Moderate Deviations Type Evaluation for Integral Functionals of Diffusion Processes

We establish a large deviations type evaluation for the family of integral functionals ε −κ T ε 0 Ψ(X ε s)g(ξ ε s)ds, ε 0, where Ψ and g are smooth functions, ξ ε t is a " fast " ergodic diffusion while X ε t is a " slow " diffusion type process, κ ∈ (0, 1/2). Under the assumption that g has zero barycenter with respect to the invariant distribution of the fast diffusion, we derive the main result from the moderate deviation principle for the family (ε −κ t 0 g(ξ ε s)ds) t≥0 , ε 0 which has an independent interest as well. In addition, we give a preview for a vector case.


Introduction
In this paper, we consider a two scaled diffusion model with independent Wiener processes V t and W t : The fast component ξ ε t is assumed to be an ergodic Markov process while the slow component X ε t is a diffusion type process governed by the fast process ξ ε = (ξ ε ) t≥0 and independent of it Wiener process W = (W t ) t≥0 .Under appropriate conditions, a stochastic version of the Bogolubov averaging principle holds (see [12]), that is, the slow process is averaged with respect to the invariant density of the fast one, say p(z).In other words, the X ε t process is approximated by a Markov diffusion process X t with respect to some Wiener process W t : with the averaged drift and diffusion parameters Let us assume functions F and G are unknown and indicate here a statistical procedure for estimation of the averaged function F from the observation of the 'slow" process X ε t .The averaging principle suggests the following recipe: to proceed with the path of X ε t as if it is the path of X t .For instance, it is well known the kernel estimate (with kernel K and bandwidth dt is taken as estimate of F (x) via X ε , 0 ≤ t ≤ T .An asymptotic analysis, as ε → 0, for F ε (x) leads to study of properties for integral functionals where F (i) (x, y) is the i -th derivative of F (x, y) in x.Namely, assuming that both T and h depend on ε : T = T ε , h = h ε with T ε ∞, h 0, we need to show that for any of functions F (x, y), F (1) (x, y), F (2) (x, y) specified as H(x, y) the integral goes to zero faster than ε κ with some κ > 0 (see [18]).For fixed x, denote by g(ξ t ) = H(x, ξ ε t ) − H(x, y) p(y) dy and by Ψ(X ε t ) = K( Then the desired property holds, if for instance with T ε ε → 0 we have We avoid a straightforward verification of this asymptotic and prefer to consider first the special case, Ψ ≡ 1 , which is of an independent interest and allows to clarify the main idea and to simplify the exposition.So, let where g = g(z) is an arbitrary function with zero barycenter with respect to the invariant density p.Our approach to the asymptotic analysis employs, so called, Poisson decomposition in the form used in [4] for proving the central limit theorem (CLT): where e ε,κ t is a negligible process and S ε t is a continuous martingale with the predictable quadratic variation S ε,κ t is "close" to a linear increasing function γt, so that the S ε,κ t process is approximated in the distribution sense by Wiener process with the diffusion parameter γ.The Poisson decomposition allows to analyze the asymptotic behavior, in ε 0, of S ε,κ t under 0 ≤ κ ≤ 1/2.We exclude two extreme points κ = 1/2 and κ = 0 and emphasize only that for κ = 1/2 the family (S ε t ) t≥0 , ε 0 obeys the CLT (see e.g [4]) while for κ = 0 a family of occupation measures of (ξ ε t ) t≥0 , ε 0, obeys the large deviation principle (LDP) (see [7] or e.g.[16]) and so, due to the contraction principle of Varadhan [25], the family (S ε,κ t ) t≥0 , ε 0, at least for bounded g, obeys the LDP as well.In contrast to both, the case 0 < κ < 1/2 preserves the large deviation type property, which is the same as for a family of Wiener processes parametrized by diffusion parameter ε 1−2κ γ.In other words, the case 0 < κ < 1/2 guarantees, so called, moderate deviation evaluation for the family (S ε,κ t ) t≥0 , ε 0. For establishing the moderate deviation principle (MDP), we use the conditions on the drift and diffusion parameters b and σ (see (A-1) in Section 2) proposed by Khasminskii, [13] and modified by Veretennikov, [26].These conditions allow to verify that both e ε,κ t and S ε,κ t − γt are "exponentially negligible" processes with the rate of speed ε 1−2κ .Formally we could apply a recent Pukhalskii's result [23], which being adapted to the case considered is reformulated as: if S ε t and γt are exponentially indistinguishable with the rate of speed ε 1−2κ , then the family (ε 1−2κ S ε t ) t≥0 , ε 0 obeys the same type of LDP as (ε 1/2−κ √ γ W t ) t≥0 does, that is with the rate of speed ε 1−2κ and the rate function of Freidlin-Wentzell's type (see [12]): for absolutely continuous function ϕ Nevertheless, we give here another proof of the same implication, in which the role of the "fast" convergence of S ε t to γt is discovered with more details and might be interested by itself.For the integral functional 1 t )dt we also apply the Poisson decomposition (5) in which the first term is the Itô integral with respect to the semimartingale e ε,κ t while the second one is the Itô integral with respect to the martingale S ε t .As for "Ψ ≡ 1", the main contribution comes from the second term and many details of proof are borrowed from "Ψ ≡ 1".
Results on the MDP for processes with independent increments are well known from Borovkov, Mogulski [2], [3] and Chen [5], Ledoux [15].For the depended case, the MDP estimations have attracted some attention as well.Some pertinent MDP results can be found in: Bayer and Freidlin [1] for models with averaging, Wu [27] for Markov processes, Dembo [8] for martingales with bounded jumps, Dembo and Zajic [9] for functional empirical processes, Dembo and Zeitouni [10] for iterates of expanding maps.The paper is organized as follows.In Section 2, we fix assumptions and formulate main results.Proofs of the main results are given in Section 3. Taking into account an interest to the vector case setting, in Section 4 we give a preview for the MDP with a vector fast process.

Acknowledgement
The authors thank S. Pergamenshchikov and A. Veretennikov for helpful remarks and suggestions leading to significant improvement of the paper.

Assumptions. Formulation of main results
For a generic positive constant, notation ' ' will be used hereafter.We fix the following assumptions.The initial conditions ξ 0 and X 0 for the Itô equations ( 1) and ( 2) respectively are deterministic and independent of ε.
(A-0) The function g(x) is continuously differentiable and

continuous, and Lipschitz continuous in
x uniformly in z.
(A-3) Function Ψ = Ψ(x) is twice continuously differentiable and bounded jointly with its derivatives (the value Ψ * = sup x |Ψ(x)| is involved in the formulation of the main result).
(A-4) For some d > 0 and every ε > 0, T ε ≥ d and for a chosen κ ∈ (0, 1 2 ) It is well known from [14], [24] that under (A-1) the process ξ ε t is ergodic with the uniquely defined invariant density p(z) = const. .Under (A-0), we have R |g(z)|p(z)dz < ∞ and in addition to (A-0) assume Then, the function is well defined and bounded.Set Our main result is formulated in the theorem below.
Let M t be a continuous local martingale with the predictable quadratic variation M t .It is well known that M t is a continuous process and (λ ∈ R) is a continuous local martingale.Being positive, the Z t process is a supermartingale (see e.g.Problem 1.4.4 in [18]) and therefore for every Markov time τ (on the set {τ = ∞}, We apply this property for the following useful Lemma 1 Let τ be a stopping time and A be an event from F .
1.If there exists a positive constant α so that 2. Let η and B be positive constants so that M τ ≥ η, M τ ≤ B on the set A. Then 3. If for fixed T > 0, B > 0 it holds M T ≤ B, then Proof: 1.By virtue of ( 10), 1 ≥ EI A Z τ (1) ≥ P (A)e α and the result holds.
3. Introduce Markov times τ ± = inf{t : ±M t ≥ η}, where inf{?} = ∞, and two sets 2B , it remains to note only that The proof is the same as for 3. with

Poisson decomposition for S ε,κ
Let S ε,κ t and v(y) be defined in ( 4) and ( 7) respectivelly.Set u(z) = z 0 v(y)dy, z ∈ R. It is well known (see, e.g.[14]) that the invariant density p(z) of the fast process satisfies the Fokker-Plank-Kolmogorov equation 0 Lemma 2 Under (A-0), (A-1), and ( 6), the Poisson decomposition holds Moreover, for every t > 0 and a suitable constant , Proof: Let us consider the conjugate to (11) equation with g from ( 4): It is clear the function v, defined in (7), is one of solutions of this equation.The Itô formula, applied to u(ξ ε t ), gives the required decomposition Under ( 6) v(z) is bounded function and so, the last statement of the lemma holds. 2

Exponential negligibility of e ε,κ
We establish here the exponential negligibility (13) is derived from Lemma 2 and the following lemma below.
Lemma 3 Under (A-1), for every η > 0 and sufficiently large L , Proof: We start with useful remarks: 1. the second statement of the lemma is valid provided that for some positive constant C and L large enough (hereafter we will use the constant C from the assumption (A-1.3)); 2. the first statement of the lemma is valid, if for all L So, only ( 14) and ( 15) will be verified below.By the Itô formula we have where where An appropriate lower bound for the right side of ( 17) is constructed as follows.The nonnegative value ψ(c ν , ξ ε T ) is excluded from the right side of (17).Then, with C from assumption (A-1), we find Due to assumption (A-1), there exists a positive constant H(ν, C), depending on ν and C, such that and thus, the first term in the right side of ( 17) is larger than −H(ν, C)T.The second term in the right side of ( 17) is evaluated below by using (A-1): Hence, with ν = ν • = 1/c , it holds Then, with chosen L, by Lemma 1 we have and ( 14) follows. 2 Remark 1 We emphasize one estimate which is useful for verifying the statement of Theorem 1.For ε small and L large enough and any T

Martingale S ε t
The process , is the continuous martingale with the predictable quadratic variation By assumption (A-1) the random process ξ ε t is ergodic in the following sense (see e.g.[14]): for every continuous and bounded function h and fixed t P − lim ε→0 t 0 h(ξ ε s )ds = R h(z)p(z)dz.Hence, with γ defined in (8), The proof of Theorem 2 requires a stronger ergodic property.

Lemma 4 Assume (A-1)
. Then for every T > 0 and η > 0 Proof: Set g(x) = v 2 (x)σ 2 (x) − γ and note that Since the function g is bounded, continuously differentiable, and R g(z)p(z)dz = 0, the function z −∞ g(y)p(y)dy (compare ( 7)) is continuously differentiable and bounded as well.Define also the function u(z) = z 0 v(y)dy.The same arguments, which have been applied for the proof of Lemma 2, provide the Poisson decomposition with With ε < 1 we have ε < ε 1−κ and so similarly to the proof of ( 13) we obtain Thus, it suffices to check that lim The process S ε t is the continuous martingale and its predictable quadratic variation fulfills √ ε S ε T ≤ ε T .Then by Lemma 1 and the required assertion follows. 2

The MDP
We are now ready to complete the proof of Theorem 2. Due to Lemma 3 the families (ε 1/2−κ S ε,κ t ) t≥0 and (ε 1/2−κ S ε t ) t≥0 are exponentially indistinguishable with the rate of speed ε 1−2κ , that is if one of family obey the MDP with the rate of speed ε 1−2κ , then the another family possesses the same property.We will examine the MDP for the family of martingales (ε 1/2−κ S ε t ) t≥0 .To this end, we apply the Dawson-Gärtner theorem (see e.g.[10] and [22]) which states that it suffices to check that the family (ε 1/2−κ S ε t ) t≤T obeys the MDP (for every T > 0) in the metric space (C [0,T ] , r T ) (r T is the uniform metric on [0, T ]) with the rate of speed ε 1−2κ and the rate function For fixed T , for the verification of the above-mentioned MDP we use well known implication (see e.g. in [17] Theorem 1.3) formally reformulated here for the MDP case: Let us recall the definitions of exponential tightness and local MDP.
Following Deushel and Stroock [6] (see also Lynch and Sethuraman [20])), the family with the rate of speed ε 1−2κ , if there exists a sequence of compacts (K j ) j≥1 : Effective sufficient conditions for ( 27) are known from Pukhalskii [22] lim where "sup" is taken over all stopping times τ ≤ T .

Verification of (28)
Introduce Markov times (with inf{?} = ∞) and note that (28) holds, if lim We consider separately cases "±".Since S ε t is the continuous martingale, the positive process is the local martingale and the supermartingale as well, that is The proof for the case "−" is similar. 2

Verification of (30)
By virtue of Lemma 4, we check only lim sup For ϕ 0 = 0 the left side of ( 34) is equal to −∞.Hence, for ϕ 0 = 0 the desired upper bound holds.
Let ϕ 0 = 0. Introduce the continuous local martingale s , where λ(t) is piece wise constant right continuous function.The process Now, we find a lower bound for the random value z T (λ) on the set Since λ(t) is right continuous having limit to the left function of bounded variation on [0, T ], the Itô formula for λ(t) S ε t is valid: Applying it, we find Hence, there is a constant , depending on T and λ, so that on the set U ε,δ,η , If ϕ t is not absolutely continuous function, one can choose a sequence of piece wise constant and right continuous functions λ n (t)'s so that the right side of (35) tends to −∞ along with n → ∞.If ϕ t is absolutely continuous function the right side of ( 35) is transformed into

Verification of (31)
We use the obvious lower bound It is clear that the verification of (36) is required only for absolutely continuous functions ϕ t with ϕ 0 = 0 and J T (ϕ) < ∞.Moreover, this class of functions can be reduced to twice continuously differentiable functions (ϕ t ) t≤T with ϕ 0 = 0.In fact, if ϕ 0 = 0 and J T (ϕ) < ∞ but ϕ t is absolutely continuous only and even φt is unbounded, then one can choose a sequence ϕ n , n ≥ 1 of twice continuously differentiable functions with ϕ n 0 ≡ 0 such that lim Thus, let ϕ t be twice continuously differentiable function with ϕ 0 = 0. Set λ • (t) = φt γ and define the martingale and so, z t (λ We use this equality to introduce new probability measure P • : dP • = z T (λ • )dP .Since z T (λ • ) > 0, P -a.s., not only P • P but also P To finish the proof, it remains to check that for every δ > 0, η > 0 It is well known (see e.g.Theorem 4.5.2 in [19]) that the random process ( S ε t ) t≤T , being Pcontinuous martingale, is transformed to P • -continuous semimartingale with the decomposition S ε t = A ε t + N ε t , where (N ε t ) t≤T is the continuous local martingale having S ε t as the quadratic variation and the drift A ε t is defined via the mutual variation z(λ • ), S ε t of martingales z(λ • ) t and S ε t as: Hence s and we find that Therefore, with a suitable constant , we obtain As was mentioned above N ε T coincides with S ε T (P − and P • -a.s.) and so, it is bounded P • -a.s.Now, by the Doob inequality (E • is the expectation with respect to P • ) The latter property allows to conclude that (38) holds provided that (39) is valid.
Let us recall that S ε t − γt obeys the Poisson decomposition (see (23) and ( 24)) To this end, we find an upper bound for sup where the mutual variation S ε , S ε t of P -martingales S ε t and S ε t is given, due to (12) and (24), by the formula The boundedness of v, v, and σ 2 implies sup t≤T |A ε t | ≤ const.and so (42) holds.
To finish the proof of (39), it remains to check ¿From the definition of e ε t it follows the existence of positive constants L 1 , L 2 so that sup The verification of (44) uses the semimartingale decomposition of ξ ε t with respect to P Hence By the Itô formula we find Due to (A-1), sup Then, by the Doob inequality Consequently (44) holds. 2

Proof of Theorem 1
As for Ψ ≡ 1, the proof uses the Poisson decomposition from Lemma 2: 1 Taking now δ = ε 1 2 (1−κ) , we arrive at the upper bound: with a suitable constant which tends to −∞ as ε → 0. Hence, it remains to prove only that for every δ > 0 (T ε ε) 1−2κ log P sup To this end, we apply Lemma 1: The proof of (46) uses the following auxiliary statement which slightly extends the result of Lemma 4. It is shown in [21] that under (A σ ) and (A b ) ξ t is an ergodic process with the unique invariant measure, µ.
We assume the function g is continuous, bounded, and As in the scalar case, we exploit the Poisson decomposition applying the result from [21]  and note that since the matrix σσ * is nonsingular γ > 0 for any u with ∇u ≡ 0.
Theorem 3 Under the setting of this Section and 1/2 < κ < 1, the family (S ε,κ t ) t≥0 , ε 0 obeys the MDP in the metric space (C , r) with the rate of speed ε 1−2κ and the rate function (9) with γ from (49).

0 2 2
LT , i.e. (15) is valid.Next, the function b obeys the following property (see (A-1)): b(y) ≤ −cy, y > C and ≥ −cy, y < −C which provides the boundedness for the positive part of the function z 0 b(y)dy.Therefore for every fixed positive ν one can choose a positive constant c ν such that the function ψ(c ν , z) = c ν − ν z 0 b(y)dy is nonnegative.Since the function b is smooth, the function ψ(c ν , z) is twice continuously differentiable in z and the Itô formula, applied to ψ(c ν , ξ ε t ), gives

where L = 1 2 a
ij (x)∂ x i ∂ x j + b i (x)∂ x i is the diffusion operator.It is shown in [21]that the Poisson equation obeys a bounded solution the gradient of which ∇u = ( ∂u(x) ∂x 1 , . . ., ∂u i (z) ∂x d ) has bounded components .Introduce γ = R d ∇u(x)σ(x)σ * (x)∇ * (x))µ(dx) (49) Due to (A-2) and (A-4) there exists a positive constant so that sup t≤T ε |U The same arguments, as were used for the proof for (i=1), yield