ASYMPTOTIC STATISTICAL EQUIVALENCE FOR SCALAR ERGODIC DIFFUSIONS. 1. Introduction

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1 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS ANAK DALALYAN AND MAKUS EISS Astract. For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam s deficiency etween statistical experiments is considered under long-time asymptotics. A local asymptotic equivalence result is estalished with an accompanying sequence of simple Gaussian shift experiments. Corresponding gloally asymptotically equivalent experiments are otained as compound experiments. The results are extended in several directions including time discretisation. An explicit transformation of decision functions from the Gaussian to the diffusion experiment is constructed. 1. Introduction Different statistical models often exhiit comparale features when they are considered under some natural asymptotics. In nonparametric statistics the prolems of estimating a signal in Gaussian white noise, a regression function or a density of i.i.d. oservations can all e handled y the same techniques, e.g. using kernel smoothers or projection methods, and the asymptotic minimax properties for the estimation risk usually coincide. The long standing experience that under an asymptotic point of view these models are statistically of the same kind has found its proper mathematical justification in 1996, when Brown and Low 1996 and Nussaum 1996 proved the asymptotic equivalence of these models in the sense of Le Cam s theory of equivalent statistical experiments. In essence this means that any decision function developed for one model can e carried over, at least in an astract way, to a decision function in the other models with exactly the same asymptotic risk properties. This is an important conceptual gain compared to the situation efore where asymptotic results had to e proved each time separately. In parametric statistics Le Cam s theory has een successfully applied to a huge variety of experiments ecause in this case it usually reduces to the property of local asymptotic normality LAN and its modifications Le Cam and Yang 2. The asymptotic equivalence for nonparametric experiments is conceptually more demanding and y now the class Date: Novemer 9, Mathematics Suject Classification. 62B15, 62C5, 62G2, 62M99. Key words and phrases. Asymptotic equivalence, statistical experiment, Le Cam distance, ergodic diffusion, local time, mixed Gaussian white noise. The authors acknowledge the financial support provided through the European Community s Human Potential Programme under contract HPN-CT-2-1, DYNSTOCH. 1

2 2 ANAK DALALYAN AND MAKUS EISS of models that are provaly asymptotically equivalent to the three core models of signal detection, regression and density estimation is still limited. Grama and Nussaum 1998 have proved asymptotic equivalence for generalised linear models, which has recently een extended to a wider nonparametric class in Grama and Nussaum 22, Jähnisch and Nussaum 23. Brown, Cai, Low, and Zhang 22 consider specifically nonparametric regression with random design and provide a constructive asymptotic equivalence result. Certain asymptotic equivalence results have already een otained for diffusion models. For asymptotically vanishing diffusion coefficients an equivalence result has een proved for diffusions oserved on a suitale random time interval y Genon-Catalot, Laredo, and Nussaum 22, while Milstein and Nussaum 1998 otain asymptotic equivalence for such a diffusion model and its Euler discretisation. More closely related to our work is the study of a null-recurrent diffusion model with long-time asymptotics y Delattre and Hoffmann 22. The authors prove asymptotic equivalence to Gaussian models, which have the same structure as ours with the exception of an additional mixing random variale, that can e explained in analogy with the parametric LAMN-property. To overcome technical difficulties for proving analogous results for further nonparametric models, the concept of asymptotic equivalence is sometimes reduced to its weak form, see Drees 21 for an application to lower ounds in extreme value theory. One class of standard models in mathematical statistics is certainly given y nonlinear autoregressive processes of the form X t+1 = fx t + ε t+1, t =,..., T, ε t t N N, 1 i.i.d. 1.1 and the corresponding continuous time diffusion models dx t = X t dt + dw t, t [, T ], W Brownian motion 1.2 with unknown drift functions f and, respectively. Under ergodicity assumptions and for large T it is well known that the methodology developed for nonparametric regression can e used for inference on the drift function, for an overview see Taniguchi and Kakizawa 2 for autoregressive processes and Kutoyants 23, Fan 24 for diffusions. In this paper we corroorate the folklore that autoregression is just regression y showing strong asymptotic equivalence of the scalar diffusion model 1.2 with a signal detection or Gaussian shift model, which can e interpreted as a regression model with random design. Our result is estalished for the scalar diffusion model ecause we need to employ tools from stochastic analysis that are neither availale for time series analysis nor for multidimensional diffusion processes. After sumission we learned aout the results y Grama and Neumann 24 who estalish directly asymptotic equivalence etween the autoregression 1.1 and a classical regression model y means of a suitale Skorohod emedding. Let us riefly introduce some asic notation such that we can announce the main results. For some fixed constants C, A, γ > we consider the nonparametric drift class { } Σ := Lip loc x C1 + x, x > A : x sgnx γ, 1.3

3 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS 3 where Lip loc denotes the set of locally Lipschitz continuous functions : and sgnx := x/ x. A standard result in the theory of stochastic differential equations asserts that for Σ and a Brownian motion W on some filtered proaility space Ω, A, At t, P there exists to a given initial value a unique strong solution Xt, [, T ] of equation 1.2, e.g. Karatzas and Shreve Moreover, the existence of a stationary solution, unique in law, is ensured with invariant marginal density x µ x = C exp 2 y dy, x, 1.4 where C > is a normalising constant. Considering in a first step drift functions in a shrinking neighourhood around Σ, we otain the local asymptotic equivalence result for T of the stationary diffusion experiment given y 1.2 with the accompanying Gaussian shift experiment dz x = x µ x dx + T 1/2 db x, x, 1.5 where B denotes a Brownian motion on the real line and µ is the invariant density from 1.4. The analogous regression experiment to 1.5 consists of oserving the function on a design with density µ, which can e considered random or deterministic in the sense that it determines the distance etween two design points. The main idea of the proof is to define a coupling of the original diffusion experiment with another diffusion-type experiment corresponding to a deterministic design. The implementation of this idea is heavily ased on the local time of the diffusion process. The local asymptotic equivalence result has already several implications for the statistical theory of diffusion processes. In particular, it can e used to otain asymptotically sharp lower risk ounds. For instance, the lower ound of Theorem 1 in Dalalyan and Kutoyants 22 follows immediately. In order to transfer also gloal results like upper risk ounds to the diffusion case, an equivalence result should e otained for all parameters Σ. We have to impose on the drift functions some minimal regularity larger than 1/2 to otain such a gloal result, cf. Brown and Zhang Furthermore, since the variance of the local time does not decay sufficiently rapidly, the general gloal equivalence result can only e estalished for drift functions that ehave nicely far away from the origin. To avoid too much technicalities we therefore consider a gloal class Σ Σ of drift functions of regularity larger than 1/2 that coincide with some known function Σ outside a compact interval I. In asence of a variance stailising transform the gloally equivalent experiments will e of compound type. The first accompanying sequence is given y the oservations dx t = X t dt + dw t, t [, S], 1.6 dz x = x µ S x dx + T S 1/2 db x, x I, 1.7 where W and B are independent Brownian motions, µ S is a suitale estimator of µ ased on the oservation Xt, t [, S] and S = ST, T satisfies lim T T

4 4 ANAK DALALYAN AND MAKUS EISS ST /ST p = for all p > 1. The second accompanying sequence is given solely in terms of Gaussian experiments: U x = µ x + T 1/4 B x + ξ, x I, 1.8 dz x = xu x dx + T 1/2 db x, x I, 1.9 where the Brownian motions B and B and the random variale ξ N, 1 are independent. The local and gloal equivalence results are derived in Section 2 and 3, respectively. In order to retain a clear presentation, a uniform variance ound on the local time and the construction of gloal estimators have een deferred to the Appendix. In Section 4 we discuss extensions of the theory developed so far. We start with the presentation of a constructive procedure for transferring a decision function from the Gaussian experiment to the diffusion experiment. In particular, it is seen that the restriction on the parameter space outside of a compact interval is not necessary for risk asymptotics with loss functions that act only on this interval. Then we discuss an even simpler Gaussian shift experiment which is gloally less informative than the diffusion model 1.2 such that risk upper ounds otained for this simple model immediately transfer to the diffusion case. Finally, we consider the diffusion model with a general, ut known diffusion coefficient σ, for which similar results are otained, and we treat the case of discrete oservations of the sample path in the diffusion model and the corresponding Euler discretisation. In order to convey concisely the main ideas and to save space, the results in this section are stated in a more informal way. 2. Local equivalence 2.1. The general idea. We shall show that for drift functions in a shrinking neighourhood of the drift function the statistical experiment induced y oserving the diffusion process 1.2 is for T asymptotically equivalent to the statistical experiment induced y the oservation dz x = x µ x dx + T 1/2 db x, x, 2.1 where µ is close to the density of the invariant measure of the diffusion process in 1.2 uniformly over the neighourhood and B denotes a Brownian motion on the real axis. The main idea of the proof is to pertur the diffusion model 1.2 in such a way that in each state x the local time, that is the amount of time spent y the process in x until time T, is at least T µ x and to provide no information on x after the local time has attained the level T µ x. At those states x, where the local time does not reach the level T µ x, additional information on x is revealed. The model thus otained can e considered as a regression model for with fixed deterministic design of density T µ. It is Gaussian and has the same likelihood process as the model in 2.1, which implies statistical equivalence of the associated statistical experiments.

5 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS 5 The intuitive explanation why this approach succeeds is that the diffusion model, like an autoregressive time series model, exhiits two sources of randomness. Firstly, the design, that is how often the states are visited y the process, is random. Secondly, the drift can merely e oserved after contamination y white noise dw. As it turns out, the first source of randomness is less severe than the second and we do not lose too much information y assuming that each state x is visited up to time T with a density according to the approximate expectation T µ x of the local time. However, it is evident that this procedure can only work for neighourhoods around that shrink with increasing T such that the true expectation T µ converges to T µ in a suitale manner Local experiments Definition. For a drift Σ and any density µ L 1 we introduce their local neighourhood with parameters ε, ζ, η > { Σ ε,η,ζ, µ := Σ 2 y 1/2 µ y dy ε, 1/2 } 2 y µ y µ y dy η, µ y µ y dy ζ. Here µ denotes the invariant density of the diffusion process with drift emark. It is natural to consider neighourhoods around, µ, ut it is y no means necessary for the calculations to enforce µ = µ. For the gloalisation the more general approach has the advantage of permitting the usage of separate estimators for the functions and µ. We now define precisely the local experiments E 1 and F 1, for which we shall prove asymptotic equivalence. Note that we define the Gaussian shift experiment on the space L2 and not on C via the natural interpretation of the differentials as integrators for L 2 - functions. Of course, the law is already characterised y the integration of the functions 1l [,y], y, which corresponds to the signal in white noise interpretation on the space C up to the knowledge of the value at zero Definition. We define the diffusion experiment localised around, µ E 1 := E 1, µ, T, ε, η, ζ := C[, T ], B C[,T ], P T Σε,η,ζ,µ, where P T denotes the law of the stationary diffusion process with drift on the canonical space C[, T ]. The Gaussian shift experiment localised around, µ is given y F 1 := F 1, µ, T, ε, η, ζ := L2, B L2, Q T Σε,η,ζ,µ,

6 6 ANAK DALALYAN AND MAKUS EISS where Q T denotes the law of the Gaussian shift experiment dz x = x µ x dx + T 1/2 db x, x, i.e., fxx µ x dx + T 1/2 fx db x, f L 2 with a Brownian motion B on the real line. In order to pursue our procedure of changing the design appropriately, we need to introduce the so-called local time of a diffusion process X. We refer to evuz and Yor 1999, Chapter VI for the details. We are going to use that the local time L y t X of the diffusion process X at the point y up to time t can e constructed such that L y t, y, t is a process which is continuous in t and càdlàg in y evuz and Yor 1999, Theorem VI.1.7. Henceforth we shall work with this process, which satisfies L y 1 t X = lim ε ε t 1l [y,y+ε X s ds, P -a.s. By assuming the usual conditions of the filtration A t t, we can suppose that L y t X is A t -measurale for A t t -adapted processes X. The main property we need is the following extended occupation time formula evuz and Yor 1999, Ex. VI.1.15: T T ft, X t, ω dt = ft, y, ω d t L y t X dy, P -a.s. 2.2 where f : + Ω + is any measurale function and d t L y t X denotes integration with respect to the increasing integrator t L y t X. We can now introduce the local experiment F 2 for which we shall show asymptotic equivalence with E 1. We riefly recall the conditions guaranteeing the existence of a weak solution of a stochastic differential equation with a functional form of the drift Proposition. Consider the stochastic differential equation dx t = X, t dt + dw t, t [, T ], with a progressively measurale functional : C + +. Then a weak solution with some prescried initial distriution µ exists if f, t K1 + f holds with a suitale constant K > for all f C + and t [, T ]. The law of the solution is otained y a change of the Wiener measure on C[, T ], B C[,T ] with initial distriution µ using the Girsanov density T Z T X = exp X, s dx s 1 T X, s 2 ds. 2 Proof. This is the generalisation of Proposition given in emark of Karatzas and Shreve 1991.

7 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS emark. Under suitale ergodicity assumptions, the linear growth condition on the drift can certainly e dropped and a corresponding uniqueness result will proaly hold, ut we do not want to deviate further into that direction. We just assume linear growth in the definition of Σ and work with the solution defined in terms of the Girsanov density Definition. We define the local experiment F 2 y F 2 := F 2, µ, T, ε, η, ζ := C[, T ] L2, B C[,T ] B L2, Q T Σε,η,ζ,µ, where Q T is uniquely defined y Q T A B := K T f, B T df, A A B C[,T ], B B L2. Here, T denotes the law of the weak solution Y of the stochastic differential equation dy t = Y t 1l Y + {L t t Y T µ Y t } Y t 1l Y {L t t Y >T µ Y t } dt + dwt, t [, T ], on the canonical space C[, T ] with initial distriution Y µ, given y Proposition 2.4. The proaility K T f, is the law of the Gaussian shift experiment dv f x = xt µ x L x T f 1/2 + dx + db x, x, where B denotes a two-sided Brownian motion on independent of W and Y and A + := maxa, emark. In the preceding definition we have to choose a measurale version of the mapping f, x L x T f on a set of functions f with T -proaility one in order to have the Markov kernel property of K T. This is certainly possile since y the equivalence of T with the Wiener measure this property is satisfied when using a càdlàg-version in x of the Brownian motion local time L x T W as discussed earlier. Finally, we need to introduce yet another experiment which is constructed so as to e equivalent to E 1, ut to e defined on the same space C[, T ] L2 as F Definition. We define the local experiment E 2 y E 2 := E 2, µ, T, ε, η, ζ := C[, T ] L2, B C[,T ] B L2, P T Σε,η,ζ,µ, where P T is uniquely defined y with the same notation as for F 2 P T A B := K T f, B P T df, A B C[,T ], B B L2. A 2.3. Likelihood ratio and equivalent experiments. In the sequel we shall often use the likelihood ratio or adon-nikodym derivative for the laws of diffusion-type processes on the space C[, T ]. The next theorem is an adaptation of Theorem 7.7 in Liptser and Shiryaev 21 to our purposes, see also Theorem IV.4.23 in Jacod and Shiryaev 23.

8 8 ANAK DALALYAN AND MAKUS EISS 2.9. Theorem. Suppose X 1 t, t [, T ] and X 2 t, t [, T ] are scalar diffusion-type processes that satisfy dx i t = α i X i, t dt + dw t, t [, T ], X i = ξ i, i = 1, 2, with progressively measurale functionals α i : C[, T ] + and with a standard Wiener process W. Then these processes have mutually asolutely continuous distriutions P X i on the canonical space C[, T ], B C[,T ] if for i = 1, 2 T P α i t X i 2 dt < T = 1 and P t α 2 t 2 X i dt < = 1, and if ξ 1 and ξ 2 are independent of W and have mutually asolutely continuous distriutions on. In this case the likelihood ratio Λ T X 1, X 2 X = dp X 1 X is dp X 2 dp T ξ 1 X exp α 1 t α 2 t X dx t 1 T 1 α t X 2 α 2 t X 2 dt, dp ξ 2 2 α 1 which under P X 2 is in law equal to dp T ξ 1 ξ 2 exp α 1 t α 2 t X 2 dw t 1 dp ξ 2 2 T α 1 t α 2 t 2 X 2 dt emark. This representation of the likelihood gives another indication why our limiting experiment F 1 is natural for the diffusion experiment E 1 : the Fisher information at in functional directions h and h is for T of order [ T ] I h,h = E hx t h X t dt + ot = T hxh xµ x dx + ot. Here and also later we employ the classical Landau symols o and O. From the definition of the Kullack-Leiler divergence or relative entropy, denoted y KL the following result is immediate, compare also with the expression for the Hellinger distance Jacod and Shiryaev 23, Theorem IV Corollary. Under the conditions of Theorem 2.9 the Kullack-Leiler divergence etween the laws of X 1 and X 2 is given y [ dpξ ] 1 KLP X 1, P X 2 = E log ξ [ T ] dp ξ 2 2 E α 1 t α 2 t 2 X 2 dt. With these tools at hand we otain the first equivalence results. We only need to know that two general dominated experiments G 1 = Ω, A, P ϑ ϑ Θ and G 2 = Ω, A, Q ϑ ϑ Θ are statistically equivalent iff the laws of the likelihood processes under the dominating measures P and Q coincide Strasser 1985, Cor. 25.9: dpσ L dp σ Θ P = L dqσ dq σ Θ Q.

9 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS Proposition. The statistical experiments E 1 and E 2 are equivalent. Proof. By Theorem 2.9 the measures P T and PT are equivalent for all, Σ such that the likelihood process for E 1 is well defined. Moreover, in experiment E 2 we use the kernel K T which is independent of such that the adon-nikodym derivative d P T d P X, V = KT X, dv P dx T K T X, dv P dx = dpt X dp T depends only on the first coordinate. Consequently, the likelihood processes coincide Proposition. The statistical experiments F 1 and F 2 are equivalent. Proof. Let us determine the likelihood process for F 2 under the dominating measure Q T. We first note that d QT log K T d Q Y, V = log Y, dv T dy dk T = log Y, d T T K T Y, dv T dy dk Y, V + log Y d T holds. Both log-likelihood functions consist of a stochastic integral with respect to a Brownian motion and its quadratic variation term under the dominating measure Q T. Let us calculate up to sets of proaility zero the quadratic variation term in the log-likelihood log dt d T given in Theorem 2.9 using the occupation time formula 2.2: T T = T = = Y t 1l Y + {L t t Y T µ Y t } Y t 1l Y {L t t Y >T µ Y t } Y t 2 Y t 1l {L Y t t Y T µ Y t } dt 2 y1l {L y t Y T µ y} d t L y t Y dy 2 y mint µ y, L y T Y dy. 2 dt Similarly, the quadratic variation term in log dkt Y, is given y dk T Y, 2 xt µ x L x T Y + dx. Putting the two identities together, we have proved that the quadratic variation term in log d Q T d Q equals T 2 yt µ y dy and is thus deterministic. The preceding calculations remain valid when is replaced y + λ for any λ. Hence using E [ d Q d Q ] = 1, we conclude that, under Q, the Laplace transform

10 1 ANAK DALALYAN AND MAKUS EISS E [expλm ] of the stochastic integral term Mh := T hy t 1l Y dw {L t t Y T µ Y t } t + hxt µ x L x T Y 1/2 + db x, h L 2, equals exp λ 2 2 yt µ 2 y dy. Therefore the random variale M is Gaussian with variance T 2 µ. The covariation etween two such stochastic integrals with replaced y 1 and 2, respectively, is y the occupation time formula again T 1 2 µ. By the Cramér-Wold device it follows that the random process M, Σ ε,η under Q T is Gaussian with zero mean. Since the likelihood process of the Gaussian shift experiment F 1 under Q T is given y exp T x µ x db x T 2 2 xµ x dx, Σ ε,η, 2 the laws of the two likelihood processes coincide and the experiments are equivalent emark. The main idea in the preceding proof was to show that the likelihood ratios associated to the experiments E 2 and F 2 as random processes indexed y have the same law. At the first look it seems that the experiment generated y the Itô process dy t = Y t 1l {t τ Y } + Y t 1l {t>τ Y } dt + dwt, t [, T ], T 1/2 dy t = T 2 yµ ydy 2 Y t dt dt + dw t, t T, T + 1, τ Y = inf Y = ξ, { t T : t } 2 Y s ds > T 2 yµ y dy, satisfies the same property and therefore can e used instead of F 2. Unfortunately, this assertion is false: for a fixed value of the log-likelihood of the process Y is a Gaussian random variale, ut the same log-likelihood as a process indexed y is not jointly Gaussian emark. For future reference we list further experiments that are equivalent to F 1 and F 2 for parameters Σ ε,η : dy x = x dx + T 1/2 µ x 1/2 db x, x, dy x = x x µ x dx + T 1/2 db x, x, dy x = F 1 µ x dx + T 1/2 db x, x, 1, where F µ x = x µy dy and db is Gaussian white noise. For the proof it suffices to check that the laws of the likelihood processes coincide. +

11 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS Asymptotic equivalence. By Corollary 59.6 in Strasser 1985 the Le Cam distance etween experiments defined on the same measurale space can e estimated y a uniform ound on the total variation distance etween the corresponding proaility measures. An application of this coupling technique allows to prove the main theorem on local asymptotic equivalence Theorem. If for T the asymptotics εt = ot 1/4, ηt = ot 1/2 and ζt = o1 hold, then the following convergence holds true uniformly over all Σ: lim E 1, µ, T, εt, ηt, ζt, F 1, µ, T, εt, ηt, ζt =. T Proof. By Propositions 2.12 and 2.13 it suffices to prove the asymptotic equivalence for the experiments E 2 and F 2. Their families of measures P T and Q T are defined on the same measurale space C[, T ] L2, B C[,T ] B L2. We infer with short-hand notation E 1, F 1 = E 2, F 2 sup P T Q T T V, Σ ε,η,ζ,µ T V denoting the total variation norm. Since the measures P T and Q T correspond to diffusion-type processes with different initial distriutions, we use the representations Kallenerg 22, Theorem 18.1 P T = P T,x µ x dx and Q T = Q T,x µ x dx with the corresponding laws for deterministic initial values x and infer y the triangle inequality P T Q T T V µ µ L 1 + P T,x Q T,x T V µ x dx. Because of ζt the first term tends to zero uniformly. Since the square of the total variation is ounded y twice the Kullack-Leiler divergence Deuschel and Stroock 1989, Eq , it suffices for the second term to prove that KL QT,x, P T,x µ x dx tends to zero uniformly. By Corollary 2.11 this expression equals up to the factor 1/2 E [ T 2 Y t 1l Y dt + {L t t Y >T µ Y t} ] 2 yt µ y L y T Y + dy [ = E 2 y L y T Y T µ y + + T µ y L y T Y ] + dy [ = 2 ye L y T Y T µ y ] dy.

12 12 ANAK DALALYAN AND MAKUS EISS Since we are in the stationary case, a ias-variance decomposition yields in comination with Proposition 5.1 from Appendix E [ L y T Y T µ y ] T µ y µ y + CT µ y 1/2. Hence, we otain the uniform convergence result over Σ εt,ηt,ζt, µ [ ] E 2 y L y T Y T µ y dy 2 y T µ y µ y + CT µ y dy which proves the assertion. T η 2 T + CT 1/2 ε 2 T T, Corollary. The preceding asymptotic equivalence result holds in particular for the local parameter suclass { Σ ε,t, µ := Σ 2 y 1/2 1/2 µ y dy ε, µ µ µ 1/2 T }, 1/2 when ε = εt = ot 1/4 for T. Proof. Just note that for any Σ ε,t, µ 2 y µ y µ y dy µ 1/2 µ µ 1/2 2 y µ y dy ε 2 T T 1/2 holds and equally µ µ T 1/2 µ 1/2 T 1/2 follows uniformly over y the uniform exponential decay of µ. Therefore Σ εt,t, µ Σ εt,ηt,ζt, µ follows with ηt = εt T 1/4 = ot 1/2 and ζt = OT 1/2 = o1. For later use we also show asymptotic equivalence with another Gaussian experiment Proposition. The statistical experiment F 1, µ, T, ε, η, ζ is for η = ot 1/2 and aritrary ε, ζ > asymptotically equivalent to the experiment induced y oserving dy x = x x µ x dx + T 1/2 db x, x, where µ is the invariant density corresponding to, db is Gaussian white noise on L 2 and the parameters elong to the the same neighourhood Σ ε,η,ζ, µ. Proof. Since the two concerned experiments are defined on the same space, the result follows if we show that the Kullack-Leiler divergence etween the likelihood ratios tends to zero. This divergence is given y T 2 x x 2 µ x µ x 2 dx. Using the general inequality A B 2 A 2 B 2 for A, B >, the condition on η yields the result.

13 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS Gloalisation 3.1. Main result. A common way of gloalising a local equivalence result makes use of the variance stailising transformation see Grama and Nussaum 1998 for the exact definition. In our case this amounts to seeking a functional T whose differential DT [h] at the point = is equal to µ h. Indeed, for such a functional the Kullack-Leiler divergence etween the laws of the Gaussian random measures dz x = x x µ x dx+ T 1/2 db x, x, and d Z x = T x T x dx + T 1/2 db x, x, is equal to T 2 T x T x DT [ ]x 2 dx and, at a heuristic level, tends to zero if the functional T is sufficiently regular. This yields the asymptotic equivalence of the two Gaussian shift experiments corresponding to Z and Z. Furthermore, it permits to infer the asymptotic equivalence of the experiments characterised y the oservations dz x = x µ x dx + T 1/2 db x, x and dz x = T x dx + T 1/2 db x, x, the latter eing independent of. Unfortunately, following Delattre and Hoffmann 22 we can show that such a transformation does not exist. Indeed, let us consider the simple case when is unknown only on a compact interval I. Then the differential of the operator S : L 2 I L 2 I, S = µ at the point h L 2 I is oviously given y µ+εh µ DS[h] = lim, ε ε where the convergence is understood in the mean square sense. We find DS[h]x = µ x hy [ ] F y 1l {y x} dy, x I, I where F is the distriution function corresponding to the invariant density µ. Therefore, the equality DT [h 1 ] = h 1 µ = h 1 S would imply that T is twice continuously differentiale and D DT [h 1 ] [h 2 ] = D DT [h 2 ] [h 1 ] for any h 1, h 2 L 2 I. This last equality can e rewritten in the form h 1 DS[h 2 ] = h 2 DS[h 1 ], h 1, h 2 L 2 I, which is evidently not true. This contradiction results essentially from the nonlocal character of the mapping S. This indicates why the gloal asymptotic equivalence with a Gaussian shift experiment of the form dz x = T x dx + T 1/2 db x might e impossile to estalish. Nevertheless, we give elow an equivalence result which is gloal and involves a mixed Gaussian white noise experiment. The main idea is to replace in the Gaussian shift experiment dz x = x x µ xdx + T 1/2 db x the invariant density µ y a random approximation, which is independent of B and has the advantage of eing oservale Definition. The parameter class Σ = Σ β, L,, I consists of drift functions Σ satisfying x = x, x I; x y L x y β, x, y I,

14 14 ANAK DALALYAN AND MAKUS EISS where I = [ D, D] is a compact interval, is a fixed known function and β, emark. Let us riefly explain why we restrict to the case when x is known for x \ I. Since the variance under P T of the local time Lx T Lx S is of order T Sµ x, condition 3.2 requires the existence of an estimator ST such that [ x E ST x ] 2 T ST µ x dx = 3.1 lim T uniformly in. Standard arguments yield that the mean squared error MSE of estimating x y a kernel method with andwidth h is of order h 2β + ST hµ x 1. Therefore the optimal choice of a andwidth is h = ST µ x 1/2β+1. Even for this oracle choice of h the MSE is of order ST µ x 2β/2β+1 and the integral 3.1 is not finite for β > 1/2. Fundamentally, this ostruction is due to the relatively slow decay of the variance of local time compared to its expectation: lim y Var[L y T Y ]/E[Ly T Y ] =. This strong restriction can certainly e relaxed y an exponential estimate of the form x x C 1 e C 2 x, x, or circumvented y a model of reflected diffusions on a compact interval. Even in the case of a regression model with known random design the transformation of a degenerate or unounded design distriution to a uniform distriution would yield a similar condition on the nonparametric class in order to give asymptotic equivalence Brown, Cai, Low, and Zhang 22, p. 69. Compare, however, the discussion in Section 4.2. Note that although the functions in Σ are locally Lipschitz continuous, the Hölder restriction of order β is of different nature: it is uniform over x I and over Σ. Q T,ϕ 3.3. Definition. For any ϕ > we denote y Z, U on the canonical space L2 I CI, where { dz x = xu x dx + T 1/2 db x, x I, U x = µ x + ϕ B x + ξ, x I. the measure induced y the process with B, B eing a two dimensional Brownian motion and ξ = ϕ 1 U N, 1 a random variale independent of B, B. The accompanying experiment is then Hϕ, T = L2 I CI, B L2 I T,ϕ B CI, { Q } Σ Definition. The statistical experiment defined y oserving a sample path of the stationary diffusion process 1.2, when the parameter set is Σ, is denoted y ET. We can now announce the main theorem of this section, whose proof is deferred to the end of the section Theorem. Let Σ and Σ e defined as aove. If β > 1/2, then the statistical experiments ET and HT 1/4, T are asymptotically equivalent as T.

15 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS emark. The inspection of the proof of Theorem 3.5, comined with the fact that the total variation is ounded y twice the square root of the Kullack-Leiler divergence, shows that the -distance etween the experiments ET and Hϕ, T tends to zero at the rate β 1 2β+1 + ϕt 4β+2 + ϕ 1 T β 2β+1. Therefore the rate-optimal choice of ϕ is ϕt = T 1/4 T 1 4 and we have ET, HT 1/4, T CT 1/2 β/4β Definition of experiments. We introduce some proaility measures that will e repeatedly used in this section. Some have already een defined in the previous section, ut for the present purposes we need to specify their dependence not only on, ut also on other parameters. In this section, the sustitution of the suscript of any proaility measure y indicates that we consider that measure for identically equal to zero, e.g. Q T,ϕ T,ϕ = Q, ut the meaning of µ has not changed. Let Q T,µ denote the law of the Gaussian shift dz x = x µ x dx + T 1/2 db x on the canonical space L2. The log-likelihood of this family of measures is defined y dq T l Q T, µ,µ, Z = log Z = T x µ dq T x dz x T 2 xµ x dx.,µ 2 Let Q T,,y,µ denote the law of the process Y, V given y Definition 2.6 with initial condition Y = y. The log-likelihood of this family of measures is d QT lq T, µ,,y, Y, V = log,µ d Q Y, V T,,y,µ = T Y t 1l {L Y t t Y T µ Y t} dy t + x T µ x L x T Y 1/2 dv + x T 2 x µ x dx. 2 It is noteworthy that this log-likelihood does not depend on y and. ecall that P T,x and P T,x are defined as in Definitions 2.3 and 2.8, except that the initial condition is X = x. The log-likelihoods of the families of measures P T,x Σ and P T,x Σ will e denoted y lt P, X and l T P, X, respectively. Note that although P,x is a measure on the product space C[, T ] L2, the log-likelihood l T P, X depends only on the first component Definition. Let Ẽ = C[, T ] L2, B C[,T ] B L2, PT Σ. Let us fix S in the interval, T and define the compound experiment G = GS, T, Σ as follows: we oserve a sample path of the stationary diffusion process X with drift up to time S, we compute an estimator µ S = µ S X, C of the invariant density µ, and then we oserve a realisation of the conditionally Gaussian process dz x = x µ S x dx+t S 1/2 db x, x. In order to avoid sutle questions of measuraility, we assume that µ S takes its values in a countale set M = {µ 1, µ 2,...} C.

16 16 ANAK DALALYAN AND MAKUS EISS 3.8. Definition. The experiment G is defined rigorously as where S,T GS, T := C[, S] L2 I, B C[,S] B L2 I, S,T Σ, is the measure characterised y S,T A B = i=1 Q T S,µ i B P S A { µ S = µ i }, A B C[,S], B B L2 I Asymptotic results. Our program in this section is as follows. We split the diffusion path oserved up to time T into two parts: a path oserved over [, S] and another over [S, T ]. We prove that y replacing the second path y a conditionally to the first path Gaussian oservation we otain an asymptotically equivalent experiment. Then we sustitute this conditionally Gaussian experiment y another one, not involving anymore the oserved path over [, S]. In the last step we apply this method in the converse direction, that is, making use of estimators ased on the Gaussian oservations, we replace the diffusion experiment over [, S] y a conditionally Gaussian one. One method of carrying out this program consists in reducing the gloal equivalence prolem to a local one via Lemma 9.3 of Nussaum 1996, or its extension in Lemma 1 of Delattre and Hoffmann 22. However, this requires a local asymptotic equivalence result etween the diffusion starting at a fixed point x and a Gaussian shift, uniformly in x. Achieving the result y this technique seems to e more technical than what we do elow Proposition. Let S = ST, T e such that for some estimator S of ased on the oservations X t, t [, S] and taking values in a countale set B = { 1, 2,...} the following condition is satisfied: lim sup E x T Σ [ ST x 2 L x T X L x ST X µ ST xt ST dx ] =. 3.2 Then the experiments ET and GST, T are asymptotically equivalent as T tends to infinity. Proof. We introduce an auxiliary compound experiment G. It is generated y the oservation of a sample path of a stationary diffusion with drift up to time S = ST and an Itô process similar to the one of Definition 2.6, except that µ and are replaced y the estimators µ S and S respectively. More precisely, the statistical experiment G = GT is defined on the space C[, T ] L2, B C[,T ] B L2 y the family of proaility measures S,T A B = i,j=1 A { µ S, S =µ i, j } Q T S, j,fs,µ i B P S df, for any A B C[,S], B B C[S,T ] B L2. The aove integral is well defined since the T S mapping y Q, j,y,µ i B is measurale in fact, it is continuous.

17 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS 17 It is easy to check see the proof of Proposition 2.13 for similar calculations that the log-likelihood log ds,t X, Z of the family of measures { S,T d S,T } Σ is given y l T,S, Z, X = l Q T S, µ SX,, Z + l P S, X, X, Z C[, S] L2. S,T Likewise, the log-likelihood of the family of measures { } Σ is given y l T,S, Y, V, X = l Q T S, µ SX,, Y, V + l P S, X, where Y, V C[S, T ] L2 and X C[, S]. We have proved in Proposition 2.13, that L l Q T S, µ i, Z Σ Q T,µ S i = L lq T S, µ i, Y, V Σ S QT, j,y,µ i for any j N and for any y. This implies L l Q T S, µ i, Z Σ Q T S, µ S f = L lq T S, µ i, Y, V Σ S QT, S f,fs, µ S f for any f C[, S], and consequently L l T,S, Z, X Σ T,S We infer that the experiments G and G are equivalent. = L l T,S, Y, V, X Σ T,S. In order to show E, G, it suffices to prove Ẽ, G ecause the experiments E and Ẽ are also equivalent: their likelihood ratios coincide see Proposition The experiments Ẽ and G are defined on the same proaility space and the Kullack-Leiler divergence etween the respective laws is see Corollary E [ T S X t S X t 2 1l X {L t t X L X dt t S X> µ SX t T S} + x S x 2 µs xt S L x T X L x SX ] dx + = 1 [ 2 E x ] S x 2 L x T X L x SX µ S xt S dx. By condition 3.2, this expression tends to zero uniformly in Σ when T. To pursue the gloalisation, we replace the Gaussian shift experiment dz x = x µ ST x dx + T ST 1/2 db x 3.3 y a simple experiment not involving the estimators ST and µ ST. Note also that since is known outside I, the oservations Z x, x I of the Gaussian shift experiment are void they do not contain any information on the unknown parameter x, x I Definition. The experiment G 1 = G 1 S, T is for S, T defined on C[, S] L2 I S,T CI y the family of product measures A B = P S T S,ϕT A Q B, for

18 18 ANAK DALALYAN AND MAKUS EISS any A B C[,S] oserving and for any B B L2 I B CI. In other words, G 1 is induced y { dz x = x U x dx + T S 1/2 db x, x I, U x = µ x + ϕt B 3.4 x + ξ, x I, where B, B are independent Brownian motions and ξ N, 1 is independent of B, B Proposition. Assume that the estimators µ S are continuously differentiale on I and satisfy the conditions of Proposition 3.9. Moreover, let for ST, T, ϕt > the assumptions lim ϕt T 2 sup Σ I [ lim ϕt T 2 sup E µ Σ lim ϕt T 2 T ST sup Σ [ E µ x µ ST x ] 2 dx =, I µ ST 2 ] =, E [ ST x x 2 ] dx = e satisfied. Then the statistical experiments GST, T and G 1 ST, T are asymptotically equivalent as T tends to infinity. Proof. In what follows we consecutively replace experiments y asymptotically equivalent ones until we reach the experiment G 1. ecall that G is defined y oserving a diffusion path up to time S and a realisation of the Gaussian process Z given y 3.3. If we replace in G the oservations Z y dz x = x S x µs x dx + T S 1/2 db x, x I, 3.5 we otain an equivalent experiment, since it has exactly the same likelihood ratio. Then we replace these oservations y dz x = x S x µ x dx + T S 1/2 db x, x I, 3.6 in view of the fact that the Kullack-Leiler divergence etween the corresponding measures is up to some multiplicative constant equal to [ x T S E S x 2 µ x µ S x ] 2 dx, I which tends to zero as T uniformly in Σ, according to the assumption of Proposition 3.9 and the inequality µ µ S 2 µ µ S. It is evident that the statistical experiment dz x = x S x µ x dx + T S 1/2 db x, x I, du x = µ S x dx + ϕ d B x, x I, U = µ 3.7 S + ϕ ξ.

19 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS 19 with independent Brownian motion B and ξ N, 1 is equivalent to 3.6, since the sample paths of the process U do not contain any information on. The first two assumptions of the proposition yield the equivalence of experiment 3.7 and dz x = x S x µ x dx + T S 1/2 db x, x I, du x = µ x dx + ϕ d B x, x I, U = µ + ϕ ξ. An equivalent form of this experiment is { dz x = x S x µ x dx + T S 1/2 db x, x I, U x = µ x + ϕ B x + ξ, x I. 3.8 Computing the Kullack-Leiler divergence and using the third assumption of the proposition, one can easily check that experiment 3.8 is asymptotically equivalent to the experiment { dz x = x S x U x dx + T S 1/2 db x, x I, U x = µ x + ϕ B 3.9 x + ξ, x I. This completes the proof of the proposition, since the laws of likelihood processes of the experiments 3.9 and 3.4 coincide. Having otained the asymptotic equivalence etween E and G 1, we aim at replacing the first part of the compound experiment G 1, which is the ergodic diffusion, y a conditionally Gaussian experiment. To do so, we assume that µ T,ST and T,ST are estimators of µ and ased on the oservations Z x, U x, x I and taking their values in countale susets of CI. We denote y y the processes in 3.4. ÊT S,ϕ the expectation with respect to the measure Q T S,ϕ induced Proposition. Let ϕ = ϕt and S = ST e such that the assumptions [ lim sup Ê T S,ϕ x T,S x 2 Sµ x + S µ x µ T,S x dx] =, 3.1 T Σ I [ lim sup Ê T S,ϕ µ x µ T,S x ] dx =, 3.11 T Σ lim sup S T Σ ÊT S,ϕ [ I x T,S x 2 µ ] 2 T,S x U x dx =, 3.12 are fulfilled. Then the statistical experiments G 1 ST, T and HϕT, T are asymptotically equivalent as T tends to infinity. Proof. ecall that the experiment G 1 is characterised y the oservations X, Z 1, U, where X is defined y 1.2 and Z 1, U are as in 3.4 with B replaced y B 1.

20 2 ANAK DALALYAN AND MAKUS EISS Let G 2 e the statistical experiment defined y the oservations Y, V, Z 1, U, where Z 1 and U are as aove, Y µ T,S and { dyt = Y t 1l Yt + {L t Y S µ T,SY t} T,S Y t 1l Y {L t t Y >S µ T,SY t} dt + dwt, t [, S], dv x = x S µ T,S x L x S Y 1/2 dx + + db x, x I. In these formulae, we assume that T,S x is equal to x = x for any x I and the Brownian motions W, B are mutually independent and independent of B 1, B, ξ. The total variation distance etween the laws descriing the experiments G 1 and G 2 is controlled y see the proof of Theorem 2.16 [ Ê T S,ϕ x ] T,S x 2 E L x S,ϕ SX S µ T,S x dx + ÊT µ T,S µ L 1. From Proposition 5.1 we know Var L x S CSµ x. By the Cauchy-Schwarz inequality E L x S X ST µ x C Sµ x. So assumptions 3.1 and 3.11 yield the asymptotic equivalence of G 1 and G 2. epeating the same arguments as those used in Proposition 3.9 for estalishing the equivalence etween G and G 1, we can prove that the experiment G 2 is equivalent to dz x 2 = x T,S x µ T,S x dx + S 1/2 db x 2, x I, dz x 1 = x U x dx + T S 1/2 db x 1, x I, U x = 3.13 µ x + ϕt B x + ξ, x I, where B 2 is a Brownian motion independent of B 1, B, ξ. Once again considering the Kullack-Leiler divergence, one checks that the statistical experiment 3.13 is asymptotically equivalent to dz x 2 = x T,S x U x dx + S 1/2 db x 2, x I, dz x 1 = x U x dx + T S 1/2 db x 1, x I, U x = µ x + ϕt B x + ξ, x I, provided that 3.12 is satisfied. This last experiment, in turn, is equivalent to dz x 2 = x U x dx + S 1/2 db x 2, x I, dz x 1 = x U x dx + T S 1/2 db x 1, x I, U x = µ x + ϕt B x + ξ, x I, 3.14 since their likelihood functions coincide. The same argument yields the equivalence of 3.14 and Hϕ, T from Definition 3.3, with B x = S B x 2 + T S B x 1 / T and Z x = S Z x 2 + T S Z x 1 /T.

21 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS Proof of Theorem 3.5. First, note that under the conditions imposed on and, all the stochastic differential equations introduced in previous sections have a weak solution. To estalish the result of the theorem, it suffices to check that for some ST [, T ] the conditions of Propositions are fulfilled with ϕt = T 1/4. Set ST = T/2. Since E L x T Lx ST = T ST µ x and the variance of the local time at x etween time instants ST and T is ounded y CT ST µ x, the estimators ST and µ ST proposed in the Appendix satisfy 3.2 as soon as ST 2β/2β+1 T ST + T ST / ST tends to zero. This convergence holds for ST = T/2 if and only if β > 1/2. To verify the conditions of Proposition 3.11 we use the ovious relation µ x = x µ x and the ounds We infer that the desired conditions are fulfilled if ϕt 2 T 2β/2β+1 and T 1 2β/2β+1 ϕt 2 tend to zero. This is oviously the case for ϕt 2 = T 1/2 and β > 1/2. The verification of the conditions of Proposition 3.12 is achieved similarly using Lemma Extensions and generalisations 4.1. Constructive local equivalence. The main interest of the statistical equivalence is the following. If the models E = Ω, A, Pϑ ε, ϑ Θ and E = Ω, A, Q ε ϑ, ϑ Θ are asymptotically equivalent when ε, then for any decision function δ in E there exists a decision function δ in E such that, for any loss function L ϑ ounded y 1, the risk of δ is ounded y the risk of δ plus a term tending to zero as ε uniformly on the parameter class, that is L ϑ δ w Q ε ϑdw L ϑ δw Pϑdw ε + o ε 1. Ω Ω Our results so far are non-constructive, in particular they do not provide an explicit procedure for constructing a decision function for the diffusion experiment from a decision function for the simpler Gaussian experiment. Nevertheless, such a procedure is hidden in the proofs and we present it riefly. Consider the local setting, that is elongs to a neighourhood of a known function. Let X = X t, t [, T ] e a sample path of the diffusion dx t = X t dt + dw t and B = B x, x e a Brownian motion independent of W. For any a set T Φ a X, V := 1 1l T {Xt [,a], L X µ t t X T µ X t } X t 1/2 dx t X t dt + 1 a 1 Lx T X 1/2 dv x, 4.1 T T µ x +

22 22 ANAK DALALYAN AND MAKUS EISS with dv x = x T µ x L x T X 1/2 dx + db + x. By 2.2 we otain where B a = T Φ a X, V = a xµ x 1/2 dx + 1 Ba, T 1l {Xt [,a], L X t t T µ X t} µ X t 1/2 dw t + a 1 Lx T X 1/2 db x. T µ x + Using the Lévy characterisation, B can e shown to e a Brownian motion. Thus Zx = Φ x X, V is a realisation of the Gaussian process dz x = x µ x dx + T 1/2 d B x, x. 4.2 This means that Φ maps the model defined y X, V to the model defined y 4.2. On the other hand, we have proved that the total variation distance etween the laws of X, V and of X, B tends to zero. In conclusion, if δz is a decision function in the Gaussian model 4.2, then δ X = δφx, B will e a randomised decision function in the diffusion model with asymptotically the same risk as δz. Moreover, if the loss function under consideration is convex, then according to Jensen s inequality the risk of the decision function δ X = E[δΦX, B X] will e smaller than the risk of δ X. Let us give a concrete example. If δz is the classical kernel estimator of in the Gaussian shift model 4.2 δz, = + 1 x K µ x 1/2 dz x, h h where K is a kernel function and h > the andwidth, then the corresponding estimators in the diffusion experiment are given y δ X, B, = + 1 T X t K 1l X T h h µ {L t t T µ X t } X t 1 dx t X t dt + 1 x T µ x L x T K X1/2 db x, h h T µ x δ X, = + 1 T X t K 1l X T h h µ {L t t T µ X t } X t 1 dx t X t dt. Hence, δ is otained y a susampling of the standard kernel estimator under localisation ˆ = + 1 T K X t µ X T h h t 1 dx t X t dt Constructive gloal equivalence for losses on a compact interval. In emark 3.6, one can check that the constant C in the inequality ET, HT 1/4, T CT α with α = 2β 1/8β + 4 > depends on only via the parameters C, A, γ entering in the definition of Σ. Therefore, we have sup Σ ET, HT 1/4, T CT α.

23 ASYMPTOTIC STATISTICAL EQUIVALENCE FO SCALA EGODIC DIFFUSIONS 23 Unfortunately, the equivalence mapping from ET to HT 1/4, T providing this inequality depends on I c = x, x \ I, and therefore cannot e used in prolems where I c is merely an unknown nuisance parameter. Nevertheless, y slightly modifying the experiment H we get an equivalence result which is uniform in Σ and is attained y an equivalence mapping independent of I c Theorem. Let Hϕ, ψ, T e defined y the parameter space Σ and the oservations { dz x = xu x dx + T 1/2 db x, x I, U x = µ x + ϕ B x + ψξ, x I, with ϕ, ψ > and B, B, ξ, β as in Definition 3.3. If ψt = T 2β+2 6β+3, then sup ET, HT 1/4, ψt, T CT 2β 1/12β Σ and there exists an equivalence mapping from ET to HT 1/4, ψt, T independent of I c and realising the ound in inequality 4.3. Proof. To prove 4.3 we follow the methodology of Section 3.3 with the only difference that ϕ is replaced y ψ in the second equality of Proposition As for the construction of the equivalence mapping, remark that the local equivalence mapping Φ in 4.1 depends only on the values of and µ on the interval I. The estimators µ S and S in the diffusion model and the estimator ˆ T,S in the Gaussian model do not depend on I c, whereas the estimator µ T,S x in the Gaussian model does depend on I c even if x I, cf. Sections 5.2, 5.3. Thus, to otain an equivalence mapping independent of I c, it suffices to modify the estimator µ T,S. Because of µ x = µ exp 2 x u du we redefine the estimator µ T,S x = U 2 exp 2 x T,S u du, where T,S is the kernel estimator from Section 5.3. For any p > and x I the moment of order 2p of µ T,S x µ x is ounded in order y ψt 2p + T S p. Therefore, with our choice of ψt, the conditions of the analogue of Proposition 3.12 with HϕT, ψt, T instead of HϕT, T are fulfilled. If I c is known the estimator µ T,S presented in Section 5.3 converges more rapidly to µ than the estimator U 2 exp 2 x T,S u du. This explains the deterioration of the convergence rate in 4.3 as compared with that given in emark 3.6 and shows that there is a price to pay for having a Markov kernel independent of I c. Note also that ψt is chosen from a trade-off etween the second equality in Proposition 3.11 and 3.1 in order to otain the est possile rate in 4.3. We riefly descrie the equivalence mapping from the diffusion experiment X t, t [, T ] to the conditionally Gaussian experiment U x, Z x, x I provided y Theorem 4.1. Let

24 24 ANAK DALALYAN AND MAKUS EISS B 1, B 2 and B e independent Brownian motions and ξ e a standard Gaussian random variale independent of everything else. Using the mapping ΦX T, B,, µ in 4.1 and the estimators from Section 5.2 and denoting y X S,T the diffusion path X t, t [S, T ], we set S = T/2 and { U x = µ S x + T 1/4 Bx + ψt ξ, x I, Z 1 x = Φ x X S,T, B 1, S, µ S + x S yu y dy, x I. Using the new oservations U x, Z x 1, x I, we define the estimator T,S as in Section 5.3 and set µ T,S x = U 2 exp x 2 T,S u du. Define Z 2 x = Φ x X S, B 2, T,S, µ T,S + x T,S yu y dy, x I. Finally, put Z x = Z x 1 + Z x 2 /2. Let us denote y Ψ the mapping that associates to X T, B 1, B 2, B, ξ the couple U, Z Corollary. Assume that we consider a statistical prolem where oth parts I = x, x I and I c = x, x I c are unknown, ut the loss function L we consider depends only on I. If L = L I is ounded, then for any decision function δ in the model HT 1/4, ψt, T we can construct a decision function δ in the diffusion model such that lim sup sup E [L I δ ] E [L I δ] =. T Σ I Σ Proof. It suffices to take δ = δ Ψ A less informative experiment. We present an accompanying sequence of simple white noise experiments that is gloally less informative for the asymptotics T than our diffusion experiment. Let us first consider the local experiment where Q T,< F < 1 := F < 1, T, ε, η, ζ := L2, B L2, Q T,< Σε,η,ζ,µ, denotes the law of the Gaussian shift experiment dz < x = x µ x dx + T 1/2 db x, x, 4.4 with a Brownian motion B on the real line and a measurale function µ : [, satisfying µ x µ x for all x, Σ ε,η,ζ, µ. Hence, F 1 with the centre, µ and F < 1 are defined on the same measurale space and only differ in the choice of µ and µ, respectively. We claim that the experiment F < 1 is less informative than F 1. In fact, it suffices to construct a Markov kernel K : L2 B L2 [, 1] such that Q T,< = K Q T holds for all