On Parameter Estimation of Hidden Telegraph Process

Transcription

1 On Parameter Estimation of Hidden elegraph Process arxiv:159.74v1 [math.s] 9 Sep 15 R. Z. Khasminskii 1 and Yu.A. Kutoyants 1 Wayne State University, Detroit, USA 1 Institute for Information ransmission Problems, Moscow, Russia National Research University MPEI, Moscow, Russia, University of Maine, Le Mans, France Abstract he problem of parameter estimation by the observations of the two-state telegraph process in the presence of white Gaussian noise is considered. he properties of estimator of the method of moments are described in the asymptotics of large samples. hen this estimator is used as preliminary one to construct the one-step MLE-process, which provides the asymptotically normal and asymptotically efficient estimation of the unknown parameters. MSC Classification: 6M5, 6F1, 6F1. Key words: elegraph process, estimator of method of moments, one-step MLE, asymptoic efficiency 1 Introduction his work is devoted to the problem of parameter estimation by the observations in continuous time X = Xt), t ) of the following stochastic process dx t = Y t) dt+dw t, X, 1) here W t, t is a standard Wiener process, X) = X is the initial value independent of the Wiener process and Y t), t is a two-state 1

2 y 1 and y ) stationary Markov process with transition rate matrix ) λ λ. µ µ We suppose that the values λ > and µ > are unknown and we have to estimate the two-dimensional parameter ϑ = λ,µ) Θ, here Θ = c,c 1 ) c,c 1 ) by the observations X t, < t, i.e., the estimator ϑ t,, < t is stochastic process. Here c < c 1 are positive constants. herefore, our goal is to construct an on-line estimator-process ϑ = ϑ t,, < t ), which can be sufficiently easy to evaluate and is asymptotically optimal in some sense, as. his estimator-process we constructe in two steps. First we introduce a learning interval [, δ], here δ 1,1) and propose a δ -consistent preliminary estimator constructed of the method of moments. hen we improve it up to asymptotically efficient one with the help of slightly modified one-step MLE procedure. Such model of observations is called Hidden Markov Model HMM) or partially observed system. here exists an extensive study of such type HMM for discrete time models of observations, see, for example, [5], [1], [] and the references therein. For continuous time observation models this problem is not so-well studied. See, for example, Elliot et al. [5], here the Part III Continuous time estimation is devoted to such models of observations. One can find there the discussion of the problems of filtration and parameter estimation in the case of observations of finite-state Markov process in the presence of White noise. he problem most close to our statement was studied by Chigansky [3], who considered the parameter estimation in the case of hidden finite-state Markov process by continuous time observations. He showed the consistency, asymptotic normality and asymptotic efficiency of the MLE of one-dimensional parameter. he case of two-state hidden telegraph process studied in our work was presented there as example but there supposed that λ = µ. he problem of parameter estimation for similar models including the identification of partially observed linear processes models of Kalman filtration) were studied by many authors. Let us mention here [1], [4], [17], [11]. he problem of asymptotically efficient estimation for the model of telegraph process observed in discrete times was studied in [6]. he proposed here one-step MLE-process is motivated by the work of Kamatani and Uchida [8] who introduced Newton-Raphson multi-step estimators in the problem of parameter estimation by discrete time observations of diffusion process. In particular, it was shown that multi-step Newton- Raphson procedure allows to improve the rate of convergence of preliminary

3 estimator up to asymptotically efficient one. Note that preliminary estimator there is constructted by all observations and they studied estimator of the unknown parameter. Applied in this work estimator-process uses the preliminary estimator constructed by the observations on the initial learning interval and follows the similar construction as that one introduced in the work [13] see as well [1]). Problem statement and auxiliary results We start with description of the MLE for this model of observations. he stochastic process 1) according to innovation theorem see [14], heorem 7.1) admits the representation dx t = mt,ϑ) dt+d W t, X, t, here mt,ϑ) is the conditional expectation mt,ϑ) = E ϑ [ Y t) F X t ] = y1 P ϑ Y t) = y1 F X t ) +y P ϑ Y t) = y F X t Here F X t is the σ-algebra generated by the observations up to time t, i.e., F X t := σx t, s t) and W t, t is an innovation Wiener process. Let us denote Hence πt,ϑ) = P ϑ Y t) = y1 F X t ), Pϑ Y t) = y F X t mt,ϑ) = y +y 1 y )πt,ϑ). ) = 1 πt,ϑ). he random process πt,ϑ), t satisfies the following equation see [14], heorem 9.1 and equation 9.3) there) dπt,ϑ) = [µ λ+µ)πt,ϑ) +πt,ϑ)1 πt,ϑ))y y 1 )y +y 1 y )πt,ϑ))]dt +πt,ϑ)1 πt,ϑ))y 1 y ) dx t. ) { } Denote by P t) ϑ,ϑ Θ the measures induced by the observations X t = X s, s t) of stochastic processes 1) with different ϑ in the space of realizations C[, t] continuous on [, t] functions). hese measures are equivalent and the likelihood ratio function L ϑ,x t) = dpt) ) ϑ X t, ϑ Θ, < t dp t) 3 ).

4 can be written as follows L ϑ,x t) { = exp ms,ϑ)dx s 1 } ms,ϑ) ds. Here P t) is the measure corresponding to X t with Y s). he MLE-process ˆϑ t, is defined by the equation ) L ˆϑt,,X t = supl ϑ,x t), < t. 3) ϑ Θ It is known that in the one-dimensional case d = 1, λ = µ = ϑ) the MLE ˆϑ, = ˆϑ is consistent, asymptotically normal ˆϑ ϑ) = N,Iϑ) 1) and asymptotically efficient see [3], [7]). Here Iϑ) is the Fisher information. Note that the construction of the MLE-process ˆϑ t,, < t according to 3) and ) is computationally difficult problem because we need to solve the family of equations ) for all ϑ Θ and 3) for all t,]. We propose the following construction. First we study an estimator ϑ of the method of moments and show that this estimator is -consistent, i.e., E ϑ ϑ ϑ ) C. hen using this estimator ϑ δ obtained by the observations on the learning interval [, δ], here 1 < δ < 1, we introduce the one-step MLE-process ϑ t, = ϑ δ + 1/ I ϑ δ) 1 t ϑ δ,x t), δ t. Here the empirical Fisher information matrix I t ϑ) = 1 t δ ṁϑ,s)ṁϑ,s) ds Iϑ), as t, δ = ot) and the vector score-function process is t ϑ,x t ) = 1 t δ ṁϑ,s) [dx s mϑ,s)ds]. Here and in the sequel dot means the derivation w.r.t. parameter ϑ, ṁϑ, t) is the vector-column of the derivatives ṁ λ ϑ,t) and ṁ µ ϑ,t). he estimator 4

5 ϑ t, is in some sense asymptotically efficient. In particular for ϑ, = ϑ we have ϑ ϑ) = N,Iϑ) 1), i.e., it is asymptotically equivalent to the MLE. Note that the calculation of the estimator ϑ t, for all t [ δ, ] requires the solution of the equation ) for one value ϑ = ϑ δ only. Recall as well the well-known properties of the elegraph stationary) process Y t),t. 1. he stationary distribution of the process Y t) is P ϑ {Y t) = y 1 } = µ λ+µ, P ϑ{y t) = y } = λ λ+µ. Let us denote P ij t) = P ϑ {Y t) = y j Y ) = y i }, then solving the Kolmogorov equation we obtain P 11 t) = µ λ+µ + λ λ+µ e λ+µ)t, P 1 t) = λ λ+µ λ λ+µ e λ+µ)t, P 1 t) = µ λ+µ µ λ+µ e λ+µ)t, P t) = λ λ+µ + µ λ+µ e λ+µ)t 5) 4) It follows from 4) and 5) that y1 µ+y λ Ks) = E ϑ Y t)y t+s) = λ+µ here +y y 1 ) λµ λ+µ) e λ+µ)s = Ȳ ) +De λ+µ)s, 6) Ȳ = E ϑ Y t) = y 1µ+y λ λ+µ, D = y y 1 ) λµ λ+µ). 7) 3. Let F Y t F be a family of σ-algebras, induced by the events ) {Y s) = y i, s t,i = 1,}. hen it follows from 5) that for some constant K > and A < and for all s > A,t > the inequality Eϑ { Y s+)y t+) F Y A } Eϑ [Y s)y t)] < Ke λ+µ) A) holds. 5 8)

6 3 Method of moments estimator Let us consider the problem of the construction of -consistent estimators of the parameter ϑ by the method of moments. Recall that we observe in continuous time the stochastic process dx t = Y t)dt+dw t, X, t, 9) here W t, t is a standard Wiener process, X is independent of W t, t initial value, Y t) = Y t,ω) is stationary Markov process with two states y 1 and y and infinitesimal rate matrix ) λ λ. µ µ he processes Y t),t and W t,t are independent. We suppose for simplicity that is an integer number. Introduce the condition λ [c,c 1 ],µ [c,c 1 ] 1) here c and c 1 are some positive constants. o introduce the estimators we need the following notations. he function he statistics Φx) = 1 x 1 x 1 e x ). 11) ζ = 1 1 [X i+1 X i ] 1. 1) i= he random variable α is defined as a solution of the equation ) X ζ = +η Φα ), 13) here η = ) X y 1 y X ). 14) he event A that the equation 13) has a solution. 6

7 he random variable β = α 1I {A } +c +c 1 ) 1I {A c } 15) he method of moments estimator is ˆϑ = ˆλ,ˆµ ), here ˆλ = X) y 1 y y 1 β ; ˆµ = y X) β. 16) y y 1 he properties of these estimators are given in the following theorem. heorem 1 Let the condition 1) holds. hen for the estimators 16) and some constante C > we have for all > E ϑ [ ˆλ λ)] < C, Eϑ [ ˆµ µ)] < C. 17) he proof is given in several steps. he next lemma gives -consistent estimator for Ȳ see 7)). Lemma 1 Let the condition 1) be fulfilled. hen the estimator X / is uniformly consistent for Ȳ and for any > ) X E ϑ Ȳ C, 18) here the constant C > does not depend on ϑ. Proof. Making use 6) we obtain the fllowing relations ) X E ϑ Ȳ 1 [ = E ϑ Y t) Ȳ ] dt+ W = [ E ϑ Y t) Ȳ ] dt 1 1+ λµ ) λ+µ) 3 y y 1 ) 1 ) 1+ c 1 y 4c 3 y 1 ). Corollary. he existence of the consistent estimator for λ λ+µ and µ λ+µ follows from 4) and Lemma 1. Indeed, from the equality Ȳ = λ λ+µ y + µ λ+µ y 1 7

8 and Lemma 1 we obtain he statistics E ϑ [ 1 X y 1 y y 1 E ϑ [ y 1 X y y 1 µ λ+µ X = 1 λ )] < C, λ+µ )] < C. 19) Y t)dt+ W is the sum of a bounded a.s. random variable and an independent of it gaussian random variable with parameters, 1 ). Hence for η defined in 14) we can write the estimate E ϑ [ η D)] < C, ) here the constant C > does not depend on and ϑ. he constant D is defined in 7). Note that from the condition 1) we have λµ λ+µ) > c 4c 1 and we easily obtain the estimate ) for the estimator { } η = max η, c 8c 1 Lemma he following equality holds 1) and under the condition 1) we have as well E ϑ ζ = Ȳ +DΦλ+µ) ) E ϑ [ ζ E ϑ ζ )] < C. 3) Proof. From stationarity of the process Y t) and 6) we obtain E ϑ ζ = E ϑ [X 1 X ] 1 = E ϑ 1 1 Y s)y t)dsdt+1 1 = Ȳ +DΦλ+µ). 4) 8

9 Denote γ i = i+1 Further, from the equality ζ E ϑ ζ = 1 1 i= follows the estimate E ϑ ζ E ϑ ζ ) 3 E ϑ i Y t)dt; W i) = W i+1 W i. γ i E ϑ γ i) E ϑ 1 1 γ i W i)+ 1 1 W i) 1 ) i= γ i E ϑ γi i= 1 i= ) ) E ϑ i= γ i W i) ) W i) 1 )) := 3J 1 +1J +3J 3. 5) i= From stationarity of Y t) we obtain J 1 = 1 E ϑ = 1 1 i,j= 1 γ i E ϑ γi i= { ) ) = ) E ϑ γ i E ϑ γi) γ j E ϑ γj i= E ϑ {Y s)y t) j= E ϑ [ Y i j +s1 )Y i j +t 1 ) F Y 1 ]} dsdtds1 dt 1 E ϑ γ }. 6) he estimate 8) allows to write [ ] Eϑ Y i j +s1 )Y i j +t 1 ) F1 Y Eϑ Y s 1 )Y t 1 ) Ke λ+µ) j i. From this estimate, 6) and 1) we obtain J 1 K 1 i,j= he following estimates are evident J = 1 1 ) E ϑ γ i W i) = 1 J 3 = 1 E ϑ i= 1 i= e λ+µ) j i K 1. 1 i= [ W i) 1 ]) = 1 1 E ϑ γ i = E ϑγ i= K, E ϑ [ W i) 1 ] K. he second proposition of the Lemma follows from these estimates and 5). 9

10 Lemma 3 he function Φx) see 11)) has the following properties Proof. From 1) we obtain the representations lim x +, 7) lim Φx) =, x 8) Φ x) <, for x >. 9) Φx) = 1 x 3! + x 4! x3 5! Φ x) = 3! x ) ) 3x 4x3... 4! 5! 6! which allow to verify the limits 7) and 8) and as well the estimate 9) for x <. For x this estimate follows from the explicit expression for this derivative Φ x) = 1 ) x x 1 x + 1 ) e x. 3 x Let us consider the equation 13) for α, here ζ and η are defined in 1) and 14) respectively. Due to Lemma 3 this equation has not more than one solution. Recall that A is the following event: the equation 13) has solution and consider the statistics β defined in 15) here c,c 1 are the constants from the condition 1)). Lemma 4 Under the condition 1) the estimate β is -consistent for λ+µ. Moreover, for some constant C > which does not depend on and ϑ we have the property E ϑ [ β λ+µ))] < C. 3) Proof. It follows from Lemmae 1 and that ζ = Ȳ +DΦλ+µ)+ε 1 ). 31) Here and below we have for ε i ),i = 1,... the estimates E ϑ εi )) < C. 1

11 By Lemma 1, estimates ), 1) and the boundedness of Φx) we obtain as well the relation ζ = Ȳ + η Φλ+µ)+ε ). 3) If we have the event A then it follows from 13) and 3) that η Φα ) = η Φλ+µ)+ε 3 ). his relation, Lemma 3 and the separation from zero by a positive constant of the estimator η see Corollary to Lemma 1) yield for ω A herefore from Lemma 3 we obtain Φα ) Φλ+µ) = ε 4 ). E ϑ {1I {A } β λ+µ))} < C. 33) If ω A c then the equation ) X ) Ȳ +DΦλ+µ)+γ 3 ) = + η Φx) 34) has no solution x [c,c 1 ]. It follows from 34), Lemma 1 and the Corollary that the equation has no solution for x [c,c 1 ]. Φx) = Φλ+µ)+ε 4 ) Hence we can write A c { ε 4) > α} = B for some positive constant α which does not depend on. his allow us to write PA c ) PB } ) = P{ ε 4 ) > α} P{ ε4 ) > α E ϑ ε4 ) < C α. his estimate and 33) prove the Lemma 4. Proof of the heorem 1. he obtained results allow us to prove that the estimators defined in 17) are -consistent. Indeed, from the obvious equality X) y 1 λ ˆλ λ = β β y y 1 λ+µ + λ λ+µ β λ+µ)) 11

12 and 19) we obtain by Lemma 4 the estimate [ )] )) β X) y E ϑ ˆλ λ 1 Eϑ λ y y 1 λ+µ ) λ [ ] + E ϑ β λ+µ)) < C. λ+µ he second inequality in 17) can be proved by the same way. herefore the estimator ˆϑ ) = ˆλ,ˆµ is -consistent. 4 One-step MLE Our goal is to construct the asymptotically efficient estimator-process of the parameter ϑ = λ, µ) Θ. We do it in two steps. First we obtain by the observations X δ = X t, t δ) on the learning interval [, δ] the method of moments estimator ˆϑ δ = ˆλ δ,ˆµ δ) studied in the preceding section. Here δ 1,1). his estimator by heorem 1 satisfies the condition: sup ϑ K δ E ϑ ˆϑ δ ϑ C, here the constant C > does not depend on and ϑ Θ. Remind that Θ = c,c 1 ) c,c 1 ). Introduce the additional condition MN). For some N c N +9 >. 35) y 1 y ) 4 Having this preliminary estimator ˆϑ δ we propose one-step MLE which is based on one modification of the score-function as follows ) t ϑ,x t = 1 ṁϑ,s) [dx s mϑ,s)ds], t ϑ t, = ˆϑ δ +t 1 I t ˆϑ δ) 1 δ ṁˆϑ δ,s) 1 δ t [ ] dx s mˆϑ δ,s)ds. 36)

13 Here the vector ṁϑ,s) = y 1 y ) π λs,ϑ) ϑ πt,ϑ) = y 1 y ), πt,ϑ) ) λ µ and the empirical Fisher information matrix I t ϑ) is I t ϑ) = 1 t δ ṁϑ,s)ṁϑ,s) ds Iϑ) as t by the law of large numbers. Here Iϑ) is the Fisher information matrix Iϑ) = y 1 y ) E ϑ πs,ϑ) ϑ πs,ϑ). ϑ Let us change the variable τ = t 1 [,1] and introduce the random process ϑ τ),τ δ τ 1, here ϑ τ) = ϑ τ, and τ δ = δ 1. heorem Suppose that ϑ Θ, δ 1,1) and the condition M) holds, then the one-step MLE-process is consistent: for any ν > and any τ,1] P ϑ { ϑ τ) ϑ > ν} 37) and it is asymptotically normal τ ϑ τ) ϑ ) = N,Iϑ ) 1). 38) Proof. Let us denote the partial derivatives π λ t,ϑ) = πt,ϑ) λ and so on., π µ t,ϑ) = πt,ϑ), π λ,λ t,ϑ) = πt,ϑ), µ λ Lemma 5 Suppose that ϑ Θ and N > 1. If the condition c y 1 y ) > N ) holds, then supe ϑ π λ t,ϑ) N + π µ t,ϑ) N) < C 1, 4) ϑ Θ 13

14 and if the condition holds, then c N +9 > y 1 y ) 4 41) supe ϑ π λ,λ t,ϑ) N + π λ,µ t,ϑ) N + π µ,µ t,ϑ) N) < C. 4) ϑ Θ Here the constants C 1 >,C > do not depend on t. Proof. For simplicity of exposition we write π λ t,ϑ) = π λ, π µ t,ϑ) = π µ, πt,ϑ) = π. By the formal differentiation of dπ = [µ λ+µ)π π1 π)y 1 y )y +y 1 y )π)]dt we obtain the equations +π1 π)y 1 y ) dx t. 43) d π λ = πdt π λ [λ+µ+1 π)y 1 y )[y +y 1 y )π] +π1 π)y 1 y ) ] dt+ π λ 1 π)y 1 y ) dx t), 44) d π µ =[1 π]dt π µ [λ+µ+1 π)y 1 y )[y +y 1 y )π] +π1 π)y 1 y ) ] dt+ π µ 1 π)y 1 y ) dx t). 45) If we denote the true value of the parameters by ϑ and πt,ϑ ) = π o etc., then these equations for ϑ = ϑ become [ d π λ o = πo dt π λ o λ +µ +π o 1 π o )y 1 y ) ] dt + π λ o 1 πo )y 1 y ) d W t), 46) here d π o µ =[1 π o ]dt π o µ[ λ +µ +π o 1 π o )y 1 y ) ] dt + π o µ1 π)y 1 y ) d W t), 47) dπ o = [µ λ +µ )π o ]dt+π o 1 π o )y 1 y ) d W t). 48) his system of linear for π λ and π µ equations can be re-written as follows x t = π o,y t = π o λ,z t = π o µ,a = λ +µ,b = y 1 y ) dx t = [µ ax t ]dt+bx t 1 x t )d W t, 49) dy t = x t dt [ a+b x t 1 x t ) ] y t dt+b1 x t )y t d W t, 5) dz t = [1 x t ]dt [ a+b x t 1 x t ) ] z t dt+b1 x t )z t d W t. 51) 14

15 Note that as λ > and µ > the process πt,ϑ ) = x t,1) is ergodic with two reflecting borders and 1. herefore the process π o t,ϑ ) is ergodic with the invariant density f ϑ,x) = [x1 x)]µ λ) y 1 y ) Gϑ ) = [x1 x)]γµ λ ) Gϑ ) { exp µ } +λ µ )x y 1 y ) x1 x) { exp γµ x γλ } 1 x here we denoted γ = y 1 y ) and Gϑ ) is the normalizing constant 1 { Gϑ ) = [x1 x)] γµ λ ) exp γµ x γλ } dx. 1 x he processes y t and z t have explicite expressions { ] y t = exp [a+b x s 1 x s ) b 1 x s) ds z t = v +b 1 x s )d W s }x v dv, 5) v { ] exp [a+b x s 1 x s ) b v 1 x s) ds } +b 1 x s )d W s [1 x v ] dv. 53) Let us put x s = 1 x s. hen we have v and y t = x s 1 x s ) 1 1 x s) = 3x s xv ) 1 ) } a+ e b 4 t v) exp {3b x s ds+b x s d W s dv. v o estimate the moments E ϑ y t N we note that xv 1 1 Hölder inequality N ) N 1 f v)gv) dv) f v) N N 1 dv gv) N dv 15 v and use the

16 with f v) = exp{ at v)ε} and { gv) = exp a1 ε)+ )t v)+3b b x 4 sds+b here ε >. his yields the estimate E ϑ y t N CN,ε) v v x s d W s }, ) e N a1 ε)+ b 4 t v) E ϑ e 3Nb v x s ds+nb v xsd W s dv, here the constant CN,ε) > does not depend on t. Further, we can write } E ϑ exp {3Nb x s ds+nb x s d W s v v { } = E ϑ exp Nb x s d W s N b x s ds v v { }) exp Nb N +3) x s ds v { } Nb exp N +3)t v) 4 because x s 1/4 and { E ϑ exp Nb herefore, We see that if E ϑ y t N CN,ε) v = CN,ε) } x s d W s N b x s ds = 1. v e N a1 ε)+ b 4 b 4 N+3) ) t v) dv e N a1 ε) b N+1) ) t v) dv. λ +µ y 1 y ) > 1 + N, then E ϑ y t N C. In particular, if in the condition 39) we put N = and choose sufficiently small ε >, then we obtain the estimate πt,ϑ ) sup E ϑ ϑ Θ λ C, 54) 16

17 here the constant C > does not depend on t. We need as well to estimate the derivatives 44), 45) for the valuesϑ ϑ. he equation for π λ becomes d π λ = πdt π λ [ λ+µ+1 π)y1 y ) π π ) +π1 π)y 1 y ) ] dt+ π λ 1 π)y 1 y ) d W t). 55) Hence if we put a = λ + µ,y t = π λ and b = y 1 y, then we obtain the equation dy t = x t dt [ a+b 1 x t ) x t x t) +b x t 1 x t ) ] y t dt +b1 x t )y t d W t. he solution of this equation can be written explicitly like 5) but with additional term b 1 x t )x t x t ) in the exponent. his term satisfies the inequality 1 x t ) x t x t) 1. Hence if we repeat the evaluation of the E ϑ y t as it was done above, then for it boundness we obtain the condition λ+µ y 1 y ) > 3 + N. For the second derivative π = π λ,λ t,ϑ) we obtain the similar estimates of the moments as follows. he equation for π is d π = y t [ b y t xt x t) +b y t 1 x t ) ] dt by t d W t π [ a+b 1 x t ) x t x t) +b x t 1 x t ) ] dt+b π1 x t )d W t. Let us write it as d π = At)dt+Bt) d W t π[a+ct)]dt+ π t Dt) d W t in obvious notations. Hence the solution of it is πt,ϑ) λ = We have the corresponding estimate Cs) Ds) e v[a+cs) 1 Ds) ]ds+ v Ds)d W s [ Av)dv +Bv)d Wv ]. = b 1 x s ) ) x s x b s + b 3b x 1 ) + b 4 = 3b 4 3b 17 [ xs 1 x s ) 1 x s ) ] x 1 ) 3b

18 because x 1 ) 1 4. herefore, and if a 3 b N +3 b = a b N b λ+µ y 1 y ) > N, then we obtain E ϑ πt,ϑ) λ N < C. Hence under condition 41) we have sup ϑ Θ E ϑ πt,ϑ) λ N < C. he similar estimates can be obtained for the other derivatives. Lemma 5 is proven. Lemma 6 he solutions x t,y t,z t ) of the equations 49)-51) have ergodic properties. In particular, we have the following mean square convergence 1 1 ṁ λ t,ϑ ) dt = b ṁ λ t,ϑ )ṁ µ t,ϑ )dt = b 1 ṁ µ t,ϑ ) dt = b y t dt I 11ϑ ), y t z t dt I 1 ϑ ), z t dt I ϑ ), Proof. For the proof of the invariant measure existence see [3], section 4.. Note that the equations 49)-51) do not coincide with that of [3], because there it is supposed that λ = µ and y 1 = 1,y =, but the arguments given there are directly applied to the system of equations 49)-51) too. Recall that the strong mixing coefficient αt) for ergodic diffusion process 49) satisfies the estimate αt) < e c t. 18

19 For the proof see heorem in [15]. o check the conditions of this theorem we change the variables in the equation 49) ξ t = gx t ), gx) = x 1/ and obtain the stochastic differential equation dv bv1 v), x,1) dξ t = Aξ t )dt+dw t, ξ = gx ), t. he process ξ t,t has ergodic properties and the drift coefficient A ) satisfies the conditions of this theorem. Now to verify the convergence E ϑ 1 y t dt 1 = E ϑ 1 ) E ϑ yt dt we can apply the result of the following lemma. ) [ ] y t E ϑ yt dt 56) Lemma 7 Let {Y t,t > } be a stochastic process with zero mean and for some m > and k 1 E Y t mk 1) < C 1, here αt) is the strong mixing the coefficient. hen E Y t dt Proof. For proof see Lemma.1 in [9]. t k 1 [αt)] m )/m dt < C, k C 3 k. herefore if we put Y y = yt E ϑ yt, m = 3 and k = 1, then we obtain the convergence 56). Let us verify the consistency of the one-step MLE-process. We can write { P ϑ { ϑ ν } τ) ϑ > ν} P ϑ ˆϑ δ ϑ > { I τ ˆϑ +P δ) 1 τ [ ] } ϑ ṁˆϑ τ δ,s) dx s mˆϑ δ,s)ds > ν. δ 19

20 For the first probability by the heorem 1 we have { ν } P ϑ ˆϑ δ ϑ > 4 C ν E ϑ ˆϑ δ ϑ ν. δ he second probability can be evaluated as follows P ϑ { I τ ˆϑ δ) 1 τ τ P ϑ { I τ ˆϑ δ) 1 τ δ τ +P ϑ { I τ ˆϑ δ) 1 τ ṁˆϑ δ,s) δ τ [ ] } dx s mˆϑ δ,s)ds > ν } > ν 4 ṁˆϑ δ,s)d W s δ ṁˆϑ δ,s) m ˆϑ δ,s) ) here mˆϑ δ,s = mϑ,s) mˆϑ δ,s). We can write I τ ˆϑ δ) 1 τ ṁˆϑ τ δ,s)d W s δ I τ ˆϑ δ) 1 1 τ ṁˆϑ δ,s)d W s, γ δ γ here γ is such that δ γ > 1. Hence { I τ ˆϑ P δ) 1 } τ ϑ ṁˆϑ τ δ,s)d W s > ν 4 δ { 1 τ } P ϑ δ γ ṁˆϑ δ,s)d W s ν > δ I τ ˆϑ δ) 1 ν +P ϑ >, γ δ } ds > ν, 4 as, because { 1 P ϑ δ γ τ δ } ṁˆϑ δ,s)d W s ν > ṁˆϑ δ,s) ds 1 ν δ γe ϑ δ C. νδ γ 1

21 Recall that δ γ 1 >. Further { I τ ˆϑ P δ) 1 } τ [ ] ϑ ṁˆϑ τ δ,s) mϑ,s) mˆϑ δ,s) ds > ν 4 δ { 1 τ 1 ) P ϑ τ 1 γ ṁˆϑ δ,s) ṁϑ v,s) dvdsˆϑ δ ϑ > δ I τ ˆϑ δ) 1 ν +P ϑ > γ } ν as, because ˆϑ δ ϑ = O δ/) and other terms are bounded in probability. Here ϑ v = ϑ +vˆϑ δ ϑ ). o prove 38) we write δ τ τ ϑ τ) ϑ ) = τ ˆϑ δ ϑ )+ I τˆϑ δ) 1 ṁˆϑ δ,s)d W s τ δ + I τˆϑ δ) 1 τ [ ] ṁˆϑ δ,s) mϑ,s) mˆϑ δ,s) ds. τ We have the estimate E ϑ 1 τ τ 1 τ [ ] ṁˆϑ δ,s) ṁϑ,s) d W s δ τ E ϑ δ ṁˆϑ δ,s) ṁϑ,s) ds as, and by the central limit theorem the convergence in distribution 1 τ ṁϑ,s)d W s = N,Iϑ )). τ δ Further, let us denote ˆv δ = ) τ ˆϑ δ ϑ, then we can write τ ˆv δ + I τˆϑ δ) 1 τ δ = I τ ˆϑ δ) 1 I τ ˆϑ δ) 1 [ ] ṁˆϑ δ,s) mϑ,s) mˆϑ δ,s) ds 1 τ τ δ here ϑ r = ˆϑ ) δ +r ˆϑ δ ϑ. he presentation ṁϑ r,s) = ṁˆϑ δ,s)+ 1 1 ) ṁˆϑ δ,s)ṁϑ r,s) dr ds ˆv δ, ) mϑ q,s)dq ˆϑ δ ϑ

22 and the equality allows us to write as, I τ ˆϑ δ) = 1 τ ˆv δ + I τˆϑ δ) 1 τ = τ τ δ τ δ ṁˆϑ δ,s)ṁˆϑ δ,s) ds [ ] ṁˆϑ δ,s) mϑ,s) mˆϑ δ,s) ds ˆϑ δ ϑ O1) = 1 δ O1), Let us verify that the Fisher information matrix is non degenerate. It is sufficient to show that the matrix Eϑ ỹt Jϑ ) =, E ) ϑ ỹ t z t E ϑ ỹ t z t, E ϑ z t, is non degenerated. Here ỹ t, z t are stationary solutions of 5) and 51) respectively. If this matrix is degenerated, then Recall that by Cauchy-Schwarz inequality E ϑ ỹ t E ϑ z t = E ϑ ỹ t z t ). 57) E ϑ ỹ t z t ) E ϑ ỹ t E ϑ z t with equality if and only if z t = cỹ t with some constant c. herefore in the case of equality we have E ϑ cỹ t z t ) =. Introduce a new process ṽ t = cỹ t z t as a solution of the equation dṽ t = [ x t 1 c) 1]dt [ a+b x t 1 x t ) ] ṽ t dt+b1 x t )ṽ t d W t, here ṽ t and x t are stationary solutions. Further, following [3], Section 4, here the similar estimate was obtained, we write this solution as ṽ t = ṽ e at + +b e at s) [ x s 1 c) 1] ds b e at s) x s 1 x s )ṽ s ds e at s) 1 x s )ṽ s d W s.

23 Hence E ϑ + 4b4 a ) e at s) [ x s 1 c) 1]ds 4 1+e at) E ϑṽt e at s) 1 16 E ϑ ṽs ds+4b e at s) E ϑ ṽs ds CE ϑ ṽt with some constant C > which does not depend on t. Recall that E ϑ ṽt does not depend on t too because ṽ t is stationary solution. herefore if we show that for all c lim E ϑ e at s) [ x s 1 c) 1]ds) >, t then the matrix Jϑ ) is non degenerate. he random process ζ t = is the solution of the equation e at s) [ x s 1 c) 1]ds dζ t dt = aζ t + x t 1 c) 1, ζ =. he elementary calculations show that for all ϑ and c lim E ϑ ζt = E ϑ [ π 1 c) 1] >, t a here π is the stationary distribution. herefore the Fisher information matrix Iϑ ) is non degenerate for all ϑ Θ. Note that the limit covariance matrix of the one-step MLE-process by the heorem coincides with the covariance of the asymptotically efficient MLE [3], therefore ϑ τ) is asymptotically efficient too. 5 Discussions he learning interval in one-step section is [, δ ], where δ 1,1), i.e., it is negligeable with respect to the whole observations time. It can be done even shorter, if we use two-step MLE-process approach, as it was proposed 3

24 in [13]. It corresponds to the learning interval [, δ ) with δ 1, 1 ]. he 4 procedure is the follows. First we obtain the preliminary estimator ˆϑ δ as before. hen we introduce the second preliminary estimator ϑ t, = ˆϑ δ +t 1 I t ˆϑ δ) 1 t ˆϑ δ,x t ), t [ δ, ] and then we define two-step MLE-process here ϑ t, = ϑ t, +t 1 I t ˆϑ δ) 1 t ˆϑ δ,ϑ t,,xt ), t [ δ, ], t ϑ1,ϑ,x t) = 1 t δ ṁϑ 1,s)[dX s mϑ,s)ds], t [ δ, ]. It can be shown that for all τ,1] and t = τ we have the asymptotic normality of the estimator ϑ τ) = ϑ τ, : τ ϑ τ) ϑ ) = N,Iϑ ) 1). See the details in [13]. Note that it can be shown that the one-step MLE-process converges in distribution to the limit Brownian motion. Let us introduce the random process η τ) = τ Iϑ ) 1/ ϑ τ) ϑ ), τ δ τ 1, here τ δ = δ 1. More detailed analysis shows that the random process η τ),τ τ 1 converges to two-dimensional standard Wiener process W τ),τ τ 1 with any τ,1]. For the details see the proof of such convergence in similar problem in [13]. Acknowledgment. his work was done under partial financial support second author) of the grant of RSF number References [1] Bickel, P.J., Ritov, Y. and Rydén,. 1998) Asymptotic normality of the maximum likelihood estimator for general hidden Markov models. Ann. Statist., 6, 4,

25 [] Cappé, O., Moulines, E. and Rydén,. 5) Inference in Hidden Markov Models. Springer, N.Y. [3] Chigansky, P. 9) Maximum likelihood estimation for hidden Markov models in continuous time. Statist. Inference Stoch. Processes, 1,, [4] Dembo, A. and Zeitouni, O. 1986) Parameter estimation for partially observed continuous time processes via the EM algorithm. Stoch. Proces. Applic. 3, [5] Elliott, R.J., Aggoun, L. and Moor, J.B. 1995) Hidden Markov Models. Springer, N.Y. [6] Iacus, S. and Yoshida, N. 9) Estimation for discretly observed telegraph process. heory Probab. Math. Statist., 78, [7] Ibragimov I.A. and Khasminskii R. 1981) Statistical Estimation - Asymptotic heory. Springer-Verlag, New York. [8] Kamatani, K. and Uchida, M. 15) Hybrid multi-step estimators for stochastic differential equations based on sampled data. Statist. Inference Stoch. Processes. 18,, [9] Khasminskii, R.Z. 1966) On stochastic processes defined by differential equations with small parameter. heory Probab. Appl., 11, [1] Kutoyants, Yu.A. 1984) Parameter Estimation for Stochastic Processes. Heldermann, Berlin. [11] Kutoyants, Yu.A. 1994) Identification of Dynamical Systems with Small Noise. Kluwer, Dordrecht. [1] Kutoyants, Yu.A. 14) On approximation of the backward stochastic differential equation. Small noise, large samples and high frequency cases. Proceedings of the Steklov Institute of Mathematics., 87, [13] Kutoyants, Yu.A. 15) On multi-step MLE-processes for ergodic diffusion. submitted. [14] Liptser, R.S. and Shiryaev, A.N. 1) Statistics of Random Processes, -nd ed., vol. 1, Springer, N.Y. [15] Veretennikov, A. Yu. 1987) Bounds for the mixing rate in the theory of stochastic equations. heory Probab. Appl., 3,,

26 [16] Wonham, W. M. 1965) Some applications of stochastic differential equations to optimal non-linear filtering. SIAM J. Contr., Ser. A,, [17] Zeitouni, O. and Dembo, A. 1988) Exact filters for the estimation of the number of transitions of finite-state continuous time Marcov processes. IEEE I, 34, 4,