#### 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: