Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

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 𝑟 and 𝑇 be positive constants, and denote an 𝑚-dimensional Brownian motion by 𝑊={𝑊(𝑡)=(𝑊¹(𝑡),…,𝑊^𝑚(𝑡));𝑡∈[0,𝑇]}. Let 𝐴_𝑖(𝑖=0,1,…,𝑚) be ℝ^𝑑-valued functions on [0,𝑇]×𝐶([−𝑟,0];ℝ^𝑑) such that, for each 𝑡∈[0,𝑇], the mapping 𝐴_𝑖(𝑡,⋅)∶𝐶([−𝑟,0];ℝ^𝑑)∋𝑓↦𝐴_𝑖(𝑡,𝑓)∈ℝ^𝑑 is smooth in the Frechét sense and all Frechét derivatives of any orders greater than 1 are bounded. Under the conditions stated above, the functions 𝐴_𝑖(𝑖=0,1,…,𝑚) satisfy the linear growth condition and the Lipschitz condition in the functional sense of the form: sup_[]^𝑚_{𝑡∈0,𝑇𝑖=0}𝐴_𝑖𝑡,𝑓≤𝐶_1,𝑇1+𝑓_∞,sup_[]^𝑚_{𝑡∈0,𝑇𝑖=0}𝐴_𝑖𝑡,𝑓−𝐴_𝑖𝑡,𝑔≤𝐶_2,𝑇𝑓−𝑔_∞, (1) for 𝑓,𝑔∈𝐶([−𝑟,0];ℝ^𝑑), where ‖𝑓‖_∞=sup_{𝑡∈[−𝑟,0]}|𝑓(𝑡)|. Denote by 𝐴=(𝐴₁,…,𝐴_𝑚).

Let 0<𝜀≤1 be sufficiently small. For a deterministic path 𝜂∈𝐶([−𝑟,0];ℝ^𝑑), we will consider the ℝ^𝑑-valued process 𝑋^𝜀={𝑋^𝜀(𝑡);𝑡∈[−𝑟,𝑇]} given by the stochastic functional differential equation of the form: 𝑋^𝜀(𝑡)=𝜂(𝑡)(𝑡∈[−𝑟,0]),𝑑𝑋^𝜀(𝑡)=𝐴₀𝑡,𝑋^𝜀_𝑡𝑑𝑡+𝜀𝐴𝑡,𝑋^𝜀_𝑡𝑑𝑊(𝑡)(𝑡∈(0,𝑇]), (2) where 𝑋^𝜀_𝑠={𝑋^𝜀(𝑠+𝑢);𝑢∈[−𝑟,0]} is the segment. Since the current state of the solution depends on its past histories, the process 𝑋^𝜀 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 𝑋^𝜀,(𝑛)={𝑋^𝜀,(𝑛)(𝑡);𝑡∈[−𝑟,𝑇]}(𝑛∈ℤ₊) of the solution process 𝑋^𝜀 to (2) as follows: 𝑋^𝜀,(0)(𝑡)(𝑡[,𝑋=𝜂)(𝑡∈−𝑟,0])_𝜀,(0)(𝑡)(0(,𝑋=𝜂)(𝑡∈0,𝑇])_𝜀,(𝑛)(𝑡)=𝜂(𝑡)(𝑡∈[−𝑟,0]),𝑑𝑋^𝜀,(𝑛)(𝑡)=𝐴₀𝑡,𝑋_𝑡^{𝜀,(𝑛−1)}𝑑𝑡+𝜀𝐴𝑡,𝑋_𝑡^{𝜀,(𝑛−1)}𝑑𝑊(𝑡)(𝑡∈(0,𝑇]), (3) for 𝑛∈ℕ (cf. Itô and Nisio [1], Mohammed [2, 12]).

Proposition 1. For any 𝑝>1, it holds that sup_0<𝜀≤1𝔼sup_{[]𝑡∈−𝑟,𝑇}|𝑋^𝜀(𝑡)|^𝑝≤𝐶_{3,𝑝,𝑇,𝜂}. (4)

Proof. Let 𝑝>2 and 𝑡∈[0,𝑇]. The Hölder inequality and the Burkholder inequality tell us to see that 𝔼sup_{[]𝜏∈−𝑟,𝑡}𝑋^𝜀(𝜏)^𝑝≤𝐶_4,𝑝‖𝜂‖^𝑝_∞+𝐶_4,𝑝𝔼sup_{[]𝜏∈0,𝑡}|𝑋^𝜀(𝜏)|^𝑝≤𝐶_4,𝑝‖𝜂‖^𝑝_∞+𝐶_5,𝑝𝔼sup_{[]𝜏∈0,𝑡}^𝜏₀𝐴₀𝑠,𝑋^𝜀_𝑠𝑑𝑠^𝑝+𝐶_5,𝑝𝜀^𝑝𝔼sup_{𝜏∈[0,𝑡]}^𝜏₀𝐴𝑠,𝑋^𝜀_𝑠𝑑𝑊(𝑠)^𝑝≤𝐶_4,𝑝‖𝜂‖^𝑝_∞+𝐶_5,𝑝𝑇^𝑝−1𝑡₀𝔼|𝐴₀𝑠,𝑋^𝜀_𝑠|^𝑝𝑑𝑠+𝐶_6,𝑝𝜀^𝑝𝑇^{𝑝/2−1𝑡}₀^𝑚_𝑖=1𝔼|𝐴_𝑖𝑠,𝑋^𝜀_𝑠|^𝑝𝑑𝑠≤𝐶_{7,𝑝,𝑇,𝜂}+𝐶_8,𝑝,𝑇^𝑡₀𝔼sup_{[]𝜏∈−𝑟,𝑠}𝑋^𝜀(𝜏)^𝑝𝑑𝑠, (5) from the linear growth condition on the coefficients 𝐴_𝑖(𝑖=0,1,…,𝑚). Hence, the Gronwall inequality enables us to obtain the assertion for 𝑝>2.
As for 1<𝑝≤2, the Jensen inequality yields us to see that sup_0<𝜀≤1𝔼sup_{[]𝑡∈−𝑟,𝑇}|𝑋^𝜀(𝑡)|^𝑝≤sup_0<𝜀≤1𝔼sup_{𝑡∈[−𝑟,𝑇]}|𝑋^𝜀(𝑡)|^2𝑝1/2, (6) 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 𝐶₀([0,𝑇];ℝ^𝑚), briefly, where 𝐶₀([0,𝑇];ℝ^𝑚) is the set of ℝ^𝑚-valued continuous functions on [0,𝑇] starting from the origin. See Di Nunno et al. [13] and Nualart [9, 14] for details. Let 𝐻 be the Cameron-Martin subspace of 𝐶₀([0,𝑇];ℝ^𝑚) with the inner product ⟨𝑔,ℎ⟩_𝐻=^𝑇₀𝑔(̇𝑡)⋅ℎ̇(𝑡)𝑑𝑡𝑔,ℎ∈𝐻. (7) Denote by 𝒮 the set of ℝ-valued random variables such that a random variable 𝐹 is represented as the following form: 𝐹(𝑊)=𝑓𝑊ℎ₁,…,𝑊ℎ_𝑛 (8) for 𝑊∈𝐶₀([0,𝑇];ℝ^𝑚), where ℎ₁,…,ℎ_𝑛∈𝐻, 𝑊[ℎ]=∫^𝑇₀ℎ(𝑠)⋅𝑑𝑊(𝑠) for ℎ∈𝐻, and 𝑓∈𝐶^∞_𝑝(ℝ^𝑛;ℝ). Here, we will denote by 𝐶^∞_𝑝(ℝ^𝑛;ℝ) the set of smooth functions on ℝ^𝑛 such that all derivatives of any orders have polynomial growth. For 𝑘∈ℕ, the 𝑘-times Malliavin-Shigekawa derivative 𝐷^𝑘𝐹={𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝐹;𝑢₁,…,𝑢_𝑘∈[0,𝑇]} for 𝐹∈𝒮 is defined by 𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝐹(𝑊)=^𝑛_𝑗∑₌₁𝜕_𝑗𝑓𝑊ℎ₁,…,𝑊ℎ_𝑛×∫^𝑢1₀ℎ_𝑗(𝑠)(),𝐷𝑑𝑠𝑘=1_𝑢1⋯𝐷_𝑢𝑘𝐹(𝑊)(𝑘≥2). (9) We will consider 𝐷⁰𝐹=𝐹, which helps us to define the operator 𝐷^𝑘 for 𝑘∈ℤ₊. For 𝑝>1 and 𝑘∈ℤ₊, let 𝔻_𝑘,𝑝 be the completion of 𝒮 with respect to the norm ‖𝐹‖_𝑘,𝑝=𝔼|𝐹|^𝑝1/𝑝(),𝔼𝑘=0|𝐹|^𝑝1/𝑝+∑^𝑘_𝑗=1𝔼𝐷^𝑗𝐹^𝑝_𝐻⊗𝑗^1/𝑝(𝑘∈ℕ). (10) Let 𝔻_𝑘,𝑝(ℝ^𝑑) be the set of ℝ^𝑑-valued random variables with the components of which belong to 𝔻_𝑘,𝑝, and set 𝔻_∞(ℝ^𝑑)=⋂_𝑝>1⋂_𝑘∈ℤ+𝔻_𝑘,𝑝(ℝ^𝑑). For 𝐹∈𝔻_1,2(ℝ^𝑑), the ℝ^𝑑⊗ℝ^𝑑-valued random variable 𝑉_𝐹 given by 𝑉_𝐹=⟨𝐷𝐹,𝐷𝐹⟩_𝐻=^𝑇₀𝑑𝐷𝑑𝑢^𝑢𝑑𝐹⋅𝐷𝑑𝑢^𝑢𝐹𝑑𝑢 (11) is well defined, which is called the Malliavin covariance matrix for 𝐹.

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

Lemma 2 (cf. Kusuoka and Stroock [4], Lemma 2.1). Let Γ be a real separable Hilbert space, and 𝛼∶[0,𝑇]×Ω→ℝ^𝑚⊗Γ be a progressively measurable process such that 𝔼^𝑇₀‖𝛼(𝑠)‖^𝑝_ℝ𝑚⊗Γ𝑑𝑠<+∞ (12) for all 𝑝>1. Then, for any 𝑝>2 and 𝜏∈[0,𝑇], it holds that 𝔼sup_{[]𝑡∈0,𝜏}^𝑡₀𝛼(𝑠)𝑑𝑊(𝑠)^𝑝_Γ≤𝐶_9,𝑝,𝜏^𝜏₀^𝑚_𝑖=1𝔼𝛼_𝑖(𝑠)^𝑝_Γ𝑑𝑠. (13)

Lemma 3 (cf. Nualart [9], Proposition 1.3.8). Let {𝛽(𝑡);𝑡∈[0,𝑇]} be a (ℱ_𝑡)-adapted, ℝ^𝑚⊗ℝ^𝑑-valued process such that 𝛽(𝑡)∈𝔻_1,2(ℝ^𝑚⊗ℝ^𝑑) for almost all 𝑡∈[0,𝑇], and that 𝔼^𝑇₀^𝑇₀|𝐷_𝑢𝛽(𝑡)|²𝑑𝑢𝑑𝑡<+∞. (14) Then, for each 𝑡∈[0,𝑇], it holds that ∫^𝑡₀𝛽(𝑠)𝑑𝑊(𝑠)∈𝔻_1,2(ℝ^𝑑), and that 𝐷_𝑢^𝑡₀𝛽(𝑠)𝑑𝑊(𝑠)=₀^𝑢∧𝑡𝛽(𝑠)𝑑𝑠+^𝑡₀𝐷_𝑢𝛽(𝑠)𝑑𝑊(𝑠). (15)

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

Proposition 4. Let 𝑛∈ℤ₊ and 0<𝜀≤1. Then, for each 𝑡∈[−𝑟,𝑇], the ℝ^𝑑-valued random variable 𝑋^𝜀,(𝑛)(𝑡) is in 𝔻_∞(ℝ^𝑑). Moreover, for each 𝑘∈ℤ₊, it holds that 𝔼sup_{𝑡∈[−𝑟,𝑇]}‖𝐷^𝑘𝑋^𝜀,(𝑛)(𝑡)‖^𝑝_{𝐻⊗𝑘⊗ℝ𝑑}≤𝐶_{10,𝑘,𝑝,𝑇,𝜂},𝔼sup_{[]𝑡∈−𝑟,𝑇}‖𝐷^𝑘𝑋^𝜀,(𝑛)(𝑡)−𝐷^𝑘𝑋^{𝜀,(𝑛−1)}(𝑡)‖^𝑝_{𝐻⊗𝑘⊗ℝ𝑑}≤𝐶_{11,𝑘,𝑝,𝑇,𝜂}2^𝑛/2. (16)

Proof. At the beginning, we will consider the case 𝑝>2 inductively on 𝑘∈ℤ₊. As for 𝑘=0, 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 𝐴_𝑖(𝑖=0,1,…,𝑚), similarly to Proposition 1. Next, we will discuss the case 𝑘=1. Let 𝑛∈ℕ, because the assertion for 𝑛=0 is trivial. Since 𝐷𝑋^𝜀,(𝑛)=0 for 𝑡∈[−𝑟,0], we have only to prove the assertion for 𝑡∈(0,𝑇]. The chain rule on the operator 𝐷 and Lemma 3 tell us to see that 𝐷_𝑢𝑋^𝜀,(𝑛)(𝑡)=𝜀^𝑢₀𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝕀_{(𝑠≤𝑡)}+𝑑𝑠_𝑡0∇𝐴₀𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝐷_𝑢𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑠+𝜀^𝑡₀∇𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝐷_𝑢𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑊(𝑠) (17) for 𝑢∈[0,𝑇] (cf. Ferrante et al. [10], Lemma 6.1), where the symbol ∇ is the Frechét derivative in 𝐶([−𝑟,0];ℝ^𝑑). Thus, the Hölder inequality and Lemma 2 enable us to get the assertions. Finally, we will discuss the general case 𝑘∈ℤ₊. Suppose that the assertions are right until the case 𝑛−1. Remark that 𝐷^𝑘_{𝑢1,…,𝑢𝑘}^𝑡₀𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑊(𝑠)=𝐷_𝑢^𝑘−1_{1,…,𝑢𝑘−1}^𝑡₀𝐷_𝑢𝑘𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑊(𝑠)+𝐷_𝑢^𝑘−1_{1,…,𝑢𝑘−1}^𝑢𝑘₀^∧𝑡𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}=𝑑𝑠=⋯_𝑡0𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}(𝑠)+𝑑𝑊_{𝜎∈𝔖𝑘𝑢𝜎(𝑘)0∧𝑡}𝐷_𝑢^𝑘−1_{𝜎(1),…,𝑢𝜎(𝑘−1)}𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑠, (18) from Lemma 3, where 𝔖_𝑘 is the set of permutations of {1,…,𝑘}. Since 𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝐴_𝑖𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}=𝐷_𝑢^𝑘−1_{1,…,𝑢𝑘−1}∇𝐴_𝑖𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝐷_𝑢𝑘𝑋_𝑠^{𝜀,(𝑛−1)}=_{𝜎∈𝔖𝑘}^𝑘−1_𝑗=0𝑗𝐷𝑘−1_{𝑢𝑘−1−𝑗𝜎(1),…,𝑢𝜎(𝑘−𝑗)}∇𝐴_𝑖𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}×𝐷_𝑢^𝑗+1_{𝜎(𝑘−𝑗+1),…,𝑢𝜎(𝑘−1),𝑢𝑘}𝑋_𝑠^{𝜀,(𝑛−1)} (19) for 𝑖=0,1,…,𝑚, and 𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝑋^𝜀,(𝑛)(𝑡)=𝐷^𝑘_{𝑢1,…,𝑢𝑘}^𝑡₀𝐴₀𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑠+𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝜀^𝑡₀𝐴𝑠,𝑋^{𝜀,(𝑛−1)}_s𝑑𝑊(𝑠)=_{𝜎∈𝔖𝑘}𝜀^{𝑢𝜎(𝑘)}₀^∧𝑡𝐷_𝑢^𝑘−1_{𝜎(1),…,𝑢𝜎(𝑘−1)}𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}+𝑑𝑠_𝑡0𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝐴₀𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑠+𝜀^𝑡₀𝐷^𝑘_{𝑢1,…,𝑢𝑘}𝐴𝑠,𝑋_𝑠^{𝜀,(𝑛−1)}𝑑𝑊(𝑠), (20) we can get the assertion by using the Hölder inequality, Lemma 2, and the assumption on the case until 𝑘−1 of the induction.
The case 1<𝑝≤2 is the direct consequence by the Jensen inequality. The proof is complete.

Proposition 5. For 𝑡∈[−𝑟,𝑇], the ℝ^𝑑-valued random variable 𝑋^𝜀(𝑡) is in 𝔻_∞(ℝ^𝑚). Moreover, for each 𝑢∈[0,𝑇], the ℝ^𝑚⊗ℝ^𝑑-valued process {𝐷_𝑢𝑋^𝜀(𝑡);𝑡∈[−𝑟,𝑇]} satisfies the equation of the form: 𝐷_𝑢𝑋^𝜀(𝑡)([]),𝐷=0𝑡∈−𝑟,0or𝑡<𝑢_𝑢𝑋^𝜀(𝑡)=𝜀₀^𝑢∧𝑡𝐴𝑠,𝑋^𝜀_𝑠𝑑𝑠+^𝑡₀∇𝐴₀𝑠,𝑋^𝜀_𝑠𝐷_𝑢𝑋^𝜀_𝑠𝑑𝑠+𝜀^𝑡₀∇𝐴𝑠,𝑋^𝜀_𝑠𝐷_𝑢𝑋^𝜀_𝑠𝑑𝑊(𝑠)(𝑡∈[𝑢,𝑇]). (21)

Proof. Let 𝑝>1 and 𝑘∈ℤ₊ be arbitrary. For each 𝑡∈[−𝑟,𝑇], the sequence {𝑋^𝜀,(𝑛)(𝑡);𝑛∈ℕ} is the Cauchy one in 𝔻_𝑘,𝑝(ℝ^𝑑), from Proposition 4. Hence, we can find the limit, denoted by 𝑋^𝜀(𝑡), in 𝔻_𝑘,𝑝(ℝ^𝑑). Then, it is a routine work to see that the process {𝑋^𝜀(𝑡);𝑡∈[−𝑟,𝑇]} satisfies (2), via the Hölder inequality and the Burkholder inequality, from the conditions on the coefficients 𝐴_𝑖(𝑖=0,1,…,𝑚), which implies 𝑋^𝜀(𝑡)=𝑋^𝜀(𝑡) for 𝑡∈[−𝑟,𝑇] from the uniqueness of the solutions. Thus, we can get 𝑋(𝑡)∈𝔻_𝑘,𝑝(ℝ^𝑑) for 𝑡∈[−𝑟,𝑇]. Similarly, we can check that {𝐷_𝑢𝑋(𝑡);𝑢∈[0,𝑇]} satisfies (21), by taking the limit in each term of (17) via the Hölder inequality and Lemma 2.

For 𝑢∈[0,𝑇], denote by {𝑍^𝜀(𝑡,𝑢);𝑡∈[−𝑟,𝑇]} the ℝ^𝑑⊗ℝ^𝑑-valued process determined by the following equation: 𝑍^𝜀()([]),𝑍𝑡,𝑢=0𝑡∈−𝑟,0or𝑡<𝑢_𝜀(𝑢,𝑢)=𝐼_𝑑,𝑑𝑍^𝜀(𝑡,𝑢)=∇𝐴₀𝑡,𝑋^𝜀_𝑡𝑍^𝜀_𝑡(⋅,𝑢)𝑑𝑡+𝜀∇𝐴𝑡,𝑋^𝜀_𝑡𝑍^𝜀_𝑡(⋅,𝑢)𝑑𝑊(𝑡)(𝑡∈(𝑢,𝑇]), (22) where 𝑍^𝜀_𝑡(⋅,𝑢)={𝑍^𝜀(𝑡+𝜏,𝑢);𝜏∈[−𝑟,0]}.

Corollary 6. 𝐷_𝑢𝑋^𝜀(𝑡)=𝜀₀^𝑢∧𝑡𝑍^𝜀(𝑡,𝑠)𝐴𝑠,𝑋^𝜀_𝑠𝑑𝑠. (23)

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 𝑋^𝜀(𝑡) with respect to the Lebesgue measure on ℝ^𝑑.

Lemma 7 (cf. Kusuoka and Stroock [4]). Suppose the uniformly elliptic condition on the coefficients 𝐴_𝑖(𝑖=1,…,𝑚) of (2) as follows: inf_{𝜁∈𝕊𝑑−1}inf_{[]𝑡∈0,𝑇}inf_{]𝑓∈𝐶[−𝑟,0;ℝ𝑑}^𝑚_𝑖=1(𝜁⋅𝐴_𝑖𝑡,𝑓)²>0. (24) Then, for each 𝑡∈(0,𝑇] and 0<𝜀≤1, there exists a smooth density 𝑝^𝜀(𝑡,𝑦) for the probability law of 𝑋^𝜀(𝑡) with respect to the Lebesgue measure over ℝ^𝑑.

Proof. Since 𝑋(𝑡)∈𝔻_∞(ℝ^𝑑) from Proposition 5, it is sufficiently to study that (det𝑉^𝜀(𝑡))⁻¹∈⋂_𝑝>1𝕃^𝑝(Ω) under the uniformly elliptic condition (24), where 𝑉^𝜀(𝑡) is the Malliavin covariance matrix for 𝑋^𝜀(𝑡). Denote by 𝑉^𝜀(𝑡)=^𝑡₀^𝑚_𝑖=1𝑍^𝜀(𝑡,𝑢)𝐴_𝑖𝑢,𝑋^𝜀_𝑢𝐴_𝑖(𝑢,𝑋^𝜀_𝑢)^∗𝑍^𝜀(𝑡,𝑢)^∗𝑑𝑢. (25) Then, 𝑉^𝜀(𝑡)=𝜀²𝑉^𝜀(𝑡), so we have only to discuss the moment estimate on 𝑉^𝜀(𝑡). As stated in Lemma 1 of Komatsu and Takeuchi [15], we will pay attention to the boundedness of sup_{𝜁∈𝕊𝑑−1}𝔼𝜁⋅𝑉^𝜀(𝑡)𝜁^−𝑝, (26) for any 𝑝>1, which is sufficient to our goal. Since 𝔼𝜁⋅𝑉^𝜀(𝑡)𝜁^−𝑝=1Γ𝑝₀^+∞𝜆^𝑝−1×𝔼exp−𝜆𝜁⋅𝑉^𝜀(𝑡)𝜁𝑑𝜆, (27) we have to study the decay order of sup_𝜁∈𝕊^𝑑−1𝔼[exp(−𝜆𝜁⋅𝑉^𝜀(𝑡)𝜁)] as 𝜆→+∞.
Let 𝜆>1 be sufficiently large. Remark that 𝔼𝑍^𝜀(𝑡,𝑢)−𝐼_𝑑^𝑝_{ℝ𝑑⊗ℝ𝑑}≤𝐶_12,𝑝,𝑇(𝑡−𝑢)^𝑝/2, (28) for any 𝑝>1, from the Burkholder inequality and the Hölder inequality. Let 𝜉>1/2, 1<𝛾<2𝜉 and 0<𝜎<(𝛾−1)/2. Write 𝑡_𝜉∶=𝑡−𝜆^−𝜉, and let 𝜁∈𝕊^𝑑−1. Then, we see that 𝔼exp−𝜁⋅𝑉^𝜀(𝑡)𝜁≤𝔼₁exp−𝜆𝜁⋅𝑉^𝜀(𝑡)𝜁+ℙ^𝑡_𝑡𝜉𝑍^𝜀(𝑡,𝑢)−𝐼_𝑑²_{ℝ𝑑⊗ℝ𝑑}𝑑𝑢≥𝜆^−𝛾+ℙsup_{[]𝑠∈−𝑟,𝑡}𝑋^𝜀(𝑠)≥𝜆^𝜎=∶𝐼₁+𝐼₂+𝐼₃, (29) where 𝔼₁[⋅]∶=𝔼⋅∶^𝑡_𝑡𝜉‖𝑍^𝜀(𝑡,𝑢)−𝐼_𝑑‖²_{ℝ𝑑⊗ℝ𝑑}𝑑𝑢<𝜆^−𝛾,sup_{[]𝑠∈−𝑟,𝑡}𝑋^𝜀(𝑠)<𝜆^𝜎. (30) The Chebyshev inequality yields that 𝐼₂≤𝜆^𝛾𝑝𝔼^𝑡_𝑡𝜉𝑍^𝜀(𝑡,𝑢)−𝐼_𝑑²_{ℝ𝑑⊗ℝ𝑑}𝑑𝑢^𝑝≤𝐶_13,𝑝,𝑇𝜆^{−(2𝜉−𝛾)𝑝}. (31) Similarly, the Chebyshev inequality leads to 𝐼₃≤𝜆^−𝜎𝑝𝔼sup_{[]𝑠∈−𝑟,𝑡}𝑋^𝜀(𝑠)^𝑝≤𝐶_{14,𝑝,𝑇,𝜂}𝜆^−𝜎𝑝, (32) from Proposition 1. On the other hand, as for 𝐼₁, we have 𝐼₁≤𝔼₁exp−𝜆^𝑡_𝑡𝜉^𝑚_𝑖=1𝜁⋅𝑍^𝜀(𝑡,𝑢)𝐴_𝑖𝑢,𝑋^𝜀_𝑢²𝑑𝑢≤𝔼₁−𝜆exp2inf_{𝜁∈𝕊𝑑−1}^𝑡_𝑡𝜉^𝑚_𝑖=1𝜁⋅𝐴_𝑖𝑢,𝑋^𝜀_𝑢²𝜆𝑑𝑢×exp^𝑡_𝑡𝜉𝑍^𝜀(𝑡,𝑢)−𝐼_𝑑²_{ℝ𝑑⊗ℝ𝑑}^𝑚_𝑖=1𝐴_𝑖𝑢,𝑋^𝜀_𝑢²𝜆𝑑𝑢≤exp^{1−𝛾+2𝜎}−𝜆×exp2inf_{𝜁∈𝕊𝑑−1}inf_{[]𝑢∈0,𝑇}inf_{[]𝑓∈𝐶−𝑟,0;ℝ𝑑}^𝑚_𝑖=1𝜁⋅𝐴_𝑖𝑢,𝑓²≤𝐶₁₅exp−𝐶₁₆𝜆. (33) Therefore, we can get 𝔼exp−𝜆𝜁⋅𝑉^𝜀(𝑡)𝜁≤𝐶_{17,𝑝,𝑇,𝜂}𝜆^{−𝐶18𝑝}, (34) so we have sup_{𝜁∈𝕊𝑑−1}𝔼𝜁⋅𝑉^𝜀(𝑡)𝜁^−𝑝=𝜀^−2𝑑𝑝sup_{𝜁∈𝕊𝑑−1}𝔼𝜁⋅𝑉^𝜀(𝑡)𝜁^−𝑝≤𝐶_19,𝑝,𝑇𝜀^−2𝑑𝑝, (35) for any 𝑝>1. The proof is complete.

Remark 8. Consider the case 𝐴_𝑖𝑡,𝑓=𝐴_𝑖𝑡,𝑓(0)(𝑖=1,…,𝑚), (36) where 𝐴_𝑖∶[0,𝑇]×ℝ^𝑑→ℝ^𝑑 with the good conditions on the boundedness and the regularity. Now, our stochastic functional differential equation is as follows: 𝑋^𝜀(𝑡)=𝜂(𝑡)(𝑡∈[−𝑟,0]),𝑑𝑋^𝜀(𝑡)=𝐴₀𝑡,𝑋^𝜀_𝑡𝑑𝑡+𝜀𝐴𝑡,𝑋^𝜀(𝑡)𝑑𝑊(𝑡)(𝑡∈(0,𝑇]), (37) where 𝐴=(𝐴₁,…,𝐴_𝑚). Then, we can get the same upper estimate of the inverse of the Malliavin covariance matrix 𝑉^𝜀(𝑡) for 𝑋^𝜀(𝑡) in the hypoelliptic situation, which means that the linear space generated by the vectors 𝐴_𝑖(𝑖=1,…,𝑚), and their Lie brackets span the space ℝ^𝑑 (cf. Takeuchi [16]).

4. Large Deviation Principles for 𝑋^𝜀

At the beginning, we will introduce the well-known fact on the sample-path large deviations for Brownian motions. See also [8]. Recall that 𝐻 is the Cameron-Martin space of 𝐶₀([0,𝑇];ℝ^𝑚).

Lemma 9 (cf. Dembo and Zeitouni [17], Theorem  5.2.3). The family {ℙ∘(𝜀𝑊)⁻¹;0<𝜀≤1} of the laws of 𝜀𝑊 over 𝐶₀([0,𝑇];ℝ^𝑚) satisfies the large deviation principle with the good rate function 𝐼, where 𝐼𝑓=𝑓²_𝐻2,.𝑓∈𝐻+∞𝑓∉𝐻 (38)

For 𝑓∈𝐻, let 𝑥^𝑓={𝑥^𝑓(𝑡);𝑡∈[−𝑟,𝑇]} be the solution to the following functional differential equation: 𝑥^𝑓(𝑡)=𝜂(𝑡)(𝑡∈[−𝑟,0]),𝑑𝑥^𝑓(𝑡)=𝐴₀𝑡,𝑥^𝑓_𝑡𝑑𝑡+𝐴𝑡,𝑥^𝑓_𝑡𝑓(̇𝑡)𝑑𝑡(𝑡∈(0,𝑇]). (39) Denote by 𝐶_𝜂[−𝑟,𝑇];ℝ^𝑑=𝑤∈𝐶[−𝑟,𝑇];ℝ^𝑑;𝑤(𝑡)=𝜂(𝑡)(𝑡∈[−𝑟,0]). (40)

Theorem 10. The family {ℙ∘(𝑋^𝜀)⁻¹;0<𝜀≤1} of the laws of 𝑋^𝜀 over 𝐶_𝜂([−𝑟,𝑇];ℝ^𝑑) satisfies the large deviation principle with the good rate function 𝐼̃, where 𝐼̃𝑔=inf𝐼𝑓;𝑓∈𝐻,𝑔=𝑥^𝑓, (41) and 𝐼 is the function given in Lemma 9.

Theorem 10 tells us to see, via the contraction principle (cf. Dembo and Zeitouni [17], Theorem 4.2.1).

Corollary 11. For each 𝑡∈[0,𝑇], the family {ℙ∘(𝑋^𝜀(𝑡))⁻¹;0<𝜀≤1} of the laws of 𝑋^𝜀(𝑡) over ℝ^𝑑 satisfies the large deviation principle with the good rate function 𝐼, where 𝐼𝑦=inf𝐼̃𝑔;𝑔∈𝐶_𝜂[−𝑟,𝑇];ℝ^𝑑,𝑦=𝑔(𝑡), (42) and 𝐼̃ 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 𝑎>0, the mapping 𝐻_𝑎∶=𝑓∈𝐻;‖𝑓‖_𝐻≤𝑎∋𝑓⟼𝑥^𝑓∈𝐶_𝜂[−𝑟,𝑇];ℝ^𝑑 (43) is continuous.

Proof. Let 𝑓,𝑔∈𝐻_𝑎. Since 𝑥^𝑓(𝑡)=𝜂(0)+^𝑡₀𝐴₀𝑠,𝑥^𝑓_𝑠𝑑𝑠+^𝑡₀𝐴𝑠,𝑥^𝑓_𝑠𝑓(̇𝑠)𝑑𝑠, (44) we see that sup_{[]𝜏∈−𝑟,𝑡}𝑥^𝑓(𝜏)≤‖𝜂‖_∞+sup_{[]𝜏∈0,𝑡}𝑥^𝑓(𝜏)≤2‖𝜂‖_∞+^𝑡₀𝐴₀𝑠,𝑥^𝑓_𝑠+𝑑𝑠_𝑡0‖𝐴𝑠,𝑥^𝑓_𝑠‖_{ℝ𝑚⊗ℝ𝑑}𝑓(̇𝑠)𝑑𝑠≤𝐶_20,𝑇,𝜂+𝐶_21,𝑇^𝑡₀𝑓(̇𝑠×1+)1+sup_{[]𝜏∈−𝑟,𝑠}𝑥^𝑓(𝜏)𝑑𝑠, (45) from the linear growth condition on 𝐴_𝑖(𝑖=0,1,…,𝑚), which tells us to see that sup_{[]𝜏∈−𝑟,𝑇}𝑥^𝑓(𝜏)≤𝐶_{22,𝑇,𝜂,𝑎}. (46) On the other hand, since 𝑥^𝑓(𝑡)−𝑥^𝑔(𝑡)=^𝑡₀𝐴₀𝑠,𝑥^𝑓_𝑠−𝐴₀𝑠,𝑥^𝑔_𝑠+𝑑𝑠_𝑡0𝐴𝑠,𝑥^𝑓_𝑠𝑓(̇𝑠)−𝐴𝑠,𝑥^𝑔_𝑠𝑔(̇𝑠)𝑑𝑠, (47) for 𝑡∈(0,𝑇], and the ℝ^𝑑-valued functions 𝐴_𝑖(𝑖=0,1,…,𝑚) satisfy the Lipschitz condition and the linear growth condition, we have sup_{[]𝜏∈−𝑟,𝑡}𝑥^𝑓(𝜏)−𝑥^𝑔(𝜏)=sup_{[]𝜏∈0,𝑡}𝑥^𝑓(𝜏)−𝑥^𝑔(𝜏≤_𝑡0)𝐴₀s,𝑥^𝑓_𝑠−𝐴₀𝑠,𝑥^𝑔_𝑠𝑑𝑠+^𝑡₀‖𝐴𝑠,𝑥^𝑓_𝑠−𝐴𝑠,𝑥^𝑔_𝑠‖_{ℝ𝑚⊗ℝ𝑑}𝑓(̇𝑠+)𝑑𝑠_𝑡0‖𝐴𝑠,𝑥^𝑔_𝑠‖_{ℝ𝑚⊗ℝ𝑑}𝑓(̇𝑠)−𝑔(̇𝑠)𝑑𝑠≤𝐶_23,𝑇^𝑡₀sup_{[]𝜏∈−𝑟,𝑠}𝑥^𝑓(𝜏)−𝑥^𝑔(𝜏)1+^𝑚_𝑖=1𝑓̇^𝑖(𝑠)𝑑s+𝐶_{24,𝑇,𝜂,𝑎}‖𝑓−𝑔‖_𝐻. (48) The Gronwall inequality tells us to see that sup_{[]𝜏∈−𝑟,𝑡}𝑥^𝑓(𝜏)−𝑥^𝑔(𝜏)≤𝐶_{24,𝑇,𝜂,𝑎}‖𝑓−𝑔‖_𝐻×exp𝐶_23,𝑇^𝑡₀1+^𝑚_𝑖=1𝑓̇^𝑖(𝑠)𝑑𝑠≤𝐶_{25,𝑇,𝜂,𝑎}‖𝑓−𝑔‖_𝐻, (49) which completes the proof.

Proposition 13. Suppose that the ℝ^𝑑-valued functions 𝐴_𝑖(𝑖=1,…,𝑚) are bounded. Then, for any 𝑓∈𝐻 and 𝜌>0, there exist 𝛼_𝜌>0 and 𝜀_𝜌>0 such that ℙsup_{𝜏∈[−𝑟,𝑇]}𝑋^𝜀(𝜏)−𝑥^𝑓(𝜏)>𝜌,sup_{𝜏∈[0,𝑇]}𝜀𝑊(𝜏)−𝑓(𝜏)≤𝛼_𝜌≤𝐶_{26,𝑇,𝑓,𝜌}exp−𝐶_27,𝑇,𝑓𝜌²𝜀², (50) for any 0<𝜀≤𝜀_𝜌.

Proof. Define a new probability measure 𝑑ℙ by 𝑑ℙ𝑑ℙ_ℱ𝑇=exp^𝑇₀^𝑚_𝑖=1𝑓̇^𝑖(𝑠)𝜀𝑑𝑊^𝑖(𝑠)−‖𝑓‖²_𝐻2𝜀². (51) The Girsanov theorem tells us to see that the ℝ^𝑚-valued process {𝑊(𝑡)∶=𝑊(𝑡)−𝑓(𝑡)/𝜀;𝑡∈[0,𝑇]} is also the 𝑚-dimensional Brownian motion under the probability measure 𝑑ℙ. Let {𝑋^𝜀,𝑓(𝑡);𝑡∈[−𝑟,𝑇]} be the ℝ^𝑑-valued process determined by the following equation: 𝑋^𝜀,𝑓(𝑡)=𝜂(𝑡)(𝑡∈[−𝑟,0]),𝑑𝑋^𝜀,𝑓(𝑡)=𝐴₀𝑡,𝑋_𝑡^𝜀,𝑓𝑑𝑡+𝐴𝑡,𝑋_𝑡^𝜀,𝑓𝜀𝑑𝑊(𝑡)+𝑓(̇𝑡)𝑑𝑡(𝑡∈(0,𝑇]). (52) Write 𝑀(𝑡)∶=∫^𝑡₀𝐴(𝑠,𝑋_𝑠^𝜀,𝑓)𝑑𝑊(𝑠). Remark that sup_{[]𝜏∈−𝑟,𝑡}𝑋^𝜀,𝑓(𝜏)−𝑥^𝑓(𝜏)=sup_{[]𝜏∈0,𝑡}𝑋^𝜀,𝑓(𝜏)−𝑥^𝑓(𝜏≤_𝑡0)𝐴₀𝑠,𝑋_𝑠^𝜀,𝑓−𝐴₀𝑠,𝑥^𝑓_𝑠𝑑𝑠+^𝑡₀𝐴𝑠,𝑋_𝑠^𝜀,𝑓−𝐴𝑠,𝑥^𝑓_{𝑠ℝ𝑚⊗ℝ𝑑}𝑓(̇𝑠)𝑑𝑠+sup_{[]𝜏∈0,𝑡}|𝜀𝑀(𝜏)|≤𝐶_28,𝑇^𝑡₀sup_{[]𝜏∈−𝑟,𝑠}𝑋^𝜀,𝑓(𝜏)−𝑥^𝑓(𝜏×)1+^𝑚_𝑖=1𝑓̇^𝑖(𝑠)𝑑𝑠+sup_{[]𝜏∈0,𝑡}|𝜀𝑀(𝜏)|. (53) The Gronwall inequality tells us to see that sup_{[]𝜏∈−𝑟,𝑡}𝑋^𝜀,𝑓(𝜏)−𝑥^𝑓(𝜏≤)sup_{[]𝜏∈0,𝑡}|(𝜏𝐶𝜀𝑀)|×exp_28,𝑇^𝑡₀1+^𝑚_𝑖=1𝑓̇^𝑖(𝑠)𝑑𝑠≤𝐶_29,𝑇,𝑓sup_{[]𝜏∈0,𝑡}|𝜀𝑀(𝜏)|. (54) For each 𝑘=1,…,𝑑, the martingale representation theorem enables us to see that there exists a 1-dimensional Brownian motion {𝐵^𝑘(𝑡);𝑡∈[0,𝑇]} starting at the origin with 𝑀^𝑘(𝑡)=𝐵^𝑘𝑀^𝑘𝑡),𝑀_𝑘((𝑡)=^𝑡₀^𝑚_𝑖=1𝐴^𝑘_𝑖𝑠,𝑋_𝑠^𝜀,𝑓2𝑑𝑠, (55) for 𝑘=1,…,𝑑. Remark that ⟨𝑀^𝑘⟩(𝑡)≤𝐶_30,𝑇, because of the boundedness of the ℝ^𝑑-valued functions 𝐴_𝑖(𝑖=1,…,𝑚). Since ℙsup_{𝜏∈0,𝐶30,𝑇}𝐵^𝑘(𝜏>𝜌)𝐶_31,𝑇,𝑓𝜀≤√−𝜌2exp²4𝐶_30,𝑇𝐶²_31,𝑇,𝑓𝜀², (56) from the reflection principle on Brownian motions, we have ℙsup_{[]𝜏∈−𝑟,𝑇}𝑋^𝜀(𝜏)−𝑥^𝑓(𝜏)>𝜌,sup_{𝜏∈[0,𝑇]}𝜀𝑊(𝜏)−𝑓(𝜏)≤𝛼_𝜌=ℙsup_{[]𝜏∈−𝑟,𝑇}𝑋^𝜀,𝑓(𝜏)−𝑥^𝑓(𝜏)>𝜌,sup_{𝜏∈[0,𝑇]}𝜀𝑊(𝜏)≤𝛼_𝜌≤ℙsup_{[]𝜏∈0,𝑇}|𝑀(𝜏>𝜌)|𝐶_29,𝑇,𝑓𝜀≤ℙsup_{𝜏∈0,𝐶30,𝑇}|𝐵(𝜏>𝜌)|𝐶_29,𝑇,𝑓𝜀≤ℙ^𝑑_𝑘=1sup_{𝜏∈0,𝐶30,𝑇}𝐵^𝑘(𝜏>𝜌)𝐶_29,𝑇,𝑓√≤√𝑑𝜀−𝜌2𝑑exp²4𝐶_30,𝑇𝐶²_29,𝑇,𝑓𝑑𝜀², (57) which completes the proof.

Proposition 14. It holds that lim_𝑅→+∞limsup_𝜀↘0𝜀lnℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)>𝑅=−∞. (58)

Proof . Let 𝑁>2 be sufficient large. From the Itô formula, we see that 1+𝑋^𝜀(𝑡)^2𝑁=1+𝜂(0)^2𝑁+^𝑡₀𝑁(1+|𝑋^𝜀(𝑠)|²)^𝑁−12𝜀𝑋^𝜀(𝑠)⋅𝐴𝑠,𝑋^𝜀_𝑠(𝑠)+𝑑𝑊_𝑡0𝑁1+𝑋^𝜀(𝑠)^2𝑁−1×2𝑋^𝜀(𝑠)⋅𝐴₀𝑠,𝑋^𝜀_𝑠+𝜀^2𝑚_𝑖=1𝐴_𝑖𝑠,𝑋^𝜀_𝑠²+2𝑁(𝑁−1)𝜀²(1+|𝑋^𝜀(𝑠)|²)^𝑁−2×^𝑚_𝑖=1𝑋^𝜀(𝑠)⋅𝐴_𝑖𝑠,𝑋^𝜀_𝑠²𝑑𝑠. (59) Define 𝜎_𝑅=inf{𝑡>0;|𝑋^𝜀(𝑡)|>𝑅}. Then, it holds that 𝔼(1+|𝑋^𝜀(𝑡∧𝜎_𝑅)|²)^𝑁≤(1+‖𝜂‖²_∞)^𝑁+𝔼^{𝑡∧𝜎𝑅}₀𝑁1+𝑋^𝜀(𝑠)^2𝑁−1×2𝑋^𝜀(𝑠)⋅𝐴₀𝑠,𝑋^𝜀_𝑠+𝜀^2𝑚_𝑖=1𝐴_𝑖𝑠,𝑋^𝜀_𝑠²+2𝑁(𝑁−1)𝜀²(1+|𝑋^𝜀(𝑠)|²)^𝑁−2×^𝑚_𝑖=1(𝑋^𝜀(𝑠)⋅𝐴_𝑖𝑠,𝑋^𝜀_𝑠)²≤𝑑𝑠1+‖𝜂‖_∞^𝑁+𝐶_32,𝑇𝑁+𝜀²𝑁+𝜀²𝑁²×𝔼^𝑡₀1+𝑋^𝜀𝑠∧𝜎_𝑅^2𝑁𝑑𝑠, (60) from the linear growth condition on the coefficients 𝐴_𝑖(𝑖=0,1,…,𝑚) of (2). Hence, the Gronwall inequality implies that 𝔼1+𝑋^𝜀𝑡∧𝜎_𝑅^2𝑁≤(1+‖𝜂‖_∞)^𝑁exp𝐶_32,𝑇𝑁+𝜀²𝑁+𝜀²𝑁²𝑡. (61) In particular, taking 𝑁=1/𝜀 yields that 𝔼1+𝑋^𝜀𝑡∧𝜎_𝑅^21/𝜀≤(1+‖𝜂‖_∞)^1/𝜀exp𝐶_33,𝑇1𝜀+1𝑡. (62) Therefore, the Chebyshev inequality leads us to see that ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡𝜎)>𝑅=ℙ_𝑅𝑋≤𝑇≤ℙ^𝜀𝑇∧𝜎_𝑅≥𝑅≤(1+𝑅²)^−1/𝜀𝔼(1+|𝑋^𝜀(𝑇∧𝜎_𝑅)|²)^𝑁≤1+‖𝜂‖²_∞1+𝑅^21/𝜀exp𝐶_33,𝑇1𝜀+1𝑇, (63) so we have limsup_𝜀↘0𝜀lnℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)>𝑅≤ln1+‖𝜂‖²_∞1+𝑅²+𝐶_33,𝑇𝑇, (64) which completes the proof.

Let 𝑅≥1. Define that 𝜎_𝑅=inf{𝑡>0;|𝑋^𝜀(𝑡)|>𝑅} and 𝑋^𝜀,𝑅(𝑡)=𝑋^𝜀(𝑡∧𝜎_𝑅).

Proposition 15. For any 𝛿>0, it holds that lim_𝑅→+∞limsup_𝜀↘0𝜀lnℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡)>𝛿=−∞. (65)

Proof . Remark that ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡)>𝛿≤ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡)>𝛿,sup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)≤𝑅+ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)>𝑅=ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡)>𝛿,𝜎_𝑅𝜎≥𝑇+ℙ_𝑅𝜎≤𝑇=ℙ_𝑅𝑋≤𝑇≤ℙ^𝜀𝑇∧𝜎_𝑅≤≥𝑅1+‖𝜂‖²_∞1+𝑅^21/𝜀exp𝐶_33,𝑇1𝜀+1𝑇, (66) as seen in the proof of Proposition 14. So, we can get limsup_𝜀↘0𝜀lnℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡)>𝛿≤ln1+‖𝜂‖²_∞1+𝑅²+𝐶_33,𝑇𝑇, (67) which completes the proof.

Proof of Theorem 10. We will prove the assertion in two steps of the form: the case where 𝐴_𝑖(𝑖=1,…,𝑚) are bounded, and the general case on 𝐴_𝑖(𝑖=1,…,𝑚).
Step 1. Suppose that the coefficients 𝐴_𝑖(𝑖=1,…,𝑚) are bounded. Propositions 12 and 13 are sufficient to our goal (cf. [17, 18]). In fact, the large deviation principle for the family {ℙ∘(𝑋^𝜀)⁻¹;0<𝜀≤1} comes from the one for {ℙ∘(𝜀𝑊)⁻¹;0<𝜀≤1} in Lemma 9.
Step 2. We will discuss the general case on 𝐴_𝑖(𝑖=1,…,𝑚). Let 𝑅≥1, and 𝐹 be a closed set in 𝐶_𝜂([−𝑟,𝑇];ℝ^𝑑). Denote by 𝐹_𝑅=𝐹∩𝐵(0;𝑅) and by 𝐹^𝛿_𝑅 the closed 𝛿-neighborhood of 𝐹_𝑅, where 𝐵(0;𝑅) is the open ball in 𝐶_𝜂([−𝑟,𝑇];ℝ^𝑑) with radius 𝑅 centered at 0∈𝐶_𝜂([−𝑟,𝑇];ℝ^𝑑). Then, it holds that ℙ𝑋^𝜀𝑋∈𝐹≤ℙ^𝜀∈𝐹,sup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)≤𝑅+ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡𝑋)>𝑅=ℙ^𝜀,𝑅∈𝐹_𝑅+ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)>𝑅. (68) As seen in Step 1, we have already obtained the large deviation principle for {ℙ∘(𝑋^𝜀,𝑅)⁻¹;0<𝜀≤1} with the good rate function 𝐼_𝑅̃, where 𝐼(𝑓) is given in Lemma 9 and 𝐼_𝑅̃𝑔=inf𝐼𝑓;𝑓∈𝐻,𝑔=𝑥^𝑓,sup_{[]𝑡∈−𝑟,𝑇}𝑥^𝑓(𝑡)≤𝑅. (69) So, we have limsup_𝜀↘0𝜀lnℙ𝑋^𝜀,𝑅∈𝐹_𝑅≤−inf_{𝑔∈𝐹𝑅}𝐼_𝑅̃𝑔. (70) Therefore, we can get limsup_𝜀↘0𝜀lnℙ𝑋^𝜀∈𝐹≤lim_𝑅→+∞−inf_{𝑔∈𝐹𝑅}𝐼_𝑅̃𝑔∨limsup_𝜀↘0𝜀lnℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)>𝑅=lim_𝑅→+∞−inf_{𝑔∈𝐹𝑅}𝐼̃𝑔≤−inf_𝑔∈𝐹𝐼̃𝑔, (71) 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 𝐺 be an open set in 𝐶_𝜂([−𝑟,𝑇];ℝ^𝑑), and take 𝑔 in 𝐺∩𝐵(0;𝑅). Then, we can find 𝛿>0 such that 𝐵(𝑔;𝛿)⊂𝐺. Thus, we have −𝐼̃𝑔=−𝐼_𝑅̃𝑔≤−inf_𝑔∈𝐵𝑔^;𝛿/2𝐼_𝑅̃𝑔≤liminf_𝜀↘0𝜀lnℙ𝑋^𝜀,𝑅𝛿∈𝐵𝑔;2≤liminf_𝜀↘0𝜀×lnℙ𝑋^𝜀∈𝐺+ℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡>𝛿)2≤liminf_𝜀↘0𝜀lnℙ𝑋^𝜀∨∈𝐺liminf_𝜀↘0𝜀lnℙsup_{[]𝑡∈−𝑟,𝑇}𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡>𝛿)2. (72) The first equality is right, because of 𝑔∈𝐵(0;𝑅), while the third inequality is the consequence of the large deviation principle for 𝑋^𝜀,𝑅 as seen in Step 1. The forth inequality is right, because 𝑋^𝜀∈𝐵(𝑔;𝛿/2)^𝑐 under 𝑋^𝜀,𝑅∈𝐵(𝑔;𝛿)^𝑐 and sup_{𝑡∈[−𝑟,𝑇]}|𝑋^𝜀(𝑡)−𝑋^𝜀,𝑅(𝑡)|≤𝛿/2. Taking the limit as 𝑅→+∞ leads us to see that −𝐼̃𝑔≤liminf_𝜀↘0𝜀lnℙ𝑋^𝜀∈𝐺, (73) 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 𝑝^𝜀(𝑡,𝑦) for the solution 𝑋^𝜀(𝑡), from the viewpoint of the Malliavin calculus.

Theorem 16 (Upper estimate). Suppose that the ℝ^𝑑-valued functions 𝐴_𝑖(𝑖=1,…,𝑚) satisfy the uniformly elliptic condition (24). Then, it holds that limsup_𝜀↘0𝜀²ln𝑝^𝜀𝑡,𝑦≤−𝐼𝑦, (74) where the function 𝐼 is given in Corollary 11.

Proof. Let 0<𝜎<1 be sufficiently small, and Λ_𝜎∈𝐶^∞₀(ℝ^𝑑;[0,1]) such that Λ_𝜎(𝑧)=1,0.𝑧−𝑦≤𝜎𝑧−𝑦>2𝜎 (75) Take 𝑈=∏^𝑑_𝑗=1[𝑎_𝑗,𝑏_𝑗]⊂ℝ^𝑑 such that 𝑈⊂Supp[Λ_𝜎]. Then, the integration by parts formula tells us to see that ℙ𝑋^𝜀(𝑡)𝕀∈𝑈=𝔼_𝑈𝑋^𝜀(𝑡)Λ_𝜎𝑋^𝜀(𝑡)=𝔼^{𝑋𝜀,1(𝑡)}_−∞⋯^{𝑋𝜀,𝑑(𝑡)}_−∞𝕀_𝑈𝑦₁,…,𝑦_𝑑𝑑𝑦₁⋯𝑑𝑦_𝑑Γ_(1,…,𝑑)𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡)=_𝑈𝔼^𝑑_𝑗=1𝕀_{(𝑦𝑗,+∞)}𝑋^𝜀,𝑗(𝑡)×Γ_(1,…,𝑑)𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡)𝑑𝑦₁⋯𝑑𝑦_𝑑, (76) where Γ_𝑗𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡Λ)=𝛿_𝜎𝑋^𝜀(𝑡)^𝑑_𝑘=1[(𝑉^𝜀(𝑡))⁻¹]_𝑗𝑘𝐷𝑋^𝜀,𝑘(𝑡,Γ_(1,…,𝑑))𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡)=Γ_𝑑𝑋^𝜀(𝑡),Γ_{(1,…,𝑑−1)}𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡), (77) and 𝛿 is the Skorokhod integral operator. Remark that, under the uniformly elliptic condition (24) on the ℝ^𝑑-valued functions 𝐴_𝑖(𝑖=1,…,𝑚), Γ_(1,…,𝑑)𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡)_𝕃𝑝(Ω)≤𝐶_{34,𝑝,𝑇,𝜂}(𝑉^𝜀(𝑡))⁻¹_𝕃𝛼(Ω)𝑋^𝜀(𝑡)_𝛽,𝛾Λ_𝜎𝑋^𝜀(𝑡)_𝜅,𝜎≤𝐶_{35,𝑝,𝑇,𝜂}𝜀^−2𝑑, (78) where 𝛼,𝛾,𝜎>1 and 𝛽,𝜅∈ℤ₊, by using Proposition 5 and the proof of Lemma 7. Hence, the density 𝑝^𝜀(𝑡,𝑦) can be estimated from the above as follows: 𝑝^𝜀𝑡,𝑦=𝔼^𝑑_𝑗=1𝕀_{(𝑦𝑗,+∞)}𝑋^𝜀,𝑗(𝑡)Γ_(1,…,𝑑)𝑋^𝜀(𝑡),Λ_𝜎𝑋^𝜀(𝑡)