Volume 2013, Article ID 537023,17pages http://dx.doi.org/10.1155/2013/537023
Research Article
Asymptotic Behavior of Densities for Stochastic Functional Differential Equations
Akihiro Kitagawa,
1and Atsushi Takeuchi
21Aikou Educational Institute, Kinuyama 5-1610-1, Ehime Matsuyama, 791-8501, Japan
2Department of Mathematics, Osaka City University, Sugimoto 3-3-138, Sumiyoshi-ku, Osaka 558-8585, Japan
Correspondence should be addressed to Atsushi Takeuchi; [email protected] Received 30 September 2012; Accepted 10 December 2012
Academic Editor: S. Mohammed
Copyright © 2013 A. Kitagawa and A. 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.
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 pro- cesses, because the current states of the process in the equa- tion depend on the past histories of the process. Such kind of equations was initiated by Itˆo 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 prob- abilistic 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 applica- ble 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 applica- tion 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. L´eandre [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 inhomoge- neous, and they are not only the drift terms, but also the diffu- sion 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. InSection 2, we will prepare some notations and introduce our stochastic func- tional differential equations.Section 3will 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 inSection 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 5are our main goals in the present paper. InSection 4, we will focus on the large deviation principles on the solution processes.
As an application of the result obtained inSection 4, we will study the asymptotic behavior on the density for the solution process. Moreover, we can also derive the short time asymp- totics on the density function, which can be interpreted as the generalization of the Varadhan-type estimate on diffusion processes (cf. [5–9]).
2. Preliminaries
Let𝑟and𝑇be positive constants, and denote an𝑚-dimen- sional Brownian motion by 𝑊 = {𝑊(𝑡) = (𝑊1(𝑡), . . . , 𝑊𝑚(𝑡)); 𝑡 ∈ [0, 𝑇]}. Let𝐴𝑖 (𝑖 = 0, 1, . . . , 𝑚)beR𝑑-valued functions on[0, 𝑇] × 𝐶([−𝑟, 0];R𝑑)such that, for each𝑡 ∈ [0, 𝑇], the mapping 𝐴𝑖(𝑡, ⋅) : 𝐶([−𝑟, 0];R𝑑) ∋ 𝑓 →
𝐴𝑖(𝑡, 𝑓) ∈R𝑑is smooth in the Frech´et sense and all Frech´et derivatives of any orders greater than1are bounded. Under the conditions stated above, the functions𝐴𝑖 (𝑖 = 0, 1, . . . , 𝑚) satisfy the linear growth condition and the Lipschitz condi- tion in the functional sense of the form:
sup
𝑡∈[0,𝑇]
∑𝑚
𝑖=0𝐴𝑖(𝑡, 𝑓) ≤ 𝐶1,𝑇 (1 + 𝑓∞) ,
𝑡∈[0,𝑇]sup
∑𝑚
𝑖=0𝐴𝑖(𝑡, 𝑓) − 𝐴𝑖(𝑡, 𝑔) ≤ 𝐶2,𝑇𝑓 − 𝑔∞, (1)
for 𝑓, 𝑔 ∈ 𝐶([−𝑟, 0];R𝑑), where ‖𝑓‖∞ = sup𝑡∈[−𝑟,0]|𝑓(𝑡)|.
Denote by𝐴 = (𝐴1, . . . , 𝐴𝑚).
Let0 < 𝜀 ≤ 1be sufficiently small. For a deterministic path𝜂 ∈ 𝐶([−𝑟, 0];R𝑑), we will consider theR𝑑-valued pro- cess𝑋𝜀 = {𝑋𝜀(𝑡); 𝑡 ∈ [−𝑟, 𝑇]}given by the stochastic func- tional differential equation of the form:
𝑋𝜀(𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0]) ,
𝑑𝑋𝜀(𝑡) = 𝐴0(𝑡, 𝑋𝜀𝑡) 𝑑𝑡 + 𝜀𝐴 (𝑡, 𝑋𝜀𝑡) 𝑑𝑊 (𝑡) (𝑡 ∈ (0, 𝑇]) , (2)
where𝑋𝜀𝑠= {𝑋𝜀(𝑠 + 𝑢); 𝑢 ∈ [−𝑟, 0]}is the segment. Since the current state of the solution depends on its past histories, the process𝑋𝜀is 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 approximation𝑋𝜀,(𝑛)= {𝑋𝜀,(𝑛)(𝑡); 𝑡 ∈ [−𝑟, 𝑇]}
(𝑛 ∈Z+)of the solution process𝑋𝜀to (2) as follows:
𝑋𝜀,(0)(𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0]) , 𝑋𝜀,(0)(𝑡) = 𝜂 (0) (𝑡 ∈ (0, 𝑇]) , 𝑋𝜀,(𝑛)(𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0]) , 𝑑𝑋𝜀,(𝑛)(𝑡) = 𝐴0(𝑡, 𝑋𝜀,(𝑛−1)𝑡 )𝑑𝑡
+ 𝜀𝐴 (𝑡, 𝑋𝜀,(𝑛−1)𝑡 )𝑑𝑊 (𝑡) (𝑡 ∈ (0, 𝑇]) , (3)
for𝑛 ∈N(cf. Itˆo and Nisio [1], Mohammed [2,12]).
Proposition 1. For any𝑝 > 1, it holds that
0<𝜀≤1supE[ sup
𝑡∈[−𝑟,𝑇]|𝑋𝜀(𝑡) |𝑝] ≤ 𝐶3,𝑝,𝑇,𝜂. (4) Proof. Let𝑝 > 2and𝑡 ∈ [0, 𝑇]. The H¨older inequality and the Burkholder inequality tell us to see that
E[ sup
𝜏∈[−𝑟,𝑡]𝑋𝜀(𝜏)𝑝]
≤ 𝐶4,𝑝‖𝜂‖𝑝∞+ 𝐶4,𝑝E[sup
𝜏∈[0,𝑡]|𝑋𝜀(𝜏) |𝑝]
≤ 𝐶4,𝑝‖𝜂‖𝑝∞+ 𝐶5,𝑝E[sup
𝜏∈[0,𝑡]
∫0𝜏𝐴0(𝑠, 𝑋𝜀𝑠) 𝑑𝑠
𝑝] + 𝐶5,𝑝 𝜀𝑝E[sup
𝜏∈[0,𝑡]
∫0𝜏𝐴 (𝑠, 𝑋𝜀𝑠) 𝑑𝑊 (𝑠)
𝑝]
≤ 𝐶4,𝑝‖𝜂‖𝑝∞+ 𝐶5,𝑝𝑇𝑝−1 ∫𝑡
0E[|𝐴0(𝑠, 𝑋𝜀𝑠) |𝑝] 𝑑𝑠 + 𝐶6,𝑝 𝜀𝑝 𝑇𝑝/2−1 ∫𝑡
0
∑𝑚 𝑖=1
E[|𝐴𝑖(𝑠, 𝑋𝜀𝑠) |𝑝] 𝑑𝑠
≤ 𝐶7,𝑝,𝑇,𝜂+ 𝐶8,𝑝,𝑇∫𝑡
0E[ sup
𝜏∈[−𝑟,𝑠]𝑋𝜀(𝜏)𝑝] 𝑑𝑠,
(5)
from the linear growth condition on the coefficients𝐴𝑖 (𝑖 = 0, 1, . . . , 𝑚). Hence, the Gronwall inequality enables us to obtain the assertion for𝑝 > 2.
As for1 < 𝑝 ≤ 2, the Jensen inequality yields us to see that
0<𝜀≤1supE[ sup
𝑡∈[−𝑟,𝑇]|𝑋𝜀(𝑡) |𝑝] ≤ (sup
0<𝜀≤1E[ sup
𝑡∈[−𝑟,𝑇]|𝑋𝜀(𝑡)|2𝑝])
1/2
, (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 Malli- avin calculus on the Wiener space 𝐶0([0, 𝑇];R𝑚), briefly, where 𝐶0([0, 𝑇];R𝑚) is the set of R𝑚-valued continuous functions on[0, 𝑇]starting from the origin. See Di Nunno et al. [13] and Nualart [9,14] for details. Let𝐻be the Camer- on-Martin subspace of𝐶0([0, 𝑇];R𝑚)with the inner product
⟨𝑔, ℎ⟩𝐻= ∫𝑇
0 ̇𝑔 (𝑡) ⋅ ̇ℎ (𝑡) 𝑑𝑡 (𝑔, ℎ ∈ 𝐻) . (7) Denote bySthe set ofR-valued random variables such that a random variable𝐹is represented as the following form:
𝐹 (𝑊) = 𝑓 (𝑊 [ℎ1] , . . . , 𝑊 [ℎ𝑛]) (8) for 𝑊 ∈ 𝐶0([0, 𝑇];R𝑚), whereℎ1, . . . , ℎ𝑛 ∈ 𝐻, 𝑊[ℎ] =
∫0𝑇ℎ(𝑠) ⋅ 𝑑𝑊(𝑠)forℎ ∈ 𝐻, and𝑓 ∈ 𝐶∞𝑝 (R𝑛;R). Here, we will denote by𝐶𝑝∞(R𝑛;R)the set of smooth functions onR𝑛such that all derivatives of any orders have polynomial growth. For 𝑘 ∈ N, the𝑘-times Malliavin-Shigekawa derivative𝐷𝑘𝐹 = {𝐷𝑢𝑘1,...,𝑢𝑘𝐹; 𝑢1, . . . , 𝑢𝑘∈ [0, 𝑇]}for𝐹 ∈Sis defined by
𝐷𝑘𝑢1,...,𝑢𝑘𝐹 (𝑊) = {{ {{ {{ {{ {{ {{ {
∑𝑛 𝑗=1
𝜕𝑗𝑓 (𝑊 [ℎ1] , . . . , 𝑊 [ℎ𝑛])
× ∫𝑢1
0 ℎ𝑗(𝑠) 𝑑𝑠 (𝑘 = 1) ,
𝐷𝑢1⋅ ⋅ ⋅ 𝐷𝑢𝑘𝐹 (𝑊) (𝑘 ≥ 2) . (9) We will consider𝐷0𝐹 = 𝐹, which helps us to define the oper- ator𝐷𝑘for𝑘 ∈ Z+. For𝑝 > 1and𝑘 ∈ Z+, letD𝑘,𝑝be the completion ofSwith respect to the norm
‖𝐹‖𝑘,𝑝= {{ {{ {{ {
(E[|𝐹|𝑝])1/𝑝 (𝑘 = 0) ,
E[|𝐹|𝑝]1/𝑝+∑𝑘
𝑗=1
E[𝐷𝑗𝐹𝑝𝐻⊗𝑗]1/𝑝 (𝑘 ∈N) . (10) LetD𝑘,𝑝(R𝑑)be the set ofR𝑑-valued random variables with the components of which belong toD𝑘,𝑝, and setD∞(R𝑑) =
⋂𝑝>1⋂𝑘∈Z+D𝑘,𝑝(R𝑑). For𝐹 ∈D1,2(R𝑑), theR𝑑⊗R𝑑-valued random variable𝑉𝐹given by
𝑉𝐹= ⟨𝐷𝐹, 𝐷𝐹⟩𝐻= ∫𝑇
0
𝑑
𝑑𝑢𝐷𝑢𝐹 ⋅ 𝑑
𝑑𝑢𝐷𝑢𝐹𝑑𝑢 (11) is well defined, which is called the Malliavin covariance matrix for𝐹.
Before studying the application of the Malliavin calculus to the solution process𝑋𝜀to (2), we will prepare two basic and well-known facts.
Lemma 2 (cf. Kusuoka and Stroock [4], Lemma 2.1). LetΓbe a real separable Hilbert space, and𝛼 : [0, 𝑇] × Ω → R𝑚⊗ Γ be a progressively measurable process such that
E[∫𝑇
0 ‖𝛼(𝑠)‖𝑝R𝑚⊗Γ 𝑑𝑠] < +∞ (12)
for all𝑝 > 1. Then, for any𝑝 > 2and𝜏 ∈ [0, 𝑇], it holds that E[sup
𝑡∈[0,𝜏]
∫0𝑡𝛼 (𝑠) 𝑑𝑊 (𝑠)
𝑝Γ] ≤ 𝐶9,𝑝,𝜏∫𝜏
0
∑𝑚 𝑖=1
E[𝛼𝑖(𝑠)𝑝Γ] 𝑑𝑠.
(13) Lemma 3 (cf. Nualart [9], Proposition1.3.8). Let{𝛽(𝑡); 𝑡 ∈ [0, 𝑇]}be a(F𝑡)-adapted,R𝑚⊗R𝑑-valued process such that 𝛽(𝑡) ∈D1,2(R𝑚⊗R𝑑)for almost all𝑡 ∈ [0, 𝑇], and that
E[∫𝑇
0 ∫𝑇
0 |𝐷𝑢𝛽(𝑡)|2𝑑𝑢𝑑𝑡] < +∞. (14) Then, for each 𝑡 ∈ [0, 𝑇], it holds that ∫0𝑡𝛽(𝑠)𝑑𝑊(𝑠) ∈ D1,2(R𝑑), and that
𝐷𝑢(∫𝑡
0𝛽 (𝑠) 𝑑𝑊 (𝑠)) = ∫𝑢∧𝑡
0 𝛽 (𝑠) 𝑑𝑠 + ∫𝑡
0𝐷𝑢𝛽 (𝑠) 𝑑𝑊 (𝑠) . (15) Now, we will return our position to study the application of the Malliavin calculus to the solution process𝑋𝜀to (2).
Proposition 4. Let𝑛 ∈Z+and0 < 𝜀 ≤ 1. Then, for each𝑡 ∈ [−𝑟, 𝑇], theR𝑑-valued random variable𝑋𝜀,(𝑛)(𝑡)is inD∞(R𝑑).
Moreover, for each𝑘 ∈Z+, it holds that E[ sup
𝑡∈[−𝑟,𝑇]‖𝐷𝑘𝑋𝜀,(𝑛)(𝑡)‖𝑝𝐻⊗𝑘⊗R𝑑] ≤ 𝐶10,𝑘,𝑝,𝑇,𝜂, E[ sup
𝑡∈[−𝑟,𝑇]‖𝐷𝑘𝑋𝜀,(𝑛)(𝑡) − 𝐷𝑘𝑋𝜀,(𝑛−1)(𝑡)‖𝑝𝐻⊗𝑘⊗R𝑑] ≤ 𝐶11,𝑘,𝑝,𝑇,𝜂 2𝑛/2 .
(16) Proof. At the beginning, we will consider the case𝑝 > 2 inductively on𝑘 ∈ Z+. As for𝑘 = 0, it is a routine work to check the assertion via the H¨older inequality and the Burkholder inequality, from the Lipschitz condition and the linear growth condition on the coefficients𝐴𝑖(𝑖 = 0, 1, . . . , 𝑚), similarly toProposition 1. Next, we will discuss the case 𝑘 = 1. Let𝑛 ∈ N, because the assertion for𝑛 = 0is trivial.
Since𝐷𝑋𝜀,(𝑛) = 0for𝑡 ∈ [−𝑟, 0], we have only to prove the assertion for𝑡 ∈ (0, 𝑇]. The chain rule on the operator𝐷and Lemma 3tell us to see that
𝐷𝑢𝑋𝜀,(𝑛)(𝑡) = 𝜀 ∫𝑢
0 𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 ) I(𝑠≤𝑡)𝑑𝑠 + ∫𝑡
0∇𝐴0(𝑠, 𝑋𝜀,(𝑛−1)𝑠 ) 𝐷𝑢𝑋𝜀,(𝑛−1)𝑠 𝑑𝑠 + 𝜀 ∫𝑡
0∇𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 ) 𝐷𝑢𝑋𝜀,(𝑛−1)𝑠 𝑑𝑊 (𝑠) (17) for𝑢 ∈ [0, 𝑇](cf. Ferrante et al. [10], Lemma 6.1), where the symbol ∇is the Frech´et derivative in 𝐶([−𝑟, 0];R𝑑). Thus, the H¨older inequality and Lemma 2 enable us to get the assertions. Finally, we will discuss the general case𝑘 ∈ Z+.
Suppose that the assertions are right until the case 𝑛 − 1.
Remark that 𝐷𝑘𝑢1,...,𝑢𝑘(∫𝑡
0𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )𝑑𝑊 (𝑠))
= 𝐷𝑘−1𝑢1,...,𝑢𝑘−1(∫𝑡
0𝐷𝑢𝑘(𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )) 𝑑𝑊 (𝑠)) + 𝐷𝑢𝑘−11,...,𝑢𝑘−1(∫𝑢𝑘∧𝑡
0 𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )𝑑𝑠)
= ⋅ ⋅ ⋅
= ∫𝑡
0𝐷𝑘𝑢1,...,𝑢𝑘(𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )) 𝑑𝑊 (𝑠) + ∑
𝜎∈S𝑘
∫𝑢𝜎(𝑘)∧𝑡
0 𝐷𝑘−1𝑢𝜎(1),...,𝑢𝜎(𝑘−1)(𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )) 𝑑𝑠, (18)
from Lemma 3, where S𝑘 is the set of permutations of {1, . . . , 𝑘}. Since
𝐷𝑘𝑢1,...,𝑢𝑘(𝐴𝑖(𝑠, 𝑋𝜀,(𝑛−1)𝑠 ))
= 𝐷𝑘−1𝑢1,...,𝑢𝑘−1(∇𝐴𝑖(𝑠, 𝑋𝜀,(𝑛−1)𝑠 ) 𝐷𝑢𝑘𝑋𝜀,(𝑛−1)𝑠 )
= ∑
𝜎∈S𝑘
𝑘−1∑
𝑗=0
(𝑘 − 1𝑗 ) 𝐷𝑘−1−𝑗𝑢𝜎(1),...,𝑢𝜎(𝑘−𝑗)(∇𝐴𝑖(𝑠, 𝑋𝜀,(𝑛−1)𝑠 ))
× 𝐷𝑗+1𝑢𝜎(𝑘−𝑗+1),...,𝑢𝜎(𝑘−1),𝑢𝑘𝑋𝜀,(𝑛−1)𝑠
(19) for𝑖 = 0, 1, . . . , 𝑚, and
𝐷𝑘𝑢1,...,𝑢𝑘𝑋𝜀,(𝑛)(𝑡)
= 𝐷𝑘𝑢1,...,𝑢𝑘(∫𝑡
0𝐴0(𝑠, 𝑋𝜀,(𝑛−1)𝑠 )𝑑𝑠) + 𝐷𝑘𝑢1,...,𝑢𝑘(𝜀 ∫𝑡
0𝐴 (𝑠, 𝑋𝜀,(𝑛−1)s )𝑑𝑊 (𝑠))
= ∑
𝜎∈S𝑘
𝜀 ∫𝑢𝜎(𝑘)∧𝑡
0 𝐷𝑘−1𝑢𝜎(1),...,𝑢𝜎(𝑘−1)(𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )) 𝑑𝑠 + ∫𝑡
0𝐷𝑢𝑘1,...,𝑢𝑘(𝐴0(𝑠, 𝑋𝜀,(𝑛−1)𝑠 )) 𝑑𝑠 + 𝜀 ∫𝑡
0𝐷𝑘𝑢1,...,𝑢𝑘(𝐴 (𝑠, 𝑋𝜀,(𝑛−1)𝑠 )) 𝑑𝑊 (𝑠) ,
(20)
we can get the assertion by using the H¨older inequality, Lemma 2, and the assumption on the case until𝑘 − 1of the induction.
The case1 < 𝑝 ≤ 2is the direct consequence by the Jensen inequality. The proof is complete.
Proposition 5. For𝑡 ∈ [−𝑟, 𝑇], theR𝑑-valued random var- iable𝑋𝜀(𝑡)is inD∞(R𝑚). Moreover, for each𝑢 ∈ [0, 𝑇], the
R𝑚 ⊗R𝑑-valued process{𝐷𝑢𝑋𝜀(𝑡); 𝑡 ∈ [−𝑟, 𝑇]}satisfies the equation of the form:
𝐷𝑢𝑋𝜀(𝑡) = 0 (𝑡 ∈ [−𝑟, 0] or𝑡 < 𝑢) , 𝐷𝑢𝑋𝜀(𝑡) = 𝜀 ∫𝑢∧𝑡
0 𝐴 (𝑠, 𝑋𝜀𝑠) 𝑑𝑠 + ∫𝑡
0∇𝐴0(𝑠, 𝑋𝜀𝑠) 𝐷𝑢𝑋𝜀𝑠𝑑𝑠 + 𝜀 ∫𝑡
0∇𝐴 (𝑠, 𝑋𝜀𝑠) 𝐷𝑢𝑋𝜀𝑠𝑑𝑊 (𝑠) (𝑡 ∈ [𝑢, 𝑇]) . (21) Proof. Let𝑝 > 1and𝑘 ∈Z+be arbitrary. For each𝑡 ∈ [−𝑟, 𝑇], the sequence{𝑋𝜀,(𝑛)(𝑡); 𝑛 ∈N}is the Cauchy one inD𝑘,𝑝(R𝑑), fromProposition 4. Hence, we can find the limit, denoted by𝑋̃𝜀(𝑡), inD𝑘,𝑝(R𝑑). Then, it is a routine work to see that the process{̃𝑋𝜀(𝑡); 𝑡 ∈ [−𝑟, 𝑇]}satisfies (2), via the H¨older inequality and the Burkholder inequality, from the conditions on the coefficients𝐴𝑖(𝑖 = 0, 1, . . . , 𝑚), which implies𝑋̃𝜀(𝑡) = 𝑋𝜀(𝑡) for𝑡 ∈ [−𝑟, 𝑇]from the uniqueness of the solutions.
Thus, we can get𝑋(𝑡) ∈ D𝑘,𝑝(R𝑑)for𝑡 ∈ [−𝑟, 𝑇]. Similarly, we can check that{𝐷𝑢𝑋(𝑡); 𝑢 ∈ [0, 𝑇]}satisfies (21), by taking the limit in each term of (17) via the H¨older inequality and Lemma 2.
For𝑢 ∈ [0, 𝑇], denote by{𝑍𝜀(𝑡, 𝑢); 𝑡 ∈ [−𝑟, 𝑇]}theR𝑑⊗ R𝑑-valued process determined by the following equation:
𝑍𝜀(𝑡, 𝑢) = 0 (𝑡 ∈ [−𝑟, 0] or𝑡 < 𝑢) , 𝑍𝜀(𝑢, 𝑢) = 𝐼𝑑,
𝑑𝑍𝜀(𝑡, 𝑢) =∇𝐴0(𝑡, 𝑋𝜀𝑡) 𝑍𝜀𝑡(⋅, 𝑢) 𝑑𝑡
+ 𝜀∇𝐴 (𝑡, 𝑋𝜀𝑡) 𝑍𝜀𝑡(⋅, 𝑢) 𝑑𝑊 (𝑡) (𝑡 ∈ (𝑢, 𝑇]) , (22) where𝑍𝜀𝑡(⋅, 𝑢) = {𝑍𝜀(𝑡 + 𝜏, 𝑢); 𝜏 ∈ [−𝑟, 0]}.
Corollary 6.
𝐷𝑢𝑋𝜀(𝑡) = 𝜀 ∫𝑢∧𝑡
0 𝑍𝜀(𝑡, 𝑠) 𝐴 (𝑠, 𝑋𝜀𝑠) 𝑑𝑠. (23) Proof. Direct consequence ofProposition 5and the unique- ness 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 𝑋𝜀(𝑡)with respect to the Lebesgue measure onR𝑑.
Lemma 7 (cf. Kusuoka and Stroock [4]). Suppose the uni- formly elliptic condition on the coefficients𝐴𝑖 (𝑖 = 1, . . . , 𝑚) of (2)as follows:
𝜁∈Sinf𝑑−1 inf
𝑡∈[0,𝑇] inf
𝑓∈𝐶([−𝑟,0];R𝑑)
∑𝑚 𝑖=1
(𝜁 ⋅ 𝐴𝑖(𝑡, 𝑓))2> 0. (24) Then, for each𝑡 ∈ (0, 𝑇]and0 < 𝜀 ≤ 1, there exists a smooth density𝑝𝜀(𝑡, 𝑦)for the probability law of𝑋𝜀(𝑡)with respect to the Lebesgue measure overR𝑑.
Proof. Since𝑋(𝑡) ∈ D∞(R𝑑)fromProposition 5, it is suf- ficiently to study that(det𝑉𝜀(𝑡))−1 ∈ ⋂𝑝>1L𝑝(Ω)under the uniformly elliptic condition (24), where𝑉𝜀(𝑡)is the Malliavin covariance matrix for𝑋𝜀(𝑡). Denote by
̃𝑉𝜀(𝑡) = ∫𝑡
0
∑𝑚 𝑖=1
𝑍𝜀(𝑡, 𝑢) 𝐴𝑖(𝑢, 𝑋𝜀𝑢) 𝐴𝑖(𝑢, 𝑋𝜀𝑢)∗𝑍𝜀(𝑡, 𝑢)∗𝑑𝑢.
(25) Then, 𝑉𝜀(𝑡) = 𝜀2 ̃𝑉𝜀(𝑡), so we have only to discuss the moment estimate oñ𝑉𝜀(𝑡). As stated in Lemma 1 of Komatsu and Takeuchi [15], we will pay attention to the boundedness of
sup
𝜁∈S𝑑−1E[(𝜁 ⋅ ̃𝑉𝜀(𝑡)𝜁)−𝑝] , (26) for any𝑝 > 1, which is sufficient to our goal. Since
E[(𝜁 ⋅ ̃𝑉𝜀(𝑡) 𝜁)−𝑝] = 1 Γ (𝑝)∫+∞
0 𝜆𝑝−1
×E[exp(−𝜆𝜁 ⋅ ̃𝑉𝜀(𝑡) 𝜁)] 𝑑𝜆, (27)
we have to study the decay order of sup𝜁∈S𝑑−1E[exp(−𝜆𝜁 ⋅
̃𝑉𝜀(𝑡)𝜁)]as𝜆 → +∞.
Let𝜆 > 1be sufficiently large. Remark that
E[𝑍𝜀(𝑡, 𝑢) − 𝐼𝑑𝑝R𝑑⊗R𝑑] ≤ 𝐶12,𝑝,𝑇 (𝑡 − 𝑢)𝑝/2, (28) for any𝑝 > 1, from the Burkholder inequality and the H¨older inequality. Let𝜉 > 1/2,1 < 𝛾 < 2𝜉and0 < 𝜎 < (𝛾 − 1)/2.
Write𝑡𝜉:= 𝑡 − 𝜆−𝜉, and let𝜁 ∈S𝑑−1. Then, we see that E[exp(−𝜁 ⋅ ̃𝑉𝜀(𝑡) 𝜁)]
≤E1[exp(−𝜆 𝜁 ⋅ ̃𝑉𝜀(𝑡) 𝜁)]
+P[∫𝑡
𝑡𝜉𝑍𝜀(𝑡, 𝑢) − 𝐼𝑑2R𝑑⊗R𝑑 𝑑𝑢 ≥ 𝜆−𝛾] +P[ sup
𝑠∈[−𝑟,𝑡]𝑋𝜀(𝑠) ≥ 𝜆𝜎]
=: 𝐼1+ 𝐼2+ 𝐼3,
(29)
where
E1[⋅] :=E[⋅ : ∫𝑡
𝑡𝜉‖𝑍𝜀(𝑡, 𝑢) − 𝐼𝑑‖2R𝑑⊗R𝑑𝑑𝑢 < 𝜆−𝛾,
𝑠∈[−𝑟,𝑡]sup 𝑋𝜀(𝑠) < 𝜆𝜎] .
(30)
The Chebyshev inequality yields that 𝐼2≤ 𝜆𝛾𝑝E[(∫𝑡
𝑡𝜉𝑍𝜀(𝑡, 𝑢) − 𝐼𝑑2R𝑑⊗R𝑑𝑑𝑢)
𝑝
]
≤ 𝐶13,𝑝,𝑇 𝜆−(2𝜉−𝛾)𝑝.
(31)
Similarly, the Chebyshev inequality leads to 𝐼3≤ 𝜆−𝜎𝑝E[ sup
𝑠∈[−𝑟,𝑡]𝑋𝜀(𝑠)𝑝] ≤ 𝐶14,𝑝,𝑇,𝜂 𝜆−𝜎𝑝, (32) fromProposition 1. On the other hand, as for𝐼1, we have
𝐼1≤E1[exp(−𝜆 ∫𝑡
𝑡𝜉
∑𝑚
𝑖=1𝜁 ⋅ 𝑍𝜀(𝑡, 𝑢) 𝐴𝑖(𝑢, 𝑋𝜀𝑢)2𝑑𝑢)]
≤E1[exp(−𝜆 2 inf
𝜁∈S𝑑−1∫𝑡
𝑡𝜉
∑𝑚
𝑖=1𝜁 ⋅ 𝐴𝑖(𝑢, 𝑋𝜀𝑢)2𝑑𝑢)
×exp(𝜆 ∫𝑡
𝑡𝜉𝑍𝜀(𝑡, 𝑢)−𝐼𝑑2R𝑑⊗R𝑑
∑𝑚
𝑖=1𝐴𝑖(𝑢, 𝑋𝜀𝑢)2𝑑𝑢)]
≤exp(𝜆1−𝛾+2𝜎)
×exp(−𝜆 2 inf
𝜁∈S𝑑−1 inf
𝑢∈[0,𝑇] inf
𝑓∈𝐶([−𝑟,0];R𝑑)
∑𝑚
𝑖=1𝜁 ⋅ 𝐴𝑖(𝑢, 𝑓)2)
≤ 𝐶15exp(−𝐶16𝜆) .
(33) Therefore, we can get
E[exp(−𝜆 𝜁 ⋅ ̃𝑉𝜀(𝑡) 𝜁)] ≤ 𝐶17,𝑝,𝑇,𝜂 𝜆−𝐶18𝑝, (34) so we have
sup
𝜁∈S𝑑−1E[(𝜁 ⋅ 𝑉𝜀(𝑡) 𝜁)−𝑝] = 𝜀−2𝑑𝑝sup
𝜁∈S𝑑−1E[(𝜁 ⋅ ̃𝑉𝜀(𝑡) 𝜁)−𝑝]
≤ 𝐶19,𝑝,𝑇 𝜀−2𝑑𝑝,
(35) for any𝑝 > 1. The proof is complete.
Remark 8. Consider the case
𝐴𝑖(𝑡, 𝑓) = ̃𝐴𝑖(𝑡, 𝑓 (0)) (𝑖 = 1, . . . , 𝑚) , (36) where𝐴̃𝑖 : [0, 𝑇] ×R𝑑 → R𝑑with the good conditions on the boundedness and the regularity. Now, our stochastic functional differential equation is as follows:
𝑋𝜀(𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0]) ,
𝑑𝑋𝜀(𝑡) = 𝐴0(𝑡, 𝑋𝜀𝑡) 𝑑𝑡 + 𝜀̃𝐴 (𝑡, 𝑋𝜀(𝑡)) 𝑑𝑊 (𝑡) (𝑡 ∈ (0, 𝑇]) , (37) wherẽ𝐴 = (̃𝐴1, . . . , ̃𝐴𝑚). Then, we can get the same upper estimate of the inverse of the Malliavin covariance matrix 𝑉𝜀(𝑡)for𝑋𝜀(𝑡)inthe hypoelliptic situation, which means that the linear space generated by the vectors̃𝐴𝑖 (𝑖 = 1, . . . , 𝑚), and their Lie brackets span the spaceR𝑑(cf. Takeuchi [16]).
4. Large Deviation Principles for 𝑋
𝜀At the beginning, we will introduce the well-known fact on the sample-path large deviations for Brownian motions.
See also [8]. Recall that𝐻is the Cameron-Martin space of 𝐶0([0, 𝑇];R𝑚).
Lemma 9 (cf. Dembo and Zeitouni [17], Theorem 5.2.3). The family{P ∘ (𝜀 𝑊)−1; 0 < 𝜀 ≤ 1} of the laws of 𝜀𝑊over 𝐶0([0, 𝑇];R𝑚)satisfies the large deviation principle with the good rate function𝐼, where
𝐼 (𝑓) ={{ {{ {
𝑓2𝐻
2 (𝑓 ∈ 𝐻) , +∞ (𝑓 ∉ 𝐻) .
(38)
For𝑓 ∈ 𝐻, let𝑥𝑓 = {𝑥𝑓(𝑡); 𝑡 ∈ [−𝑟, 𝑇]}be the solution to the following functional differential equation:
𝑥𝑓(𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0]) ,
𝑑𝑥𝑓(𝑡) = 𝐴0(𝑡, 𝑥𝑓𝑡)𝑑𝑡 + 𝐴 (𝑡, 𝑥𝑓𝑡) ̇𝑓 (𝑡) 𝑑𝑡 (𝑡 ∈ (0, 𝑇]) . (39) Denote by
𝐶𝜂([−𝑟, 𝑇] ;R𝑑) = {𝑤 ∈ 𝐶 ([−𝑟, 𝑇] ;R𝑑) ;
𝑤 (𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0])} . (40) Theorem 10. The family{P∘ (𝑋𝜀)−1; 0 < 𝜀 ≤ 1}of the laws of 𝑋𝜀 over𝐶𝜂([−𝑟, 𝑇];R𝑑)satisfies the large deviation principle with the good rate functioñ𝐼, where
̃𝐼(𝑔) =inf{𝐼 (𝑓) ; 𝑓 ∈ 𝐻, 𝑔 = 𝑥𝑓} , (41) and𝐼is the function given inLemma 9.
Theorem 10 tells us to see, via the contraction principle (cf. Dembo and Zeitouni [17], Theorem4.2.1).
Corollary 11. For each𝑡 ∈ [0, 𝑇], the family{P∘ (𝑋𝜀(𝑡))−1; 0 < 𝜀 ≤ 1}of the laws of𝑋𝜀(𝑡)overR𝑑satisfies the large devia- tion principle with the good rate function𝐼, where
𝐼 (𝑦) =inf{̃𝐼 (𝑔) ; 𝑔 ∈ 𝐶𝜂([−𝑟, 𝑇] ;R𝑑) , 𝑦 = 𝑔 (𝑡)} , (42) and̃𝐼is the function given inTheorem 10.
Now, we will proveTheorem 10, according to Azencott [18] and L´eandre [5–8]. Our strategy stated here is almost parallel to [10,11].
Proposition 12. For any𝑎 > 0, the mapping
𝐻𝑎 := {𝑓 ∈ 𝐻; ‖𝑓‖𝐻≤ 𝑎} ∋ 𝑓 → 𝑥𝑓∈ 𝐶𝜂([−𝑟, 𝑇] ;R𝑑) (43) is continuous.
Proof. Let𝑓, 𝑔 ∈ 𝐻𝑎. Since 𝑥𝑓(𝑡) = 𝜂 (0) + ∫𝑡
0𝐴0(𝑠, 𝑥𝑓𝑠)𝑑𝑠 + ∫𝑡
0𝐴 (𝑠, 𝑥𝑓𝑠) ̇𝑓 (𝑠) 𝑑𝑠, (44) we see that
𝜏∈[−𝑟,𝑡]sup 𝑥𝑓(𝜏) ≤ ‖𝜂‖∞+ sup
𝜏∈[0,𝑡]𝑥𝑓(𝜏)
≤ 2‖𝜂‖∞+ ∫𝑡
0𝐴0(𝑠, 𝑥𝑠𝑓)𝑑𝑠 + ∫𝑡
0‖𝐴 (𝑠, 𝑥𝑓𝑠) ‖R𝑚⊗R𝑑 ̇𝑓 (𝑠)𝑑𝑠
≤ 𝐶20,𝑇,𝜂 + 𝐶21,𝑇 ∫𝑡
0(1 + ̇𝑓(𝑠))
× (1 + sup
𝜏∈[−𝑟,𝑠]𝑥𝑓(𝜏)) 𝑑𝑠,
(45)
from the linear growth condition on𝐴𝑖 (𝑖 = 0, 1, . . . , 𝑚), which tells us to see that
sup
𝜏∈[−𝑟,𝑇]𝑥𝑓(𝜏) ≤ 𝐶22,𝑇,𝜂,𝑎. (46) On the other hand, since
𝑥𝑓(𝑡) − 𝑥𝑔(𝑡) = ∫𝑡
0{𝐴0(𝑠, 𝑥𝑓𝑠) − 𝐴0(𝑠, 𝑥𝑔𝑠)} 𝑑𝑠 + ∫𝑡
0{𝐴 (𝑠, 𝑥𝑓𝑠) ̇𝑓 (𝑠) − 𝐴 (𝑠, 𝑥𝑔𝑠) ̇𝑔 (𝑠)} 𝑑𝑠, (47) for𝑡 ∈ (0, 𝑇], and theR𝑑-valued functions𝐴𝑖(𝑖 = 0, 1, . . . , 𝑚)satisfy the Lipschitz condition and the linear growth con- dition, we have
𝜏∈[−𝑟,𝑡]sup 𝑥𝑓(𝜏) − 𝑥𝑔(𝜏)
= sup
𝜏∈[0,𝑡]𝑥𝑓(𝜏) − 𝑥𝑔(𝜏)
≤ ∫𝑡
0𝐴0(s, 𝑥𝑓𝑠) − 𝐴0(𝑠, 𝑥𝑔𝑠)𝑑𝑠 + ∫𝑡
0‖𝐴 (𝑠, 𝑥𝑓𝑠) − 𝐴 (𝑠, 𝑥𝑔𝑠) ‖R𝑚⊗R𝑑 ̇𝑓 (𝑠)𝑑𝑠 + ∫𝑡
0‖𝐴 (𝑠, 𝑥𝑔𝑠) ‖R𝑚⊗R𝑑 ̇𝑓 (𝑠) − ̇𝑔 (𝑠)𝑑𝑠
≤𝐶23,𝑇∫𝑡
0 sup
𝜏∈[−𝑟,𝑠]𝑥𝑓(𝜏)−𝑥𝑔(𝜏) (1+
∑𝑚
𝑖=1 ̇𝑓𝑖(𝑠)) 𝑑s + 𝐶24,𝑇,𝜂,𝑎 ‖𝑓 − 𝑔‖𝐻.
(48)
The Gronwall inequality tells us to see that
𝜏∈[−𝑟,𝑡]sup 𝑥𝑓(𝜏) − 𝑥𝑔(𝜏)
≤ 𝐶24,𝑇,𝜂,𝑎‖𝑓 − 𝑔‖𝐻
×exp[𝐶23,𝑇∫𝑡
0(1 +∑𝑚
𝑖=1 ̇𝑓𝑖(𝑠)) 𝑑𝑠]
≤ 𝐶25,𝑇,𝜂,𝑎‖𝑓 − 𝑔‖𝐻,
(49)
which completes the proof.
Proposition 13. Suppose that theR𝑑-valued functions𝐴𝑖 (𝑖 = 1, . . . , 𝑚)are bounded. Then, for any𝑓 ∈ 𝐻and𝜌 > 0, there exist𝛼𝜌> 0and𝜀𝜌> 0such that
P[ sup
𝜏∈[−𝑟,𝑇]𝑋𝜀(𝜏) − 𝑥𝑓(𝜏)>𝜌, sup
𝜏∈[0,𝑇]𝜀 𝑊(𝜏)−𝑓(𝜏)≤𝛼𝜌]
≤ 𝐶26,𝑇,𝑓,𝜌exp[−𝐶27,𝑇,𝑓𝜌2 𝜀2] ,
(50) for any0 < 𝜀 ≤ 𝜀𝜌.
Proof. Define a new probability measure 𝑑̃Pby 𝑑̃P
𝑑PF𝑇
=exp[∫𝑇
0
∑𝑚 𝑖=1
̇𝑓𝑖(𝑠)
𝜀 𝑑𝑊𝑖(𝑠) −‖𝑓‖2𝐻
2𝜀2 ] . (51)
The Girsanov theorem tells us to see that the R𝑚-valued process{̃𝑊(𝑡) := 𝑊(𝑡) − 𝑓(𝑡)/𝜀; 𝑡 ∈ [0, 𝑇]}is also the𝑚- dimensional Brownian motion under the probability measure 𝑑 ̃P. Let {𝑋𝜀,𝑓(𝑡); 𝑡 ∈ [−𝑟, 𝑇]} be the R𝑑-valued process determined by the following equation:
𝑋𝜀,𝑓(𝑡) = 𝜂 (𝑡) (𝑡 ∈ [−𝑟, 0]) , 𝑑𝑋𝜀,𝑓(𝑡) =𝐴0(𝑡, 𝑋𝜀,𝑓𝑡 )𝑑𝑡
+ 𝐴 (𝑡, 𝑋𝜀,𝑓𝑡 ){𝜀𝑑̃𝑊 (𝑡) + ̇𝑓 (𝑡) 𝑑𝑡} (𝑡 ∈ (0, 𝑇]) . (52)
Write𝑀(𝑡) := ∫0𝑡𝐴(𝑠, 𝑋𝜀,𝑓𝑠 )𝑑̃𝑊(𝑠). Remark that sup
𝜏∈[−𝑟,𝑡]𝑋𝜀,𝑓(𝜏) − 𝑥𝑓(𝜏)
= sup
𝜏∈[0,𝑡]𝑋𝜀,𝑓(𝜏) − 𝑥𝑓(𝜏)
≤ ∫𝑡
0𝐴0(𝑠, 𝑋𝜀,𝑓𝑠 ) − 𝐴0(𝑠, 𝑥𝑓𝑠)𝑑𝑠 + ∫𝑡
0𝐴 (𝑠, 𝑋𝜀,𝑓𝑠 ) − 𝐴 (𝑠, 𝑥𝑠𝑓)R𝑚⊗R𝑑 ̇𝑓 (𝑠)𝑑𝑠 + sup
𝜏∈[0,𝑡]|𝜀𝑀 (𝜏)|
≤ 𝐶28,𝑇 ∫𝑡
0 sup
𝜏∈[−𝑟,𝑠]𝑋𝜀,𝑓(𝜏) − 𝑥𝑓(𝜏)
× (1 +∑𝑚
𝑖=1 ̇𝑓𝑖(𝑠)) 𝑑𝑠 + sup
𝜏∈[0,𝑡]|𝜀𝑀 (𝜏)| .
(53)
The Gronwall inequality tells us to see that
𝜏∈[−𝑟,𝑡]sup 𝑋𝜀,𝑓(𝜏) − 𝑥𝑓(𝜏)
≤ (sup
𝜏∈[0,𝑡]|𝜀𝑀 (𝜏)|)
×exp[𝐶28,𝑇∫𝑡
0(1 +∑𝑚
𝑖=1 ̇𝑓𝑖(𝑠)) 𝑑𝑠]
≤ 𝐶29,𝑇,𝑓(sup
𝜏∈[0,𝑡]|𝜀𝑀 (𝜏)|) .
(54)
For each𝑘 = 1, . . . , 𝑑, the martingale representation theorem enables us to see that there exists a1-dimensional Brownian motion{𝐵𝑘(𝑡); 𝑡 ∈ [0, 𝑇]}starting at the origin with
𝑀𝑘(𝑡) = 𝐵𝑘(⟨𝑀𝑘⟩ (𝑡)) ,
⟨𝑀𝑘⟩ (𝑡) = ∫𝑡
0
∑𝑚
𝑖=1𝐴𝑘𝑖(𝑠, 𝑋𝜀,𝑓𝑠 )2𝑑𝑠, (55) for𝑘 = 1, . . . , 𝑑. Remark that⟨𝑀𝑘⟩(𝑡) ≤ 𝐶30,𝑇, because of the boundedness of theR𝑑-valued functions𝐴𝑖(𝑖 = 1, . . . , 𝑚).
Since
P̃[ [
sup
𝜏∈[0,𝐶30,𝑇]𝐵𝑘(𝜏) > 𝜌 𝐶31,𝑇,𝑓𝜀]
]
≤ √2 exp[− 𝜌2
4 𝐶30,𝑇𝐶231,𝑇,𝑓𝜀2] ,
(56)
from the reflection principle on Brownian motions, we have
P[ sup
𝜏∈[−𝑟,𝑇]𝑋𝜀(𝜏) − 𝑥𝑓(𝜏) > 𝜌, sup
𝜏∈[0,𝑇]𝜀𝑊(𝜏) − 𝑓(𝜏) ≤ 𝛼𝜌]
=̃P[ sup
𝜏∈[−𝑟,𝑇]𝑋𝜀,𝑓(𝜏)−𝑥𝑓(𝜏)>𝜌, sup
𝜏∈[0,𝑇]𝜀𝑊 (𝜏)̃ ≤𝛼𝜌]
≤ ̃P[ sup
𝜏∈[0,𝑇]|𝑀 (𝜏)| > 𝜌 𝐶29,𝑇,𝑓𝜀]
≤ ̃P[ [
sup
𝜏∈[0,𝐶30,𝑇]|𝐵 (𝜏)| > 𝜌 𝐶29,𝑇,𝑓𝜀]
]
≤ ̃P[ [
⋃𝑑 𝑘=1
{{ {
sup
𝜏∈[0,𝐶30,𝑇]𝐵𝑘(𝜏) > 𝜌 𝐶29,𝑇,𝑓 √𝑑𝜀
}} } ] ]
≤ √2 𝑑 exp[− 𝜌2
4 𝐶30,𝑇 𝐶29,𝑇,𝑓2 𝑑𝜀2] ,
(57)
which completes the proof.
Proposition 14. It holds that
𝑅 → +∞lim lim sup
𝜀↘0 𝜀lnP[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅] = −∞. (58) Proof. Let𝑁 > 2be sufficient large. From the Itˆo formula, we see that
(1 + 𝑋𝜀(𝑡)2)𝑁
= (1 + 𝜂 (0)2)𝑁 + ∫𝑡
0𝑁(1 + |𝑋𝜀(𝑠)|2)𝑁−1 2𝜀𝑋𝜀(𝑠) ⋅ 𝐴 (𝑠, 𝑋𝜀𝑠) 𝑑𝑊 (𝑠) + ∫𝑡
0{𝑁 (1 + 𝑋𝜀(𝑠)2)𝑁−1
× (2𝑋𝜀(𝑠) ⋅ 𝐴0(𝑠, 𝑋𝜀𝑠) + 𝜀2∑𝑚
𝑖=1𝐴𝑖(𝑠, 𝑋𝜀𝑠)2) + 2𝑁 (𝑁 − 1) 𝜀2(1 + |𝑋𝜀(𝑠) |2)𝑁−2
×∑𝑚
𝑖=1(𝑋𝜀(𝑠) ⋅ 𝐴𝑖(𝑠, 𝑋𝜀𝑠))2} 𝑑𝑠.
(59)
Define𝜎𝑅=inf{𝑡 > 0; |𝑋𝜀(𝑡)| > 𝑅}. Then, it holds that
E[(1 + |𝑋𝜀(𝑡 ∧ 𝜎𝑅)|2)𝑁]
≤ (1 + ‖𝜂‖2∞)𝑁 +E[∫𝑡∧𝜎𝑅
0 {𝑁(1 + 𝑋𝜀(𝑠)2)𝑁−1
×(2𝑋𝜀(𝑠)⋅𝐴0(𝑠, 𝑋𝜀𝑠)+𝜀2∑𝑚
𝑖=1𝐴𝑖(𝑠, 𝑋𝜀𝑠)2) + 2 𝑁 (𝑁 − 1) 𝜀2(1 + |𝑋𝜀(𝑠)|2)𝑁−2
× ∑𝑚
𝑖=1
(𝑋𝜀(𝑠) ⋅ 𝐴𝑖(𝑠, 𝑋𝜀𝑠))2} 𝑑𝑠]
≤ (1 + ‖𝜂‖∞)𝑁+ 𝐶32,𝑇 (𝑁 + 𝜀2𝑁 + 𝜀2𝑁2)
×E[∫𝑡
0(1 + 𝑋𝜀(𝑠 ∧ 𝜎𝑅)2)𝑁𝑑𝑠] ,
(60) from the linear growth condition on the coefficients𝐴𝑖 (𝑖 = 0, 1, . . . , 𝑚)of (2). Hence, the Gronwall inequality implies that
E[(1 + 𝑋𝜀(𝑡 ∧ 𝜎𝑅)2)𝑁]
≤ (1 + ‖𝜂‖∞)𝑁exp[𝐶32,𝑇 (𝑁 + 𝜀2 𝑁 + 𝜀2 𝑁2)𝑡] . (61) In particular, taking𝑁 = 1/𝜀yields that
E[(1 + 𝑋𝜀(𝑡 ∧ 𝜎𝑅)2)1/𝜀]
≤ (1 + ‖𝜂‖∞)1/𝜀exp[𝐶33,𝑇(1
𝜀+ 1) 𝑡] .
(62)
Therefore, the Chebyshev inequality leads us to see that
P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅]
=P[𝜎𝑅≤ 𝑇]
≤P[𝑋𝜀(𝑇 ∧ 𝜎𝑅) ≥ 𝑅]
≤ (1 + 𝑅2)−1/𝜀E[(1 + |𝑋𝜀(𝑇 ∧ 𝜎𝑅)|2)𝑁]
≤ (1 + ‖𝜂‖2∞ 1 + 𝑅2 )
1/𝜀
exp[𝐶33,𝑇(1
𝜀 + 1) 𝑇] , (63)
so we have
lim sup
𝜀↘0 𝜀lnP[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅]
≤ln(1 + ‖𝜂‖2∞
1 + 𝑅2 ) + 𝐶33,𝑇 𝑇,
(64)
which completes the proof.
Let𝑅 ≥ 1. Define that𝜎𝑅 =inf{𝑡 > 0; |𝑋𝜀(𝑡)| > 𝑅}and 𝑋𝜀,𝑅(𝑡) = 𝑋𝜀(𝑡 ∧ 𝜎𝑅).
Proposition 15. For any𝛿 > 0, it holds that
𝑅 → +∞lim lim sup
𝜀↘0 𝜀lnP[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) − 𝑋𝜀,𝑅(𝑡) > 𝛿] = −∞.
(65) Proof. Remark that
P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) − 𝑋𝜀,𝑅(𝑡) > 𝛿]
≤P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) − 𝑋𝜀,𝑅(𝑡) > 𝛿, sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) ≤ 𝑅]
+P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅]
=P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) − 𝑋𝜀,𝑅(𝑡) > 𝛿, 𝜎𝑅≥ 𝑇]
+P[𝜎𝑅≤ 𝑇]
=P[𝜎𝑅≤ 𝑇]
≤P[𝑋𝜀(𝑇 ∧ 𝜎𝑅) ≥ 𝑅]
≤ (1 + ‖𝜂‖2∞ 1 + 𝑅2 )
1/𝜀
exp[𝐶33,𝑇 (1
𝜀+ 1) 𝑇] ,
(66) as seen in the proof ofProposition 14. So, we can get
lim sup
𝜀↘0 𝜀lnP[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) − 𝑋𝜀,𝑅(𝑡) > 𝛿]
≤ln(1 + ‖𝜂‖2∞
1 + 𝑅2 ) + 𝐶33,𝑇𝑇,
(67)
which completes the proof.
Proof ofTheorem 10.We will prove the assertion in two steps of the form: the case where𝐴𝑖 (𝑖 = 1, . . . , 𝑚)are bounded, and the general case on𝐴𝑖 (𝑖 = 1, . . . , 𝑚).
Step 1. Suppose that the coefficients𝐴𝑖 (𝑖 = 1, . . . , 𝑚)are bounded. Propositions12and13are sufficient to our goal (cf.
[17,18]). In fact, the large deviation principle for the family
{P∘ (𝑋𝜀)−1; 0 < 𝜀 ≤ 1}comes from the one for{P∘ (𝜀𝑊)−1; 0 < 𝜀 ≤ 1}inLemma 9.
Step2. We will discuss the general case on𝐴𝑖(𝑖 = 1, . . . , 𝑚).
Let𝑅 ≥ 1, and𝐹be a closed set in𝐶𝜂([−𝑟, 𝑇];R𝑑). Denote by 𝐹𝑅= 𝐹 ∩ 𝐵(0; 𝑅)and by𝐹𝑅𝛿the closed𝛿-neighborhood of𝐹𝑅, where𝐵(0; 𝑅)is the open ball in𝐶𝜂([−𝑟, 𝑇];R𝑑)with radius 𝑅centered at0 ∈ 𝐶𝜂([−𝑟, 𝑇];R𝑑). Then, it holds that
P[𝑋𝜀∈ 𝐹]
≤P[𝑋𝜀∈ 𝐹, sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) ≤ 𝑅]
+P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅]
=P[𝑋𝜀,𝑅∈ 𝐹𝑅] +P[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅] . (68)
As seen in Step 1, we have already obtained the large deviation principle for{P∘(𝑋𝜀,𝑅)−1; 0 < 𝜀 ≤ 1}with the good rate func- tioñ𝐼𝑅, where𝐼(𝑓)is given inLemma 9and
̃𝐼𝑅(𝑔) =inf{𝐼 (𝑓) ; 𝑓 ∈ 𝐻, 𝑔 = 𝑥𝑓, sup
𝑡∈[−𝑟,𝑇]𝑥𝑓(𝑡) ≤ 𝑅} . (69) So, we have
lim sup
𝜀↘0 𝜀lnP[𝑋𝜀,𝑅∈ 𝐹𝑅] ≤ −inf
𝑔∈𝐹𝑅̃𝐼𝑅(𝑔) . (70) Therefore, we can get
lim sup
𝜀↘0 𝜀lnP[𝑋𝜀∈ 𝐹]
≤ lim
𝑅 → +∞{(−inf
𝑔∈𝐹𝑅̃𝐼𝑅(𝑔))
∨ (lim sup
𝜀↘0 𝜀lnP[ sup
𝑡∈[−𝑟,𝑇]𝑋𝜀(𝑡) > 𝑅])}
= lim
𝑅 → +∞(−inf
𝑔∈𝐹𝑅̃𝐼(𝑔))
≤ −inf
𝑔∈𝐹̃𝐼(𝑔) ,
(71) fromProposition 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
