the beginning, we will consider the case inductively on . As for , it is a routine work to check the assertion via the Hölder inequality and the Burkholder inequality, from the Lipschitz condition and the linear growth condition on the coefficients , similarly to Proposition 1. Next, we will discuss the case . Let , because the assertion for is trivial. Since for , we have only to prove the assertion for . The chain rule on the operator and Lemma 3 tell us to see that for (cf. Ferrante et al. [10], Lemma 6.1), where the symbol is the Frechét derivative in . Thus, the Hölder inequality and Lemma 2 enable us to get the assertions. Finally, we will discuss the general case . Suppose that the assertions are right until the case . Remark that from Lemma 3, where is the set of permutations of . Since for , and we can get the assertion by using the Hölder inequality, Lemma 2, and the assumption on the case until of the induction.The case is the direct consequence by the Jensen inequality. The proof is complete.

Proposition 5. For , the -valued random variable is in . Moreover, for each , the -valued process satisfies the equation of the form:

Proof. Let and be arbitrary. For each , the sequence is the Cauchy one in , from Proposition 4. Hence, we can find the limit, denoted by , in . Then, it is a routine work to see that the process satisfies (2), via the Hölder inequality and the Burkholder inequality, from the conditions on the coefficients , which implies for from the uniqueness of the solutions. Thus, we can get for . Similarly, we can check that satisfies (21), by taking the limit in each term of (17) via the Hölder inequality and Lemma 2.

For , denote by the -valued process determined by the following equation: where .

Corollary 6.

Proof. Direct consequence of Proposition 5 and the uniqueness of the solution to (21).

Finally, we will introduce the well-known criterion on the existence of the smooth density for the probability law of with respect to the Lebesgue measure on .

Lemma 7 (cf. Kusuoka and Stroock [4]). Suppose the uniformly elliptic condition on the coefficients of (2) as follows: Then, for each and , there exists a smooth density for the probability law of with respect to the Lebesgue measure over .

Proof. Since from Proposition 5, it is sufficiently to study that under the uniformly elliptic condition (24), where is the Malliavin covariance matrix for . Denote by Then, , so we have only to discuss the moment estimate on . As stated in Lemma 1 of Komatsu and Takeuchi [15], we will pay attention to the boundedness of for any , which is sufficient to our goal. Since we have to study the decay order of as .Let be sufficiently large. Remark that for any , from the Burkholder inequality and the Hölder inequality. Let , and . Write , and let . Then, we see that where The Chebyshev inequality yields that Similarly, the Chebyshev inequality leads to from Proposition 1. On the other hand, as for , we have Therefore, we can get so we have for any . The proof is complete.

Remark 8. Consider the case where with the good conditions on the boundedness and the regularity. Now, our stochastic functional differential equation is as follows: where . Then, we can get the same upper estimate of the inverse of the Malliavin covariance matrix for in the hypoelliptic situation, which means that the linear space generated by the vectors , and their Lie brackets span the space (cf. Takeuchi [16]).

4. Large Deviation Principles for

At the beginning, we will introduce the well-known fact on the sample-path large deviations for Brownian motions. See also [8]. Recall that is the Cameron-Martin space of .

Lemma 9 (cf. Dembo and Zeitouni [17], Theorem  5.2.3). The family of the laws of over satisfies the large deviation principle with the good rate function , where

For , let be the solution to the following functional differential equation: Denote by

Theorem 10. The family of the laws of over satisfies the large deviation principle with the good rate function , where and is the function given in Lemma 9.

Theorem 10 tells us to see, via the contraction principle (cf. Dembo and Zeitouni [17], Theorem ).

Corollary 11. For each , the family of the laws of over satisfies the large deviation principle with the good rate function , where and is the function given in Theorem 10.

Now, we will prove Theorem 10, according to Azencott [18] and Léandre [5–8]. Our strategy stated here is almost parallel to [10, 11].

Proposition 12. For any , the mapping is continuous.

Proof. Let . Since we see that from the linear growth condition on , which tells us to see that On the other hand, since for , and the -valued functions satisfy the Lipschitz condition and the linear growth condition, we have The Gronwall inequality tells us to see that which completes the proof.

Proposition 13. Suppose that the -valued functions are bounded. Then, for any and , there exist and such that for any .

Proof. Define a new probability measure by The Girsanov theorem tells us to see that the -valued process is also the -dimensional Brownian motion under the probability measure . Let be the -valued process determined by the following equation: Write . Remark that The Gronwall inequality tells us to see that For each , the martingale representation theorem enables us to see that there exists a -dimensional Brownian motion starting at the origin with for . Remark that , because of the boundedness of the -valued functions . Since from the reflection principle on Brownian motions, we have which completes the proof.

Proposition 14. It holds that

Proof . Let be sufficient large. From the Itô formula, we see that Define . Then, it holds that from the linear growth condition on the coefficients of (2). Hence, the Gronwall inequality implies that In particular, taking yields that Therefore, the Chebyshev inequality leads us to see that so we have which completes the proof.

Let . Define that and .

Proposition 15. For any , it holds that

Proof . Remark that as seen in the proof of Proposition 14. So, we can get which completes the proof.

Proof of Theorem 10. We will prove the assertion in two steps of the form: the case where are bounded, and the general case on .Step 1. Suppose that the coefficients are bounded. Propositions 12 and 13 are sufficient to our goal (cf. [17, 18]). In fact, the large deviation principle for the family comes from the one for in Lemma 9.Step 2. We will discuss the general case on . Let , and be a closed set in . Denote by and by the closed -neighborhood of , where is the open ball in with radius centered at . Then, it holds that As seen in Step 1, we have already obtained the large deviation principle for with the good rate function , where is given in Lemma 9 and So, we have Therefore, we can get from Proposition 14, which completes the proof on the upper estimate of the large deviation principle.Next, we will pay attention to the lower estimate of the large deviation principle. Let be an open set in , and take in . Then, we can find such that . Thus, we have The first equality is right, because of , while the third inequality is the consequence of the large deviation principle for as seen in Step 1. The forth inequality is right, because under and . Taking the limit as leads us to see that from Proposition 15, which completes the proof on the lower estimate of the large deviation principle. The proof of Theorem 10 is complete.

5. Density Estimates

In this section, we will consider the estimate of the density for the solution , from the viewpoint of the Malliavin calculus.

Theorem 16 (Upper estimate). Suppose that the -valued functions satisfy the uniformly elliptic condition (24). Then, it holds that where the function is given in Corollary 11.

Proof. Let be sufficiently small, and such that Take such that . Then, the integration by parts formula tells us to see that where and is the Skorokhod integral operator. Remark that, under the uniformly elliptic condition (24) on the -valued functions , where and , by using Proposition 5 and the proof of Lemma 7. Hence, the density can be estimated from the above as follows: where such that . From Corollary 11, we have Since the function is a lower semicontinuous, taking the limit as and enables us to see that which is the conclusion of Theorem 16.

Remark 17. As stated in Remark 8, a similar problem can be also studied under the hypoelliptic condition, in the case where with the good conditions on the boundedness and the regularity (cf. [16]).

Now, we will study the lower estimate of the density for the solution process to (2). Before doing it, we will prepare some arguments.

Proposition 18. Let , and assume the uniformly elliptic condition (24) on the functions . Then, it holds that for each , where is the Gram matrix for .

Proof. Let , and be the -valued mappings given by the following functional differential equation: where . From the condition on the coefficients , we see that Remark that Hence, the Gronwall inequality tells us to see that On the other hand, remark that we have already seen in the proof of Proposition 12 that Now, we will pay attention to the lower estimate of . Since, for each , satisfies the equation we have similarly to Corollary 6. Hence, the Gram matrix can be expressed as follows: Let be sufficiently close to . So, we see that which is strictly positive. Here, we will remark that there exists the constant with because the functions satisfy the uniformly elliptic condition (24), and is sufficiently close to , which justifies the sixth inequality.

For , let be the -valued process determined by the following equation: Let be the -valued process determined by the following equation:

Lemma 19. Let . It holds that for any and , where .

Proof. We will prove the statement along the following procedure.Step 1. For any , In fact, since for , and the coefficients satisfy the Lipschitz condition and the linear growth condition, we can get the assertion of Step 1 by using the Hölder inequality, the Burkholder inequality, and the Gronwall inequality. Step 2. For any , which tells us to see that the assertion of Lemma 19 holds in the case of .In fact, we will remark that from the Taylor theorem for , where is the constant. Here, for each and is the bilinear mapping on . Since for , and the coefficients are in with respect to the second variable in for each , we can get the assertion in Step 2 via the Hölder inequality, the Burkholder inequality, and the Gronwall inequality.Step 3. Let . Then, for any , Remark that for . Since as seen in Proposition 18, we have Moreover, similarly to Proposition 5, we have for any . Then, the assertion in Step 3 can be justified by using the Hölder inequality, the Burkholder inequality, and the Gronwall inequality.Step 4. Let . Then, for any , In fact, since for , and the coefficients are in with respect to the second variable in for each , the assertion can be obtained via the Hölder inequality, the Burkholder inequality, and the Gronwall inequality. Here .Step 5. Let be arbitrary, and . Then, for any , We have already proved the case of in Step 4. Remark that for adapted processes and with nice properties. Then, we can get the assertion by induction on . Then, the assertion is the direct consequences of Step 2 and Step 5. The proof of Lemma 19 is complete.

Theorem 20 (Lower estimate). Suppose that the -valued functions satisfy the uniformly elliptic condition (24). Then, it holds that where the function is given in Corollary 11.

Proof. Since the assertion of Theorem 20 is trivial in the case of , we will suppose that . Let be nonnegative. For sufficiently small , recall the function as introduced in the proof of Theorem 16: such that Then, the Girsanov theorem tells us to see that Here, the third inequality comes from the nonnegativity in the exponent while the forth inequality holds because of and on the complement of . Thus, the limiting argument enables us to see that where is the Dirac delta function. Since from Lemma 19, we have Moreover, from the definition of the function , we can find with such that Hence, it holds that Taking the limit as completes the proof.

Corollary 21. Suppose that the -valued functions satisfy the uniformly elliptic condition (24). Then, it holds that as , where the function is given in Corollary 11.

Proof. Direct consequences of Theorems 16 and 20.

Finally, we will study the asymptotic behavior of the density for in a short time. Let be a constant, and . We will consider the case where such that , and , for . Suppose that the functions satisfy the uniformly elliptic condition of the form: For , let , and let be the -valued processes determined by the equations of the form: where . Remark that . Denote by (or, ) the density for the probability law of (, resp.), whose existence can be justified under the uniformly elliptic condition (122) on the coefficients . Then, we have the following.

Corollary 22. Suppose that the functions satisfy the uniformly elliptic condition (122). Then, it holds that

Proof. Recall that where such that . Here, the second equality holds from the scaling property on the Brownian motion , while the third equality follows from . On the other hand, recall that because of . From the uniqueness of the solutions, we have in the sense of the probability law. Hence, we can get As for the density , we have already obtained the asymptotic behavior of the form: as , in Corollary 21. Taking completes the proof.

Remark 23. In particular, consider the case of where such that the functions satisfy the uniformly elliptic condition of the form: Then, our equation can be written as follows: where . Although our settings include the effect of the time-delay parameter , the effect of the parameter in (131) can be ignored. Hence, the solution is the diffusion process, so we have only to choose in the starting point of our study. Moreover, the choice of tells us to see that Corollary 22 is the well-known fact, that is, the Varadhan-type estimate, on the asymptotic behavior of the density function for diffusion processes. Hence, Corollary 22 can be also regarded as the generalization of the short-time estimate of the density for diffusion processes.

Remark 24. Ferrante et al. in [10] discussed the large deviation principle for the solution process and the asymptotic estimate of the density, in the case of where with for each . Moreover, suppose that the functions satisfy the uniformly elliptic condition of the form:

On the other hand, Mohammed and Zhang in [11] studied the large deviation principle for the solution process , in the case of where with .

Since the special forms of the coefficients on the diffusion terms are quite essential in their arguments [10, 11], our situation cannot be included in their frameworks at all.

Acknowledgments

The authors are grateful to an anonymous referee for valuable comments and suggestions. This work is partially supported by Ministry of Education, Culture, Sports, Science, and Technology and Grant-in-Aid for Encouragement of Young Scientists, 23740083. This work is also partially supported from the Research Council of Norway, 219005/F11. This work was largely carried out while the second author stayed at the Center of Mathematics for Applications, University of Oslo, from August to September, 2012, on his leave from Osaka City University.

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