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Maximum Likelihood Drift Estimation for Multiscale Diffusions

We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.


Introduction
Fitting stochastic differential equations (SDEs) to time-series data is often a useful way of extracting simple model fits which capture important aspects of the dynamics [9]. However, whilst the data may well be compatible with an SDE model in many respects, it is often incompatible with the desired model at small scales. Since many commonly applied statistical techniques see the data at small scales this can lead to inconsistencies between the data and the desired model fit. This phenomenon appears quite often in econometrics [1,2,13], where the term market microstructure noise is used to describe the high frequency/small scale part of the data as well as in molecular dynamics [19]. In essence, the problem that we are facing is that there is an inconsistency between the coarse-grained model that we are using and the microscopic dynamics from which the data is generated, at small scales. Similar problems appear quite often in statistical inference, in the context of parameter estimation for misspecified or incorrect models [11,Sec. 2.6].
The aim of this paper is to create a theoretical framework in which it is possible to study this issue, in order to gain better insight into how it is manifest in practice, and how to overcome it. In particular our goal is to investigate the following problem: how can we fit data obtained from the high-dimensional, multiscale full dynamics to a low-dimensional, coarse grained model which governs the evolution of the resolved ("slow") degrees of freedom? We will study this question for a class of stochastic systems for which we can derive rigorously a coarse grained description for the dynamics of the resolved variables. More specifically, we will work in the framework of coupled systems of multiscale SDEs for a pair of unknown functions (x(t), y(t)). We assume that y(t) is fast, relative to x(t), and that the equations average or homogenize to give a closed equation for X(t) to which x(t) converges in the limit of infinite scale separation. The function X(t) then approximates x(t), typically in the sense of weak convergence of probability measures [7,20]. We then ask the following question: given data for x(t), from the coupled system, can we correctly identify parameters in the averaged or homogenized model for X(t)?
Fast/slow systems of SDEs of this form have been studied extensively over the last four decades [4,14,20] and the references therein. Recently, various methods have been proposed for solving numerically these SDEs [6,8,23]. In these works, the coefficients of the limiting SDE are calculated "on the fly" from simulations of the fast/slow system. There is a direct link between these numerical methods and our approach in that our goal is also to infer information about the coefficients in the coarse-grained equation using data from the multiscale system. However, our interest is mainly in situations where the "microscopic" multiscale system is not known explicitly. From this point of view, we merely use the multiscale stochastic system as our "data generating process"; our goal is to fit this data to the coarse-grained equation for X(t), the limit of the slow variable x(t).
A first step towards the understanding of this problem was taken in [19]. There, the data generating process x(t) was taken to be the path of a particle moving in a multiscale potential under the influence of thermal noise. The goal was to identify parameters in the drift as well as the diffusion coefficient in the homogenized model for X(t), the weak limit of x(t). It was shown that the maximum likelihood estimator is asymptotically biased and that subsampling is necessary in order to estimate the parameters of the homogenized limit correctly, based on a time series (i.e. single observation) of x(t).
In this paper we extend the analysis to more general classes of fast/slow systems of SDEs for which either an averaging or homogenization principle holds [20]. We consider cases where the drift in the averaged or homogenized equation contains parameters which we want to estimate using observations of the slow variable in the fast/slow system. We show that in the case of averaging the maximum likelihood function is asymptotically unbiased and that we can estimate correctly the parameters of the drift in the averaged model from a single path of the slow variable x(t). On the other hand, we show rigorously that the maximum likelihood estimator is asymptotically biased for homogenization problems. In particular, an additional term appears in the likelihood function in the limit of infinite scale separation. We show then that this term vanishes, and hence that the maximum likelihood estimator becomes asymptotically unbiased, provided that we subsample at an appropriate rate.
To be more specific, in this paper we will consider fast/slow systems of SDEs of the form We will refer to equations (1.1) as the averaging problem and to equations (1.2) as the homogenization problem. In both cases our assumptions on the coefficients in the SDEs are such that a coarse-grained (averaged or homogenized) equation exists, which is of the form dX dt = F (X; θ) + K(X) dW dt . (1. 3) The slow variable x(t) converges weakly, in the limit as ǫ → 0, to X(t), the solution of (1.3). We assume that the vector field F (X; θ) depends on a set of parameters θ that we want to estimate based on data from either the averaging or the homogenization problem. We suppose that the actual drift compatible with the data is given by F (X) = F (X; θ 0 ). We ask whether it is possible to correctly identify θ = θ 0 by finding the maximum likelihood estimator (MLE) when using a statistical model of the form (1.3), but given data from (1.1) or (1.2). Our main results can be stated, informally, as follows. Precise statements of the above results can be found in Theorems 3.10, 3.12 and 3.13.
The failure of the MLE when applied to the homogenization problem is due to the presence of high frequency data. Naturally, in order to be able to identify correctly the parameter θ = θ 0 in (1.3) using data from (1.2a) subsampling at an appropriate rate is necessary.
Theorem 1.2. The MLE for the homogenization problem becomes asymptotically unbiased if we subsample at an appropriate rate.
Roughly speaking, the sampling rate should be between the two characteristic time scales of the fast/slow SDEs (1.2), 1 and ǫ 2 . The precise statement of this result can be found in Theorems 4.1 and 4.5. IIn practice real data will not come explicitly from a scale-separated model like (1.1a) or (1.2a). However real data is often multiscale in character. Thus the results in this paper shed light on the pitfalls that may arise when fitting simplified statistical models to multiscale data. Furthermore the results indicate the central, and subtle, role played by subsampling data in order to overcome mismatch between model and data at small scales.
The rest of the paper is organized as follows. In Section 2 we study the fast/slow stochastic systems introduced above, and prove appropriate averaging and homogenization theorems. In Section 3 we introduce the maximum likelihood function for (1.3) and study its limiting behavior, given data from the averaging and homogenization problems (1.1a) and (1.2a). In Section 4 we show that, when subsampling at an appropriate rate, the maximum likelihood estimator for the homogenization problem becomes asymptotically unbiased. In Section 5 we present examples of fast/slow stochastic systems that fit into the general framework of this paper. Section 6 is reserved for conclusions. Various technical results are proved in the appendices.

Set-Up
We will consider fast/slow systems of SDEs for the variables (x, y) ∈ X × Y. We can take, for example, In the second case, where the state space is compact, all of the assumptions that we need for the proofs of our results can be justified using elliptic PDEs theory.
Let ϕ t ξ (y) denote the Markov process which solves the SDE Here ξ ∈ X is a fixed parameter and, for each t ≥ 0, ϕ t ξ (y) ∈ Y, g 0 : X × Y → R d−l , β : X × Y → R (d−l)×m and V is a standard Brownian motion in m dimensions. 1 The generator of the process is with B(ξ, y) := β(ξ, y)β(ξ, y) T . Notice that L 0 (ξ) is a differential operator in y alone, with ξ a parameter. Our interest is in data generated by the projection onto the x coordinate of systems of SDEs for (x, y) in X × Y. In particular, for U a standard Brownian motion in R n we will consider either of the following coupled systems of SDEs: or the SDEs Here f i : X ×Y → R l , α 0 : X ×Y → R l×n , α 1 : X ×Y → R l×m , g 1 : X ×Y → R d−l and g 0 , β and V are as above.
• The functions f i , g i , α i , β and all derivatives are uniformly bounded in X × Y.
• If h(y; ξ) and all its derivatives with respect to y, ξ are uniformly bounded in X × Y then the same is true of Θ solving the Poisson equation above.

Remark 2.2.
In the case where the state space of the fast process is compact, Y = T d−ℓ , and the diffusion matrix B(ξ, y) is positive definite the above assumptions can be easily proved using elliptic PDE theory [20,Ch. 6]. Similar results can also be proved without the compactness and uniform ellipticity assumptions [15,16,17].
The first assumption essentially states the the process (2.1) is ergodic, for each ξ ∈ X . Let L 0 = L 0 (x) and define The generators for the Markov processes defined by equations (2.3) and (2.4) respectively are with the understanding that f 0 ≡ 0 and g 1 ≡ 0 in the case of L av . We let Ω denote the probability space for the pair of Brownian motions U, V . In (2.3) (resp. (2.4)) the dynamics for y with x viewed as frozen has solution ϕ t/ǫ x (y(0)) (resp. ϕ t/ǫ 2 x (y(0))). Of course x is not frozen, but since it evolves much more slowly than y, intuition based on freezing x and considering the process (2.1) is useful in understanding how averaging and homogenization arise for equations (2.3) and (2.4) respectively. Specifically, for (2.3) on timescales long compared with ǫ and short compared to 1, x will be approximately frozen and y will traverse its invariant measure with density ρ(y; x). We may thus average over this measure and eliminate y. Similar ideas hold for equation (2.4), but are complicated by the presence of the term ǫ −1 f 0 . These ideas underly the averaging and homogenization results contained in the next two subsections.
Note that K(x)K(x) T is positive semidefinite and hence K(x) is well defined via, for example, the Cholesky decomposition.
where W is ca standard l-dimensional Brownian motion.
We use the notation Ω 0 to denote the probability space for the Brownian motion W .

Proof. Consider the Poisson equation
. Applying Itô's formula to Ξ we obtain From this we obtain Thus, by Assumptions 2.1 and the Burkholder-Davis-Gundy inequality, Hence The quadratic variation process for M (t) is By use of the Poisson equation technique applied above to show that f 0 (x, y) can be approximated by F (x) (its average against the fast y process), we can show similarly that where, as above, Let F (x(s))ds + e 0 (t), Then where M (t) and M (t)M (t) T −q(t) are F t martingales, where F t is the filtration generated by σ((U (s), V (s)), s ≤ t). Let C ∞ c (X ) denote the space of compactly supported C ∞ functions. The martingale problem for is well posed and x(s), y(s) and X(s) are continuous. By L 2 convergence of the e i to 0 in C([0, T ], X ) we deduce convergence to 0 in probability, in the same space. Hence by a slight generalization of Theorem 4.1 in Chapter 7 of [7] we deduce the desired result.

Homogenization
In order for the equations (2.4) to produce a sensible limit as ǫ → 0 it is necessary to impose a condition on f 0 . Specifically we assume the following which, roughly, says that f 0 (x, y) averages to zero against the invariant measure of the fast y process, with x fixed.

Assumptions 2.4. The function f 0 satisfies the centering condition
which is unique by Assumptions 2.4. Define

Also define
Note that K(x)K(x) T is positive semidefinite by construction so that K(x) is well defined by, for example, the Cholesky decomposition.

Theorem 2.5. Let Assumptions 2.1, 2.4 hold. Then
where W is a standard l-dimensional Brownian motion.
Proof. We consider three Poisson equations: that for Φ given above and All of these equations have a unique solution since the right hand sides average to zero against the density ρ(y; x) by assumption (Φ) or by construction (χ, Ψ).
By the Itô formula we obtain From this we obtain, using arguments similar to those in the proof of Theorem 2.3, and where, recall, Ω is the probability space for (U, V ). Applying Itô's formula to χ, the solution of (2.10a), we may show that Thus and By applying Itô's formula to Ψ, the solution of (2.10b) we obtain Thus and Here

By use of the Poisson equation technique we can show that
where, as above, The remainder of the proof proceeds as in Theorem 2.3.

Parameter Estimation
Recall that Ω 0 is the probability space for W . Imagine that we try to fit data {x(t)} t∈[0,T ] from (2.3) or (2.4) to a homogenized or averaged equation of the from (2.7) or (2.9), but with unknown parameter θ ∈ Θ, where Θ is an open subset of R k , in the drift: Suppose that the actual drift compatible with the data is given by F (X) = F (X; θ 0 ). We ask whether it is possible to correctly identify θ = θ 0 by finding the maximum likelihood estimator (MLE) when using a statistical model of the form (3.1), but given data from (2.3) or (2.4). Recall that the averaging and homogenization techniques from the previous section show that x(t) from (2.3) and (2.4) converges weakly to the solution of an equation of the form (3.1). We make the following assumptions concerning the model equations (3.1) which will be used to fit the data.
Assumptions 3.1. We assume that K is uniformly positive-definite on X . We also assume that Given data {z(t)} t∈[0,T ] , the log likelihood function for θ satisfying (3.1) is given by To be precise where P is the path space measure for (3.1) and P 0 the pathspace measure for As a preliminary to understanding the effect of using multiscale data, we start by exhibiting an underlying property of the log-likelihood when confronted with data from the model (3.1) itself. The following theorem shows that, in this case: (i) in the limit T → ∞ the log-likelihood is asymptotically independent of the particular sample path of (3.1) chosen -it depends only on the invariant measure π; (ii) as a consequence we see that, asymptotically, time-ordering of the data is irrelevant to parameter estimation; (iii) under some additional assumptions, the large T expression also shows that choosing data from the model (3.1) leads to the correct estimation of drift parameters, in the limit T → ∞.

Proof. By Lemmas A.2 and A.3 in the appendix we deduce that, with all limits in
Completing the square provides the proof.
In the particular case where the parameter θ appears linearly in the drift it can be viewed as an R l×l matrix Θ and The correct value for Θ is thus the R l×l identity matrix I. The maximum likelihood estimator isΘ is not invertible then we setΘ(z; T ) = 0. A result closely related to Theorem 3.2 is the following 2 : Proof. We observe that B(X; T ) = A(X; T ) + J 1 where and where E|J 1 | 2 = O(1/T ) by Lemma A.2. By ergodicity, and Lemma A.3, we have that where E|J 2 | 2 = O(1/T ) and A ∞ is given by (3.2). By Assumption 3.1 and for T sufficiently large, A(z; T ) is invertible and we havê and the result follows.
Remark 3.4. The invertibility of A ∞ is necessary in order to be able to successfully estimate the drift of the linear system.
In order to prove an analogue of Theorem 3.3 when the drift depends nonlinearly on the parameter θ we need to make additional assumptions.

Assumptions 3.5.
• We assume that When (3.7) holds we will say that the system is identifiable.
• There exist an α > 0 andF : Under the above assumption we can prove convergence of the MLE to the correct value θ 0 . Proof. It is a straightforward application of the results in [22].
We now ask whether the likelihood behaves similarly when confronted with data {x(t)} from the underlying multiscale systems (2.3) or (2.4). To address this issue we make the following natural assumptions regarding the invariant measure for these underlying multiscale systems.
• The limiting SDE (2.7) or (2.9) is ergodic with invariant measure ν(dx) which is absolutely continuous with respect to the Lebesgue measure on X with smooth density π(x).
• The invariant measure µ ǫ (dxdy) = ρ ǫ (x, y)dxdy satisfies a Poincaré inequality with a constant independent of ǫ: there exists a constant C p independent of ǫ such that for every mean zero where ∇ represents the gradient with respect to (x T , y T ) T and · denotes the L 2 (X × Y; µ ǫ (dxdy)) norm.
We also need to assume that the fast/slow SDEs (2.3) and (2.4) are uniformly elliptic.

Assumption 3.8. Define the matrix field
Then there is C γ > 0, independent of ǫ → 0 such that

Remark 3.9.
It is straightforward to show that, when X = T ℓ , Y = T ℓ−d , Assumptions 3.7 follow from Assumption 3.8, using properties of periodic functions [19], together with the compactness of the state space. When X = R ℓ , Y = R ℓ−d more work is needed in order to prove that the invariant measure satisfies Poincaré's inequality with an ǫ independent constant, since this, essentially, requires to prove that the generator of the fast/slow system has an ǫ-independent spectral gap. In this case where the fast/slow system has a gradient structure with a smooth potential V (x, y), then simple criteria on the potential have been derived that facilitate determination of whether or not the invariant measure satisfies the Poincaré inequality. We refer to [24,3] and the references therein for more details.

Averaging
We now ask what happens when the MLE for the averaged equation (3.1) is confronted with data from the original multiscale equation (2.3). The following result shows that, in this case, the estimator will behave well, for large time and small ǫ. Large time is always required for convergence of drift parameter estimation, even when model and data match. In the limit ǫ → 0, X(t) from (3.1) approximates x(t) from (2.3). Proof. We start by observing that, by Lemma A.3 and Assumptions 3.7, where the limits are in L 2 (Ω). Now, from Equation (2.3) it follows that The last two integrals tend to zero in L 2 (Ω) as T → ∞ by Lemma A.2. In order to analyze the first integral on the right hand side we consider solution of the Poisson equation This has a unique solution Λ(y; x) ∈ L 2 ρ (Y; x) by construction of F . Applying Itô's formula to Λ gives The stochastic integrals tend to zero in L 2 (Ω) as T → ∞. By assumption Λ is bounded. Furthermore, in L 2 (Ω), Hence we deduce that The result follows.
In the particular case of linear parameter dependence, when the MLE is given by (3.6) we have the following result, showing that the MLE recovers the correct answer from high frequency data compatible with the statistical model in an appropriate asymptotic limit. Proof. Using equation (2.3) we find that Here, for fixed ǫ > 0,  In fact, the following theorem is true for every ǫ > 0. Proof. Let g T (ω, θ) := 1 T L(θ; x) and It is straightforward to see that arg max θ g ∞ (θ) = θ 0 by completing the square. We apply Lemma A.4, replacing ǫ by 1 T , g ǫ by g T and g 0 by g ∞ . The result follows, provided that conditions (A.2), (A.3) and (A.4) are satisfied. Condition (A.2) follows from Theorem 3.10. The identifiability condition (A.4) follows from Assumptions 3.5 and the absolute continuity of π ǫ (x)dx = Y ρ ǫ (x, y)dy dx with respect to π(x)dx. Finally, we can verify that (A.3) holds, following the proof in [22] and using the fact that functions f 1 , α 0 and α 1 are uniformly bounded.

Homogenization
We now ask what happens when the MLE for the homogenized equation (3.1) is confronted with data from the multiscale equation (2.4), which homogenizes to give (3.1). The situation differs substantially from the case where data is taken from the multiscale equations (2.3) which averages to give (3.1): the two likelihoods are not identical in the large T limit.
In order to state the main result of this subsection we need to introduce the Poisson equation which has a unique solution Γ(y; x) ∈ L 2 ρ (Y; x). Note that where Φ solves (2.8). Define The following theorem shows that the correct limit of the log likelihood is not obtained unless E ∞ = 0, something which will not be true in general. However in the case where f 0 , g 1 ≡ 0 we do obtain E ∞ = 0 and in this case we recover the averaging situation covered in the Theorems 2.3 and Theorem 3.10 (with ǫ replaced by ǫ 2 ). a sample path of (3.1) at θ = θ 0 . Then the following limits, to be interpreted in L 2 (Ω) and L 2 (Ω 0 ) respectively, and almost surely with respect to x(0), y(0), X(0), are identical: Proof. As in the averaging case of Theorem 3.10 we have Techniques similar to those used in the proof of Theorem 3.10 show that From this we deduce that Thus where, in L 2 (Ω), By the methods used in the proof of Theorem 3.10 we deduce that Putting together all the estimates we deduce that, in L 2 ,

Subsampling
In the previous section we studied the behavior of estimators when confronted with multiscale data. The data is such that, in an appropriate asymptotic limit ǫ → 0, it behaves weakly as if it comes from a single scale equation in the form of the statistical model. By considering the behavior of continuous time estimators in the limit of large time, followed by taking ǫ → 0, we studied the behavior of estimators which do not subsample the data. We showed that in the averaging set-up this did not cause a problem -the likelihood behaves as if confronted with data from the statistical model itself; but in the homogenization set-up the likelihood function was asymptotically biased for large time. In this section we show that subsampling the data can overcome this issue, provided the subsampling rate is chosen appropriately. In the following we use E π to denote expectation on X with respect to measure with density π and E ρ ǫ to denote expectation on X × Y with respect to measure with density ρ ǫ . Recall that, by Assumption 3.7 the latter measure has weak limit with density π(x)ρ(y; x). Let Ω ′ = Ω × X × Y and consider the probability measure induced on paths x, y solving (2.4) by choosing initial conditions distributed according to the measure π(x)ρ(y; x)dxdy. With expectation E under this measure we will also use the notation · p := (E| · | p ) 1/p .
We define the discrete log likelihood function found from applying the likelihood principle to the Euler-Marayama approximation of the statistical model (3.1). Let z = {z n } N −1 n=0 denote a time series in X . We obtain the likelihood Let x n = x(nδ), noting that x(t) depends on ǫ, and set x = {x n } N −1 n=0 . The basic theorem in this section proves convergence of the log likelihood function, provided that we subsample (i.e. choose δ) at an appropriate ǫ-dependent rate. We state and prove the theorem, relying on a pair of intuitively reasonable propositions which we then prove at the end of the section.  Let Assumptions 2.1, 2.4, 3.1, 3.7 and 3.8 hold. Let {x(t)} t∈[0,T ] be a sample path of (2.4) and X(t) a sample path of (3.1) at θ = θ 0 . Let δ = ǫ α with α ∈ (0, 1) and let N = [ǫ −γ ] with γ > α. Then the following limits, to be interpreted in L 2 (Ω ′ ) and L 2 (Ω 0 ) respectively, and almost surely with respect to X(0), are identical: The proof of this theorem is based on the following two technical results, whose proofs are presented in the appendix.

Proposition 4.2.
Let (x(t), y(t)) be the solution of (2.4) and assume that Assumptions 2.1 and 2.4 hold. Then, for ǫ, δ sufficiently small, the increment of the process x(t) can be written in the form where M n denotes the martingale term where the convergence is in L 2 with respect to the measure on initial conditions with density π(x)ρ(y; x).

By Proposition 4.3 we have that
We use Proposition 4.2 to deduce that F (x n ), M n a(xn) Again using Proposition 4.3 we have that Furthermore, using the fact that M n is independent of x n and has quadratic variation of order δ it follows that Here Q is defined to obtain the correct quadratic variation of the M n . Consequently, and since γ > α, as ǫ → 0. Similarly, using martingale moment inequalities [10, Eq. (3.25) p. 163] we obtain J 2 p ≤ o(1).
Finally, again using Proposition 4.2, we have, for q −1 + p −1 = 1, as ǫ → 0, since we have assumed that α ∈ (0, 1). We thus have By completing the square we obtain (4.1). As before, we would like to use this theorem in order to prove the consistency of our estimator. The theory developed in [22] no longer applies because it is based on the assumption that the function we are maximizing (i.e. the log likelihood function) is a continuous semimartingale, which is not true for the discrete semimartingale L N,δ (θ; x). The most difficult part in proving consistency is to prove that the martingale converges uniformly to zero (Assumption A.3 in Lemma A.4). To avoid this difficulty, we make some extra assumptions that allow us to get rid of the martingale part: 1. There exists a function V : X × Θ → R such that for each θ ∈ Θ, V (·, θ) ∈ C 3 (X ) and (4.4) 2. Define G : X × Θ → R as follows: where D 2 V denotes the Hessian matrix of V . Then there exist an β > 0 and G : X → R that is square integrable with respect to the invariant measure, such that Suppose that the above assumption is true and {X(t)} t∈[0,T ] is a sample path of (3.1). Then, if we apply Itô's formula to function V , we get that for every θ ∈ Θ: dV (X(t); θ) = ∇V (X(t); θ), dX(t) + 1 2 G(X(t); θ)dt.
Using this version of the log-likelihood function , we definẽ Now we can prove asymptotic consistency of the MLE, provided that we subsample at the appropriate sampling rate. Proof. We apply Lemma A.4 with g ǫ (x, θ) = 1 N δL N,δ (θ; x) and g 0 (θ) its limit. Note that lim by Proposition 4.3 and the fact that which follows from the ergodicity of X. As in Theorem 4.1, the limits are interpreted in L 2 (Ω ′ ) and L 2 (Ω 0 ) respectively, and almost surely with respect to X(0). As we have already seen, the maximizer of g 0 (θ) is θ 0 . So, Assumption (A.2) is satisfied. Also, Assumption 3.5 is equivalent to (A.4). To prove consistency, we need to prove (A.3), which can be viewed as uniform ergodicity. The proof is again similar to that in [22]. First, we note that by Assumptions 3.5 and 4.4, both g ǫ (·, θ) and g 0 (θ) are continuous with respect to θ, so it is sufficient to prove (A.3) on a countable dense subset Θ ⋆ of Θ. Then, uniform ergodicity follows from [5, Thm. 6.1.5] , provided that i.e. the number of balls of radius ǫ with respect to · L 1 (π) needed to cover is finite. As demonstrated in [22], this follows from the Hölder continuity of |F (z; θ)| 2 a(z) and G(z; θ).

Examples
Numerical experiments, illustrating the phenomena studied in this paper, can be found in the paper [19]. The experiments therein are concerned with a particular case of the general homogenization framework considered in this paper and illustrate the failure of the MLE when the data is sampled too frequently, and the role of subsampling to ameliorate this problem. In this section we construct two examples which identify the term E ∞ responsible for the failure of the MLE.

Langevin Equation in the High Friction Limit
We consider the Langevin equation in the high friction limit: 3 where V (q; θ) is a smooth confining potential depending on a parameter θ ∈ Θ ⊂ R ℓ , 4 β stands for the inverse temperature and W (t) is standard Brownian motion on R d . We write this equation as a first order system In the notation of the general homogenization set-up we have (x, y) = (q, p) and The fast process is simply an Ornstein-Uhlenbeck process with generator The unique square integrable (with respect to the invariant measure of the OU process) solution of the Poisson equation (2.8) is Φ = p. Therefore, Hence the homogenized equation is 5 Consider now the parameter estimation problem for "full dynamics" (5.1) and the "coarse grained" model (5.3): We are given data from (5.1) and we want to fit it to equation (5.3). Theorem 3.13 implies that for this problem the maximum likelihood estimator is asymptotically biased. 6 In fact, in this case we can compute the term E ∞ , responsible for the bias and given in equation (3.11). We have the following result.
Proof. We have that The invariant measure of the process is ǫ-independent and we write it is ρ(q, p; θ) dqdp = Z −1 e −βH(p,q;θ) dqdp.
Furthermore, since the homogenized diffusion matrix is 2β −1 I, where ·, · stands for the standard Euclidean inner product. We readily check that Thus, which is precisely (5.4).

Motion in a Multiscale Potential
Consider the equation [19] dx dt where the fluctuating part of the potential p(·) is taken to be a smooth 1-periodic function. Setting y = x/ǫ we obtain In the notation of the general homogenization set-up we have and The fast process has generator The invariant density is ρ(y) = Z −1 p exp(−βp(y)) with Z p = T d exp(−βp(y)) dy. The Poisson equation for Φ is L 0 Φ(y) = ∇ y p(y).
Notice that Φ is a function of y only. The homogenized equation is Suppose now that the potential contains parameters, V = V (x, θ), θ ∈ Θ ⊂ R ℓ . We want to estimate the parameter θ, given data from (5.5) and using the homogenized equation Theorem 3.13 implies that, for this problem, the maximum likelihood estimator is asymptotically biased and that subsampling at the appropriate rate is necessary for the accurate estimation of the parameter θ. As in the example presented in the previous section, we can calculate explicitly the error term E ∞ . For simplicity we will consider the problem in one dimension.
Proof. Equations (5.6) in one dimension becomė The invariant measure of this system is (notice that it is independent of ǫ) The homogenized equation iṡ The cell problem is L 0 φ = ∂ y p and the homogenized coefficient is We have that The error in the likelihood is We have that Consequently In addition, The error in the likelihood is In above derivation we used various integrations by parts, together with the formula for the derivative of the solution of the Poisson equation ∂ y φ = −1 + Z −1 p e βp(y) , [20, p. 213]. The fact that E ∞ is nonpositive follows from the inequality Z −1 p Z −1 p < 1 (for p(y) not identically equal to 0), which follows from the Cauchy-Schwarz inequality.

Conclusions
The problem of parameter estimation for fast/slow systems of SDEs which admit a coarse-grained description in terms of an SDE for the slow variable was studied in this paper. It was shown that, when applied to the averaging problem, the maximum likelihood estimator (MLE) is asymptotically unbiased and we can use it to estimate accurately the parameters in the drift coefficient of the coarse-grained model using data from the slow variable in the fast/slow system. On the contrary, the MLE is asymptotically biased when applied to the homogenization problem and a systematic asymptotic error appears in the log-likelihood function, in the long time/infinite scale separation limit. The MLE can lead to the correct estimation of the parameters in the drift coefficient of the homogenized equation provided that we subsample the data from the fast/slow system at the appropriate sampling rate.
The averaging/homogenization systems of SDEs that we consider in this paper are of quite general form and have been studied quite extensively in the last several decades since they appear in various applications, e.g. molecular dynamics, chemical kinetics, mathematical finance, atmosphere/ocean science-see the references in [20]. Thus, we believe that our results show that great care has to be taken when using maximum likelihood in order to infer information about parameters in stochastic systems with multiple characteristic time scales.
There are various problems, both of theoretical and of applied interest, that remain open and that we plan to address in future work. We list some of them below.
• Bayesian techniques for parameter estimation of multiscale diffusion processes.
• The development of efficient algorithms for estimating the parameters in the coarse-grained model of a fast/slow stochastic system. Based on the work that has been done to similar models in the context of econometrics [13,2] one expects that such an algorithm would involve the estimation of an appropriate measure of scale separation ǫ, and of the optimal sampling rate, averaging over all the available data and a bias reduction step.
• Investigate whether there is any advantage in using random sampling rates.
• Investigate similar issues for deterministic fast/slow systems of differential equations.
Then the equation has a unique solution Φ : Z → R, with Φ and ∇Φ bounded.
Then there exists a constant C > 0: E|I| 2 ≤ C for all T > 0.
Proof. Use the Itô isometry and invoke the boundedness of ψ.
The result concerning L 2 (Ω) convergence follows from boundedness of Φ, ∇Φ and γ, together with Lemma A.2. Almost sure convergence follows from the ergodic theorem.

A.3 Proof of Propositions 4.2 and 4.3
In this section we present the proofs of Propositions 4.2 and 4.3 which we repeat there, for the reader's convenience.
Proposition A.5. Let (x(t), y(t)) be the solution of (2.4) and assume that Assumptions 2.1 and 2.4 hold. Then, for ǫ, δ sufficiently small, the increment of the process x(t) can be written in the form where M n denotes the martingale term Proposition A.6. Let g ∈ C 1 (X ) and let Assumptions 3.7 hold. Assume that ǫ and N are related as in Theorem 4.1. Then where the convergence is in L 2 with respect to the measure on initial conditions with density π(x)ρ(y; x).
For the proofs of Propositions A.5 and A.6, both used in the proof of Theorem 4.1, we will need the following two technical lemmas. We start with a rough estimate on the increments of the process x(t).
Estimate (A.7) follows from these estimates.
Using this lemma we can prove the following estimate.
Lemma A.8. Let h(x, y) be a smooth, bounded function, let (x(t), y(t)) be the solution of (2.4) and assume that Assumption 2.1 holds. Define h(x, y) ρ(y; x)dy.
Hence ψ ǫ , Γ∇ψ ǫ ≤ CE ρ ǫ |ĝ| 2 , where the notation introduced in (A.11) was used. The constant C in the above inequalities is independent of ǫ. Thus Since the measure with density ρ ǫ converges to the measure with density π(x)ρ(y; x) the desired result follows.