Pchelintsev, Evgeny A.

Perelevskiy, Svyatoslav

English

Tomsk State University Repository

Applied Methods of Statistical AnalysisAsymptotically efficient estimation ofa drift coefficient in diffusion processesEvgeny Pchelintsev and Svyatoslav PerelevskiyTomsk State University, Tomsk, Russiae-mail: evgen-pch@yandex.ru, slavaperelevskiy@mail.ruAbstractThe article studies the asymptotic properties of an adaptive model selectionprocedure for estimation an unknown drift coefficient in diffusion processes. Itis shown that the procedure is asymptotically efficient, i.e. it is establishedthat the asymptotic quadratic risk of the procedure coincides with the Pinskerconstant, which provides an exact lower bound of the quadratic risk for allpossible estimates.Keywords: improved estimation, stochastic diffusion process, mean-squareaccuracy, oracle inequalities, Pinsker constant, asymptotic efficiency.IntroductionConsider the problem of asymptotically efficient estimation of the unknown drift coef-ficient in diffusion process, described by the following stochastic differential equation:dyt = S(yt) dt+ dwt , 0 ≤ t ≤ T , (1)where (wt)t≥0 is a scalar standard Wiener process, the initial value y0 is some givenconstant, and S(·) is an unknown function. Note that such models are widely usedin financial markets, radio-physics, etc. [1]. The problem is to estimate the functionS(x), x ∈ [a, b], from the observations (yt)0≤t≤T . The main goal of this paper toprove the asymptotic efficiency property of the improved model selection procedureproposed in [2] for estimating the function S in (1). The concept of asymptoticefficiency is associated with the optimal rate of convergence of the minimax risk, i.e.An important issue in the optimality results is the study of the exact asymptoticof the minimax risk. The problem of asymptotic non-parametric estimation in themodel of heteroscedastic regression was studied by Efroimovich [3] and Pinsker [4].To prove the asymptotic efficiency of the procedure, it is necessary to show thatits asymptotic quadratic risk coincides with the lower bound defined by the Pinskerconstant [5, 6]. In this paper, the problem is solved using an approach based on themodel selection methods and sharp oracle inequalities. Recall that the model selectionmethod appeared in the pioneering works of Akaike [7] and Mallows [8], in whichproposed to introduce a penalization term in the criteria of maximum likelihood.Further, Barron, Birgé and Massart [9], Massart [10] and Kneip [11] developed thismethod to obtain non-asymptotic oracle inequalities in non-parametric regressionmodels with Gaussian noise in discrete time. Unfortunately, this method cannot beapplied in our case to prove an asymptotic efficiency property, since the coefficient235Novosibirsk, 18-20 September, 2019in main term of the resulting oracle inequalities is greater then one. For this reason,in this paper we will use the method proposed in [12]. This paper deals with theestimating the unknown function S(x), a ≤ x ≤ b, in the sense of the mean squareriskR(ŜT , S) = ES‖ŜT − S‖2 , ‖S‖2 =∫ baS2(x)dx , (2)where ŜT is some estimate of S by observations (yt)0≤t≤T , a < b are some realnumbers. Here ES is the expectation with respect to the distribution PS of therandom process (yt)0≤t≤T given the drift function S. To obtain a reliable estimatorof function S, it is necessary that the process (1) has the ergodicity property. Forthis we suppose that unknown function S belongs to the following functional class:ΣL,N = {S ∈ LipL(R) : |S(N)| ≤ L ; ∀|x| ≥ N, ∃ S˙(x) ∈ C(R)such that− L ≤ inf|x|≥NS˙(x) ≤ sup|x|≥NS˙(x) ≤ −1/L} , (3)where L > 1, N > |a| + |b|, S˙(x)− derivative S(x). For estimating the drift Sin (1) Galtchouk and Pergamenshchikov [13] have proposed to apply the sequentialapproach. First step is a passage to a discrete time regression model by making useof the truncated sequential procedure introduced in [5]. To this end, at any pointxk of an equidistant partition of the interval [a, b], we define a sequential procedure(τk, S∗k) with a stopping rule τk and an estimators S∗k. For Yk = S∗kwith 1 ≤ k ≤ n,we come to the regression equation on some set Γ ⊆ Ω (supS∈ΣL,N PS(Γc) ≤ ΠT ,where limT→∞ Tm ΠT = 0 for any m > 0):Yk = S(xk) + ζk . (4)Here, in contrast with the classical regression model, the noise sequence (ζk)1≤k≤nhas a complicated structure, namely,ζk = σk ξk + δk , (5)where (σk)1≤k≤n is a sequence of some observed random variables, (δk)1≤k≤n is asequence of bounded random variables and (ξk)1≤k≤n is a sequence of i.i.d. randomvariables N (0, 1) which are independent of (σk)1≤k≤n.In order to estimate the function S in model (4) we make use of the model selectionmethod based on improved weighted least squares estimates proposed [18]. Improvedestimation method in nonparametric regression models has been developed in [15, 16,17].1 Oracle inequalitiesTo estimate the unknown function in model (4), we use improved weighted leastsquares estimates, defined in [2],S∗λ(xl) =n∑j=1λ(j) θ∗j,nφj(xl) 1Γ , 1 ≤ l ≤ n , (6)236Applied Methods of Statistical Analysiswhere (φj)j≤1 is an orthonormal functions system, the vector of weight coefficientsλ = (λ1, ..., λn) belongs some finite set Λ from [0, 1]n,θ∗j,n=(1− c(d)‖θ˜n‖1{1≤j≤d})θ̂j,n, ‖θ˜n‖2 =d∑j=1θ̂2j,n, θ̂j,n =b− ann∑l=1Ylφj(xl) .Here the coefficient d ≈ n/ lnn, 0 <  < 1, c(d) ≈ d/n. Now we define the estimatefor S in (1). We set for any a ≤ x ≤ bS∗λ(x) = S∗λ(x1)1{a≤x≤x1} +n∑l=2S∗λ(xl)1{xl−1<x≤xl} . (7)In order to obtain a good estimator, we have to write a rule to choose a weight vectorλ ∈ Λ in (7). It is obvious, that the best way is to minimize the empirical squarederror with respect to λ:Errn(λ) = ‖S∗λ − S‖2n → min .Making use of (7) and the Fourier transformation of S implyErrn(λ) =n∑j=1λ2(j)θ∗2j,n− 2n∑j=1λ(j)θ∗j,nθj,n +n∑j=1θ2j,n.Since the coefficient θj,n is unknown, we need to replace the term θ∗j,nθj,n by some itsestimator which we choose asϑ˜j,n = θ̂j,nθ∗j,n− b− ansj,n with sj,n =b− ann∑l=1σ2lφ2j(xl) .One has to pay a penalty for this substitution in the empirical squared error. Finally,we define the cost function of the formJn(λ) =n∑j=1λ2(j)θ∗2j,n− 2n∑j=1λ(j) ϑ˜j,n + ρPn(λ) ,where the penalty term is defined asPn(λ) =b− ann∑j=1λ2(j)sj,nand 0 < ρ < 1 is some positive constant which will be chosen later. We setλ̂ = argminλ∈Λ Jn(λ)and define an estimator of S of the form (7):S∗(x) = S∗λ̂(x) for a ≤ x ≤ b . (8)Now we obtain the non asymptotic upper bound for the quadratical risk of the esti-mator (8).237Novosibirsk, 18-20 September, 2019Theorem 1. Let Λ ⊂ [0, 1]n be any finite set such that the first d ≤ n componentsof the weight vector λ are equal to 1. Then, for any n ≥ 3 and 0 < ρ < 1/6, theestimator (8) satisfies the following oracle inequalityES‖S∗ − S‖2n ≤1 + 6ρ1− 6ρ minλ∈Λ ES‖Ŝλ − S‖2n+Ψn(ρ)n,where limn→∞Ψn(ρ)/n = 0.Now we consider the estimation problem (1) via model (4). We apply the estimat-ing procedure (8) with special weight set introduced in [5] to the regression scheme(4). Denoting S∗α= S∗λαwe setS∗ = S∗α̂with α̂ = argminα∈Aε Jn(λα) .Theorem 2. Assume that S ∈ ΣL,N and the number of the points n = n(T ) in themodel (4) . Then the procedure S∗ satisfies, for any T ≥ 32, the following inequalityR(S∗, S) ≤ (1 + ρ)2(1 + 6ρ)1− 6ρ minα∈AεR(S∗α, S) +BT (ρ)n,where limT→∞ BT (ρ)/n(T ) = 0.2 Asymptotic efficiencyIn order to study the asymptotic efficiency we define the following functional SobolevballW 0k,r= {f ∈ Ck0([a, b]) :k∑j=0‖f (j)‖2 ≤ r} , (9)where r > 0 and k ≥ 1 are some unknown parameters, Ck0([a, b]) is the space of ktimes differentiable functions f : R→ R such thatf (i)(x) = 0 , for 0 ≤ i ≤ k − 1 and x /∈ [a, b] .We will call such functions periodic on the interval [a, b]. Let S0 be a fixed k+1 timescontinuously differentiable function from ΣL,N . We setΘk,r = {S = S0 + f , f ∈ W 0k,r} . (10)In order to formulate our asymptotic results we define the following normalizingcoefficientγ(S) = ((1 + 2k)r)1/(2k+1)(J(S)kpi(k + 1))2k/(2k+1)(11)withJ(S) =∫ ba1qS(x)dx , qS(x) =exp{2 ∫ x0S(z)dz}∫ +∞−∞ exp{2∫ y0S(z)dz}dy .238Applied Methods of Statistical AnalysisIt is well known that for any S ∈ Θk,r the optimal rate of convergence is T−2k/(2k+1)(see, for example, [18]). On the basis of the model selection procedure (8) in the nextsection we construct the adaptive procedure S∗ for which we obtain the followingasymptotic upper bound for the quadratic risk.Theorem 3. The quadratic risk (2) for the estimating procedure S∗ has the followingasymptotic upper boundlim supT→∞T 2k/(2k+1) supS∈Θk,rR(S∗, S)γ(S)≤ 1 . (12)Moreover, we show that this upper bound is sharp in the following sense.Theorem 4. For any estimator Ŝ of S measurable with respect to FyT ,lim infT→∞infŜT 2k/(2k+1) supS∈Θk,rR(Ŝ, S)γ(S)≥ 1 , (13)where FyT is a σ−field generated by observations (yt)0≤t≤T .Remark 1. It should be noted that the choice of the functional class Θk,r in the formof (10) is related to the ergodicity of the process (1). This property is provided whenthe drift derivative is negative on the outside of a finite interval. The last excludesthe choice of periodic functions as a class of admissible drifts. For this reason, weuse the Sobolev ball of periodic functions with a non periodic center S0 as a class ofadmissible drift functions.Remark 2. Note that the inequalities (14) and (13) imply that the function (11) isthe Pinsker constant in this case (cf. [4]).Corollary 1. From Theorems 2 and 3 it follows that the procedure for choosing amodel S∗, defined in (8), is asymptotically efficient, i.e.limT→∞T 2k/(2k+1) supS∈Θk,rR(S∗, S)γ(S)= 1 . (14)3 Numerical simulationsWe suppose that in the model (1)S(x) = x2 sin(2pix) + x2(1− x) cos(4pix).For weight coefficients we choose n = T ,k∗ = 100 +√lnn, ε =1lnn, m = ln2 n, ωα = 100 + (Aβtn)12β+1 .The empirical risk:R(S∗, S) = 110001000∑m=1‖S∗m − S‖2n.Table 1 shows the results of the behavior of empirical mean-square risks for theproposed estimation procedure (8).239Novosibirsk, 18-20 September, 2019Table 1: Empirical quadratic asymptotic risksn 501 1001 2001 10001T 2k/(2k+1)R(S∗,S)γ(S)) 4.7257 2.0856 1.0072 0.9012From Table 1 it is clear that with an increase in the number of observations n,the normalized empirical mean-square risks tend to unity, which confirm numericallythe Corollary 1.The figures show the behavior of observation processes (yt)0≤t≤1, function S (redline), and improved estimate S∗ (green line):Figure 1: n=501Figure 2: n=1001AcknowledgementsThis work was supported by the RSF grant no 17-11-01049.240Applied Methods of Statistical AnalysisFigure 3: n=10001References[1] Kutoyants Yu. 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Asymptotically efficient estimation of a drift coefficient in diffusion processes

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Asymptotically efficient estimation of a drift coefficient in diffusion processes

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