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Volume 2012 |Article ID 342705 | https://doi.org/10.1155/2012/342705

Xiu Kan, Huisheng Shu, Yan Che, "Asymptotic Parameter Estimation for a Class of Linear Stochastic Systems Using Kalman-Bucy Filtering", Mathematical Problems in Engineering, vol. 2012, Article ID 342705, 15 pages, 2012. https://doi.org/10.1155/2012/342705

Asymptotic Parameter Estimation for a Class of Linear Stochastic Systems Using Kalman-Bucy Filtering

Academic Editor: Jun Hu
Received21 Jun 2012
Accepted21 Jul 2012
Published01 Sep 2012

Abstract

The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameter šœƒāˆ¶š‘‘š‘‹š‘”=(šœƒš›¼(š‘”)+š›½(š‘”)š‘‹š‘”)š‘‘š‘”+šœŽ(š‘”)š‘‘š‘Šš‘”. Continuous-time Kalman-Bucy linear filtering theory is first used to estimate the unknown parameter Īø based on Bayesian analysis. Then, some sufficient conditions on coefficients are given to analyze the asymptotic convergence of the estimator. Finally, the strong consistent property of the estimator is discussed by comparison theorem.

1. Introduction

Stochastic differential equations (SDEs) are a natural choice to model the time evolution of dynamic systems which are subject to random influences. Such models have been used with great success in a variety of application areas, including biology, mechanics, economics, geophysics, oceanography, and finance. For instance, refer to [1ā€“8]. In reality, it is unavoidable that a stochastic system contains unknown parameters. Since 1962, Arato et al. [10] first applied parameter estimation to geophysical problem. Parameter estimation for SDEs has attracted the close attention of many researchers, and many parameter estimation methods for various advanced models have been studied, such as maximum likelihood estimation (MLE), Bayes estimation (BE), maximum probability estimation (MPE), minimum distance estimation (MDE), minimum contrast estimation (MCE), and M-estimation (ME). See [10ā€“15] for details.

In practice, most stochastic systems cannot be observed completely, but the development of filtering theory provides an effective method to solve this problem. Over the past few decades, a lot of effective approaches have been proposed to overcome the difficulties in parameter estimation for stochastic models by filtering methods. It turns out to be helpful both in computability and asymptotic studies. See [9, 16ā€“26]. In particular, the parameter estimation has been studied based on filtering observation, and the strong consistency property has also been shown in [27, 28]. In [29], a large deviation inequality has been obtained which implies the strong consistency, local asymptotic normality, and the convergence of moments. The asymptotic properties of estimators have been studied for a class of special Gaussian ItĆ“ processes with noisy observations in [30]. It should be pointed out that, so far, although the parameter estimation problem has been widely investigated for SDEs, the parameter estimation problem for stock price model has gained much less research attention due probably to the mathematical complexity.

Stock return volatility process is an important topic in options pricing theory. During the past decades, many SDEs have been modeled to solve the financial problems. For instance, refer to [2, 31ā€“35]. Particularly, the so-called Hull-White model has been established by Hull and White [34] to analyze European call options prices under stochastic volatility at 1987. Using Taylor series expansion, an accurate formula for call options has been derived where stock returns and stock volatilities are uncorrelated. In addition, the Hull-White model readily lends itself to the estimation of underlying stochastic process parameters. Since the Hull-White formula is an effective options pricing model, it has been widely used to model the practice stock price problem. Therefore, it is reasonable to study the parameter estimation problem for Hull-White model with unknown parameter. Unfortunately, to the best of the authorsā€™ knowledge, the parameter estimation for Hull-White model with unknown parameter based on Kalman-Bucy linear filtering theory has not been fully studied despite its potential in practical application, and this situation motivates our present investigation.

Summarizing the above discussions, in this paper, we aim to investigate the parameter estimation problem for a general class of linear stochastic systems. The main contributions of this paper lie in the following aspects. (1) Kalman-Bucy linear filtering is used to solve the parameter estimation problem. (2) The asymptotic convergence of the estimator is investigated by analyzing Riccati equation. (3) The strong consistent property is studied by comparison theorem. The rest of this paper is organized as follows. In Section 2, we formulate the problem and state the well-known fact which would be used later. In Section 3, we study the asymptotic convergence of the estimator. In Section 4, the strong consistent of estimator is given. In Section 5, some conclusions are drawn.

Notation. The notation used here is fairly standard except where otherwise stated. ā„=(āˆ’āˆž,+āˆž) and ā„+=[0,+āˆž). For a vector š‘„=āˆˆā„, |š‘„| is the Euclidean norm (or šæ2 norm) with āˆš|š‘„|=š‘„ā‹…š‘„. š‘€š‘‡ and š‘€āˆ’1 represent the transpose and inverse of the matrix š‘€. det(š‘€) denotes the determinant of the matrix š‘€. š¼ denotes the identity matrix of compatible dimension. Moreover, let (Ī©,ā„±,š) be a complete probability space with a natural filtration {ā„±š‘”}š‘”ā‰„0 satisfying the usual conditions (i.e., it is right continuous, and ā„±0 contains all š-null sets). š”¼[š‘„] stands for the expectation of the stochastic variable š‘„ with respect to the given probability measure š. š¶(ā„+) denotes the class of all continuous time on š‘”āˆˆā„+.

2. Problem Statement

Hull-White model is a continuous-time, real stochastic process as follows: š‘‹š‘”=š‘‹0+ī€œš‘”0ī€·š›¼(š‘ )+š›½(š‘ )š‘‹š‘ ī€øī€œš‘‘š‘ +š‘”0šœŽ(š‘ )š‘‘š‘Šš‘ (2.1) with initial value š‘‹0 as a Gaussian random variable, where š›¼,š›½,šœŽ are deterministic continuous functions on time š‘”, š‘Šš‘” is a Brownian motion independent of the initial value š‘‹0. Obviously, Hull-White model (2.1) is a general continuous-time linear SDE for š‘‹š‘”, and we assume that the coefficient š›¼ contains an unknown parameter šœƒāˆˆš‘… as follows: š‘‘š‘‹š‘”=ī€·šœƒš›¼(š‘”)+š›½(š‘”)š‘‹š‘”ī€øš‘‘š‘”+šœŽ(š‘”)š‘‘š‘Šš‘”š‘”ā‰„0,(2.2) and we observe the process š‘‹š‘” by the following filtering observations: š‘‘š‘Œš‘”=šœ‡(š‘”)š‘‹š‘”š‘‘š‘”+š›¾(š‘”)š‘‘š‘‰š‘”š‘”ā‰„0,(2.3) where šœ‡,š›¾ are deterministic bounded continuous functions on time š‘”, and š‘‰š‘” is a Brownian motion independent of š‘Šš‘”.

Now, our aim is to estimate šœƒ in (2.2) based on the observation of (2.3). First, we can use Bayesian analysis to deal with the unknown parameter šœƒ. We model šœƒ as a random variable and denoted it as šœƒ0. We assume šœƒ0 normally distributed and independent of šœŽ(š‘Šš‘”,š‘‰š‘”,š‘”ā‰„0). Then, we can rewrite (2.2) as a two-component system for (š‘‹š‘”,šœƒš‘”) as follows: āŽ›āŽœāŽœāŽš‘‘š‘‹š‘”š‘‘šœƒš‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽš‘‹š›½(š‘”)š›¼(š‘”)00š‘”šœƒš‘”āŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽ0āŽžāŽŸāŽŸāŽ š‘‘š‘”+šœŽ(š‘”)š‘‘š‘Šš‘”š‘”ā‰„0.(2.4) Similarly, filtering observations system (2.3) can be expressed as follows: š‘‘š‘Œš‘”=ī‚€ī‚āŽ›āŽœāŽœāŽš‘‹šœ‡(š‘”)0š‘”šœƒš‘”āŽžāŽŸāŽŸāŽ š‘‘š‘”+š›¾(š‘”)š‘‘š‘‰š‘”š‘”ā‰„0.(2.5) Therefore, we can use the Kalman-Bucy linear filtering theory to estimate šœƒ0 as follows: Ģ‚šœƒš‘”ī€ŗšœƒ=š”¼0āˆ£š‘Œš‘ ī€»,,0ā‰¤š‘ ā‰¤š‘”(2.6) and moreover, we also have īš‘‹š‘”=š”¼[š‘‹š‘”|š‘Œš‘ ,0ā‰¤š‘ ā‰¤š‘”].

For given Gaussian initial conditions š‘‹0 and šœƒ0, it is well known from Kalman-Bucy linear filtering theory that error covariance matrix š‘†(š‘”) satisfies the following Riccati equation: Ģ‡š‘†(š‘”)=š“š‘†+š‘†š“š‘‡āˆ’š‘†š¶š‘‡ī€·š·š·š‘‡ī€øāˆ’1š¶š‘†+šµšµš‘‡,(2.7) where ī€·š“=š›½(š‘”)š›¼(š‘”)00ī€øī€·,šµ=0šœŽ(š‘”)ī€ø,š¶=(šœ‡(š‘”)0),š·=š›¾(š‘”), and as we all know the error covariance matrix š‘†(š‘”) is defined as follows: āŽ›āŽœāŽœāŽš‘†š‘†(š‘”)=š‘„š‘„(š‘”)š‘†š‘„šœƒš‘†(š‘”)šœƒš‘„(š‘”)š‘†šœƒšœƒāŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽš”¼ī‚øī‚€š‘‹(š‘”)š‘”āˆ’īš‘‹š‘”ī‚2ī‚¹š”¼š‘‹ī‚ƒī‚€š‘”āˆ’īš‘‹š‘”ī‚ī€·šœƒ0āˆ’Ģ‚šœƒš‘”ī€øī‚„š”¼š‘‹ī‚ƒī‚€š‘”āˆ’īš‘‹š‘”ī‚ī€·šœƒ0āˆ’Ģ‚šœƒš‘”ī€øī‚„š”¼ī‚ƒī€·šœƒ0āˆ’Ģ‚šœƒš‘”ī€ø2ī‚„āŽžāŽŸāŽŸāŽŸāŽ .(2.8) Set š‘Ž=š‘†š‘„š‘„,š‘=š‘†š‘„šœƒ=š‘†šœƒš‘„, and š‘=š‘†šœƒšœƒ. From Riccati equation (2.7), one can get the following system: Ģ‡š‘Ž=2š›½š‘Ž+2š›¼š‘+šœŽ2āˆ’šœ‡2š›¾2š‘Ž2,Ģ‡šœ‡š‘=š›½š‘+š›¼š‘āˆ’2š›¾2šœ‡š‘Žš‘,Ģ‡š‘=āˆ’2š›¾2š‘2.(2.9)

Remark 2.1. Equation (2.9) is a nontrivial nonlinear ordinary differential equation system, and it is well known from the Kalman-Bucy linear filtering theory that such Riccati equations have unique solutions for all š‘”āˆˆā„+.

Remark 2.2. From the equation Ģ‡š‘=āˆ’(šœ‡2/š›¾2)š‘2, we can see that the error variance š”¼[(šœƒ0āˆ’Ģ‚šœƒš‘”)2] is monotonically decreasing.

3. Asymptotic Convergence Analysis

Assume that the initial conditions š‘‹0 and šœƒ0 are independent and have nonvariances, so that š‘(0)=0 and š‘Ž(0)=š”¼[š‘‹20]>0,š‘(0)=š”¼[šœƒ20]>0; thus, š‘†(0) is a regular matrix. For the property of continuity of š‘†(š‘”), š‘†āˆ’1(š‘”) exists at least for small times. In order to obtain the rate of convergence of the estimator, š‘†(š‘”) should satisfy the regularity conditions. The following Theorem certifies the regularity of š‘†(š‘”).

Theorem 3.1. ā€‰(a1) Assume the initial conditions š‘‹0 and šœƒ0 for system (2.2) are independent and have nonvanishing variances.
ā€‰ā€‰(a2) Let š›¼(š‘”),š›½(š‘”),šœŽ(š‘”),šœ‡(š‘”),š›¾(š‘”)āˆˆš¶(ā„+).
ā€‰Then, the error covariance matrix š‘†(š‘”) satisfies det(S(t))>0 for all š‘”ā‰„0, and š‘†š‘„š‘„(š‘”)>0,š‘†šœƒšœƒ(š‘”)>0āˆ€š‘”ā‰„0.(3.1)

Proof. By Kalman-Bucy linear filtering theory, we know that det(š‘†(š‘”))>0 for all š‘”ā‰„0. Furthermore, it is not difficult to show that (3.1) holds for all š‘”ā‰„0.
Since det(š‘†(š‘”))>0, it follows that š‘†āˆ’1(š‘”) exists. Set š‘…(š‘”)=š‘†āˆ’1āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ (š‘”)=š‘’(š‘”)š‘“(š‘”)š‘“(š‘”)š‘”(š‘”).(3.2) As we know that š‘…=1/š‘† implies that Ģ‡š‘…=āˆ’(1/š‘†2)Ģ‡š‘†, one can easily have that Ģ‡Ģ‡š‘…=āˆ’š‘…š‘†š‘….(3.3) It follows readily form (2.9) and (3.3) that Ģ‡š‘…=āˆ’š‘…š“āˆ’š“š‘‡š‘…+š¶š‘‡ī€·š·š·š‘‡ī€øāˆ’1š¶āˆ’š‘…šµšµš‘‡š‘….(3.4) Using a similar computation as (2.9), we can get šœ‡Ģ‡š‘’=2š›¾2āˆ’2š›½š‘’āˆ’šœŽ2š‘’2,Ģ‡š‘“=āˆ’š›¼š‘’āˆ’š›½š‘“āˆ’šœŽ2š‘’š‘“,Ģ‡š‘”=āˆ’2š›¼š‘“āˆ’šœŽ2š‘“2.(3.5) The condition (a1) shows that š‘Ž(0)>0,š‘(0)=0, and š‘(0)>0, which implies that š‘’(0)>0,š‘“(0)=0, and š‘”(0)>0. Since the Riccati equations (2.9) have unique solutions on š‘…+, thus the nonlinear system (3.5) has a unique solution on ā„+. Furthermore, the first equation Ģ‡š‘’=šœ‡2/š›¾2āˆ’2š›½š‘’āˆ’šœŽ2š‘’2 with initial condition š‘’(0)>0 has a unique solution on a maximal time interval [0,š‘‡), where š‘‡āˆˆā„+. Assume that there exists a smallest time āˆ’š‘”āˆˆ(0,š‘‡) such that š‘’(āˆ’š‘”)=0. By the property of continuity of š‘’(š‘”), we have š‘’(š‘”)>0, for 0ā‰¤š‘”<āˆ’š‘”. Thus, Ģ‡š‘’(š‘”)=limĪ”š‘”ā†’0š‘’ī‚€āˆ’š‘”ī‚ī‚€āˆ’š‘’āˆ’š‘”ī‚āˆ’Ī”š‘”Ī”š‘”<0,(3.6) this contradicts with Ģ‡š‘’(š‘”)=šœ‡2(āˆ’š‘”)/š›¾2(āˆ’š‘”)āˆ’2š›½(āˆ’š‘”)š‘’(āˆ’š‘”)āˆ’šœŽ2(āˆ’š‘”)š‘’2(āˆ’š‘”)ā‰¤šœ‡2(āˆ’š‘”)/š›¾2(āˆ’š‘”) for all š‘”āˆˆ[0,š‘‡). Therefore, š‘’(š‘”)>0, for š‘”āˆˆ[0,š‘‡).
As long as Ģ‡š‘’(š‘”)=šœ‡2(āˆ’š‘”)/š›¾2(āˆ’š‘”)āˆ’2š›½(āˆ’š‘”)š‘’(āˆ’š‘”)āˆ’šœŽ2(āˆ’š‘”)š‘’2(āˆ’š‘”)ā‰¤šœ‡2(āˆ’š‘”)/š›¾2(āˆ’š‘”) for all š‘”āˆˆ[0,š‘‡) and šœ‡(š‘”),š›¾(š‘”) are bounded, we have Ģ‡š‘’(š‘”)ā‰¤š¶, where š¶ is a constant. So that š‘’(š‘”) is bounded from below by 0 and from above by š‘’(0)+š‘”, which implies that š‘’(š‘”) cannot explode in finite time, thus š‘‡=+āˆž. This shows that system (3.5) has a unique solution on ā„+ because the second equation is a linear equation for š‘“ which can be solved analytically on ā„+, and š‘” can get by integration.
Define ā„Ž(š‘”)āˆ¶=det(š‘…(š‘”))=š‘’(š‘”)š‘”(š‘”)āˆ’š‘“2(š‘”). Since det(š‘†(š‘”))>0 for all š‘”ā‰„0, thus ā„Ž(š‘”)=det(š‘…(š‘”))=1/det(š‘†(š‘”))>0 for all š‘”ā‰„0, moreover, š‘†šœƒšœƒ>0 for all š‘”ā‰„0. Finally, we assume that there exists š‘”0 such that, š‘†š‘„š‘„(š‘”0)=0, then š‘”(š‘”0)=š‘†š‘„š‘„(š‘”0)ā„Ž(š‘”0)=0, so that ā„Ž(š‘”0)=š‘’(š‘”0)š‘”(š‘”0)āˆ’š‘“2(š‘”0)ā‰¤0, and this contradicts ā„Ž(š‘”0)>0. Hence, š‘†š‘„š‘„>0 for all š‘”ā‰„0.
The proof is complete.

In order to obtain the convergence rate, the Riccati equation must be solved, and we just need the solution of (3.5). Now, we solve the equation Ģ‡š‘’=šœ‡2/š›¾2āˆ’2š›½š‘’āˆ’šœŽ2š‘’2 when š›½,šœŽ,šœ‡,š›¾ are equal to constants.

In the case š‘’(0)ā‰ š‘™2, we get š‘’š‘™(š‘”)=1+š‘™2š‘™šæexpī€ŗī€·1+š‘™2ī€øšœŽ2š‘”ī€»š‘™šæexpī€ŗī€·1+š‘™2ī€øšœŽ2š‘”ī€»,āˆ’1(3.7) where šæ=(š‘’(0)+š‘™1)/(š‘’(0)āˆ’š‘™2), š‘™1=(2š›½/šœŽ2+āˆš4š›½2/šœŽ4+4šœ‡2/šœŽ2š›¾2)/2, š‘™2=(āˆ’(2š›½/šœŽ2āˆš)+4š›½2/šœŽ4+4šœ‡2/šœŽ2š›¾2)/2.

In the other case š‘’(0)=š‘™2, the solution shows that š‘’(š‘”)=š‘™2 for all š‘”ā‰„0.

Thus, for each š›¼>0, š›½>0, šœŽ>0, šœ‡>0, š›¾>0, the solution š‘’(š‘”) obviously satisfies š‘’(š‘”)āŸ¶š‘™2asš‘”āŸ¶+āˆž.(3.8)

The convergence rate of the estimator is given by following theorem.

Theorem 3.2. Assume that š›¼,š›½,šœŽ,šœ‡,š›¾āˆˆš¶(ā„+), are all bounded, and there are constants š›¼1, š›¼2, š›½1, š›½2, šœŽ1, šœŽ2, šœ‡1, šœ‡2, š›¾1, š›¾2, and š‘”0, such that(b1): 0<š›¼1ā‰¤|š›¼(š‘”)|ā‰¤š›¼2 for all š‘”ā‰„š‘”0;(b2): 0<š›½1ā‰¤|š›½(š‘”)|ā‰¤š›½2 for all š‘”ā‰„š‘”0;(b3): 0<šœŽ1ā‰¤|šœŽ(š‘”)|ā‰¤šœŽ2 for all š‘”ā‰„š‘”0;(b4): 0<šœ‡2ā‰¤|šœ‡(š‘”)|ā‰¤šœ‡1 for all š‘”ā‰„š‘”0;(b5): 0<š›¾1ā‰¤|š›¾(š‘”)|ā‰¤š›¾2 for all š‘”ā‰„š‘”0;(b6): 2š›¼1(š›½1+šœŽ21š‘™22)>šœŽ22š‘™21 where š‘™2š‘–=(āˆ’2š›½š‘–/šœŽ2š‘–+ī”(4š›½2š‘–)/(šœŽ4š‘–)+(4šœ‡2š‘–)/(šœŽ2š‘–š›¾2š‘–))/2,š‘–=1,2.
Then, for arbitrary šœ–>0 and š‘‡>0, we have š‘ƒī€·||šœƒ0āˆ’Ģ‚šœƒš‘”||ī€øā‰¤1>šœ–šœ–2š¶š‘‡āˆ’1,(3.9) where š¶ is a positive constant independent of šœ– and š‘‡.

Proof. Let š‘’š‘– be the solution to Ģ‡š‘’š‘–=šœ‡2š‘–/š›¾2š‘–āˆ’2š›½š‘–š‘’š‘–āˆ’šœŽ2š‘–š‘’2š‘–,š‘–=1,2, and š‘’š‘–(š‘”0)=š‘’(š‘”0).
Since šœ‡22/š›¾22āˆ’2š›½2š‘’āˆ’šœŽ22š‘’2ā‰¤Ģ‡š‘’=šœ‡2/š›¾2āˆ’2š›½š‘’āˆ’šœŽ2š‘’2ā‰¤šœ‡21/š›¾21āˆ’2š›½1š‘’āˆ’šœŽ21š‘’2 for all š‘”ā‰„š‘”0, by the comparison theorem [2, 36], we obtain that š‘’2(š‘”)ā‰¤š‘’(š‘”)ā‰¤š‘’1(š‘”)āˆ€š‘”ā‰„š‘”0.(3.10) It follows from (3.7) that š‘’ is bounded, and for any given š›æāˆˆ(0,1), there is a š‘”1ā‰„š‘”0 such that 0<š‘™22(1āˆ’š›æ)ā‰¤š‘’(š‘Ÿ)ā‰¤š‘™21(1+š›æ)āˆ€š‘Ÿā‰„š‘”1.(3.11) For š‘”ā‰„š‘”1, we can obtain from (3.5) and š‘“(0)=0 that ī€œš‘“(š‘”)=āˆ’š‘”0ī‚øāˆ’ī€œexpš‘”š‘ ī€·š›½(š‘Ÿ)+šœŽ2(ī€øī‚¹ī‚øāˆ’ī€œš‘Ÿ)š‘’(š‘Ÿ)š‘‘š‘Ÿš›¼(š‘ )š‘’(š‘ )š‘‘š‘ =āˆ’expš‘”0ī€·š›½(š‘Ÿ)+šœŽ2(ī€øī‚¹ī€œš‘Ÿ)š‘’(š‘Ÿ)š‘‘š‘Ÿš‘”10ī‚øī€œexpš‘ 0ī€·š›½(š‘Ÿ)+šœŽ2(ī€øī‚¹āˆ’ī€œš‘Ÿ)š‘’(š‘Ÿ)š‘‘š‘Ÿš›¼(š‘ )š‘’(š‘ )š‘‘š‘ š‘”š‘”1ī‚øāˆ’ī€œexpš‘”š‘ ī€·š›½(š‘Ÿ)+šœŽ2ī€øī‚¹(š‘Ÿ)š‘’(š‘Ÿ)š‘‘š‘Ÿš›¼(š‘ )š‘’(š‘ )š‘‘š‘ .(3.12) As š›½(š‘Ÿ)+šœŽ2(š‘Ÿ)š‘’(š‘Ÿ)ā‰„š›½1+šœŽ21š‘™22(1āˆ’š›æ) holds for all š‘”ā‰„š‘”1, thus, the first term in (3.12) goes to 0 as š‘”ā†’āˆž. For the second term in (3.12), we have ||||ī€œš‘”š‘”1ī‚øāˆ’ī€œexpš‘”š‘ ī€·š›½(š‘Ÿ)+šœŽ2ī€øī‚¹||||ā‰¤ī€œ(š‘Ÿ)š‘’(š‘Ÿ)š‘‘š‘Ÿš›¼(š‘ )š‘’(š‘ )š‘‘š‘ š‘”0ī€ŗāˆ’ī€·š›½exp1+šœŽ21š‘™22ī€øī€»š‘™(1āˆ’š›æ)(š‘”āˆ’š‘ )21=š‘™(1+š›æ)š‘‘š‘ 21(1+š›æ)š›½1+šœŽ21š‘™22ī€œ(1āˆ’š›æ)š‘”0ī€ŗāˆ’ī€·š›½exp1+šœŽ21š‘™22ī€øī€»š‘‘ī€·š›½(1āˆ’š›æ)(š‘”āˆ’š‘ )1+šœŽ21š‘™22ī€øš‘ =š‘™(1āˆ’š›æ)21(1+š›æ)š›½1+šœŽ21š‘™22ī€·ī€ŗāˆ’ī€·š›½(1āˆ’š›æ)1āˆ’exp1+šœŽ21š‘™22ī€øš‘”ā‰¤š‘™(1āˆ’š›æ)ī€»ī€ø21(1+š›æ)š›½1+šœŽ21š‘™22.(1āˆ’š›æ)(3.13) By similar arguments, we obtain that ||||ī€œš‘”š‘”1ī‚øāˆ’ī€œexpš‘”š‘ ī€·š›½(š‘Ÿ)+šœŽ2ī€øī‚¹||||ā‰„š‘™(š‘Ÿ)š‘’(š‘Ÿ)š‘‘š‘Ÿš›¼(š‘ )š‘’(š‘ )š‘‘š‘ 22(1āˆ’š›æ)š›½2+šœŽ22š‘™21.(1+š›æ)(3.14) Therefore, for any šœ‰>0, there exists š‘”(šœ‰)>0 such that š‘™22(1āˆ’š›æ)š›½2+šœŽ22š‘™21ā‰¤||||ā‰¤š‘™(1+š›æ)š‘“(š‘”)21(1+š›æ)š›½1+šœŽ21š‘™22(1āˆ’š›æ)āˆ€š‘”ā‰„š‘”(šœ‰).(3.15) For all š‘”ā‰„š‘”(šœ‰), we can get from (3.5) that ī€·Ģ‡š‘”=2|š›¼|āˆ’šœŽ2||š‘“||ī€ø||š‘“||ā‰„īƒ©2š›¼1āˆ’šœŽ22š‘™21(1+š›æ)š›½1+šœŽ21š‘™22īƒŖš‘™(1āˆ’š›æ)22(1āˆ’š›æ)š›½2+šœŽ22š‘™21=īƒ©(1+š›æ)2š›¼1ī€·š›½1+šœŽ21š‘™22ī€øāˆ’šœŽ22ī€·š‘™21ī€ø(1+š›æ)š›½1+šœŽ21š‘™22(īƒŖš‘™1āˆ’š›æ)22(1āˆ’š›æ)š›½2+šœŽ22š‘™21(.1+š›æ)(3.16) By assumption (b6), we get Ģ‡š‘”>0 for a sufficiently small šœ‰>0. This implies that š‘”(š‘”) goes to infinity at least as a linear function. Thus, there exists a constant š¶>0, such that š”¼ī€·šœƒ0āˆ’Ģ‚šœƒš‘”ī€ø2=š‘†šœƒšœƒ=š‘’ā„Žā‰¤š¶š‘”āˆ’1.(3.17) Hence, for arbitrary šœ–>0 and all š‘‡>0, it follows from Chebyshevā€™s inequality that š‘ƒī€·||šœƒ0āˆ’Ģ‚šœƒš‘”||ī€øā‰¤1>šœ–šœ–2š¶š‘‡āˆ’1.(3.18)
The proof is complete.

Remark 3.3. From the proof of Theorem 3.2, we can see that šœƒ0āˆ’Ģ‚šœƒš‘” goes to 0 in šæ2-sense under the given conditions. In other words, Ģ‚šœƒš‘” is asymptotically unbiased.

Remark 3.4. It is well known that Kalman-Bucy linear filtering theory remains valid if one replaces the Brownian motion (š‘Šš‘”,š‘‰š‘”) in systems (2.2) and (2.3) by an arbitrary centered orthogonal increment process of the same covariance structure. Thus, Theorem 3.2 remains valid under this replacement.

4. Strong Consistency

In last section, we give the conditions for the convergence rate of the estimator. Furthermore, we use the comparison theorem to proof the strong consistency in this section. As we all know, if the parameter šœƒ is, a genuine Gaussian random variable, then we can have a clear statistical interpretation for the convergence rate. Firstly, we pick šœƒ0 at random; secondly, let system (2.2) run up to time š‘” and simultaneously observe š‘Œ by system (2.3); finally, compute Ģ‚šœƒš‘” as the following form.

The Kalman-Bucy linear filtering theory shows us āŽ›āŽœāŽœāŽš‘‘š‘‹š‘”š‘‘šœƒš‘”āŽžāŽŸāŽŸāŽ =ī‚µš¶š“(š‘”)āˆ’š‘‡(š‘”)š¶(š‘”)š·2ī‚¶āŽ›āŽœāŽœāŽš‘‹(š‘”)š‘†(š‘”)š‘”šœƒš‘”āŽžāŽŸāŽŸāŽ š‘‘š‘”+š¶(š‘”)š·2(š‘”)š‘†(š‘”)š‘‘š‘Œš‘”=āŽ›āŽœāŽœāŽœāŽšœ‡š›½(š‘”)āˆ’2(š‘”)š›¾2š‘†(š‘”)š‘„š‘„āˆ’šœ‡(š‘”)š›¼(š‘”)2(š‘”)š›¾2(š‘†š‘”)šœƒš‘„āŽžāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽš‘‹(š‘”)0š‘”šœƒš‘”āŽžāŽŸāŽŸāŽ šœ‡š‘‘š‘”+2(š‘”)š›¾2āŽ›āŽœāŽœāŽš‘†(š‘”)š‘„š‘„š‘†(š‘”)šœƒš‘„āŽžāŽŸāŽŸāŽ (š‘”)š‘‘š‘Œš‘”(4.1) with initial conditions īš‘‹0=š”¼[š‘‹0] and Ģ‚šœƒ0=š”¼[šœƒ0]. If we denote that Ī¦(š‘”) is the matrix fundamental solution of the deterministic linear system āŽ›āŽœāŽœāŽĢ‡š‘„š‘”Ģ‡š‘¦š‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽšœ‡š›½(š‘”)āˆ’2(š‘”)š›¾2š‘†(š‘”)š‘„š‘„āˆ’šœ‡(š‘”)š›¼(š‘”)2(š‘”)š›¾2š‘†(š‘”)šœƒš‘„āŽžāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ (š‘”)0š‘„(š‘”)š‘¦(š‘”),(4.2) then the solution to (4.1) is given by āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =Ī¦(š‘”)Ī¦āˆ’1āŽ›āŽœāŽœāŽš”¼ī€ŗš‘‹(0)0ī€»š”¼ī€ŗšœƒ0ī€»āŽžāŽŸāŽŸāŽ +ī€œš‘”0Ī¦(š‘”)Ī¦āˆ’1āŽ›āŽœāŽœāŽš‘†(š‘ )š‘„š‘„š‘†(š‘”)šœƒš‘„(āŽžāŽŸāŽŸāŽ š‘”)š‘‘š‘Œš‘ .(4.3) And for every particular experiment šœ”, the quantity (šœƒ0Ģ‚šœƒ(šœ”)āˆ’š‘”(šœ”))2 would be the squared estimation error.

But in this paper šœƒ is a fixed parameter, so we can only choose šœƒ0(šœ”)=šœƒ, and then the statistical mean over different values of šœƒ0(šœ”) has no experimental meaning. The true estimation error is given by Ģ‚šœƒšœƒāˆ’š‘”, not šœƒ0āˆ’Ģ‚šœƒš‘”. It is therefore desirable that estimator Ģ‚šœƒš‘” converges to šœƒ0 for ā€œall fixed values šœ=šœƒ0" a.s. To establish such an assertion we work with a product space (š‘…Ć—Ī©,ā„¬(š‘…)āŠ—ā„±,šœ‚āŠ—š‘ƒ), where šœ‚ denotes the law of šœƒ0, and (Ī©,ā„±,š‘ƒ) is the underlying probability space for Brownian motion (š‘Šš‘”,š‘‰š‘”)š‘”ā‰„0. This space is most appropriate because one can make š‘ƒ a.s. statements for fixed šœāˆˆā„. Notice that in this representation we have šœƒ0(šœ,šœ”)=šœ for all (šœ,šœ”)āˆˆā„Ć—Ī©. Assuming this underlying probability space, we use the comparison theorem to get the following consistency result.

In the proof of Theorem 3.2, we know that š‘’,š‘“ is bonded and š‘” is monotonically increasing, moreover, š‘†š‘„š‘„(š‘”)=š‘Ž=š‘”/ā„Ž=š‘”/(š‘’š‘”āˆ’š‘“2)=(š‘”āˆ’š‘“2/š‘’+š‘“2/š‘’)/(š‘’š‘”āˆ’š‘“2)=1/š‘’+š‘“2/š‘’(š‘’š‘”āˆ’š‘“2) and š‘†šœƒš‘„(š‘”)=š‘=š‘“/ā„Ž=š‘“/(š‘’š‘”āˆ’š‘“2). Thus, there exist positive constants š‘Ž1,š‘Ž2,š‘1, and š‘2 such that š‘Ž1ā‰¤š‘Žā‰¤š‘Ž2 and š‘1ā‰¤š‘ā‰¤š‘2.

Theorem 4.1. Assume that the following two conditions are satisfied:(c1): Ģ‚šœƒš‘” converges to šœƒ0 in šæ2(šœ‚āŠ—š‘ƒ);(c2): š›½2āˆ’šœ‡22/š›¾22<0;(c3): (š›½2āˆ’(šœ‡22/š›¾22)š‘Ž2)2āˆ’4š›¼2(šœ‡22/š›¾22)š‘2<0.
Then, for all fixed šœāˆˆā„, we have Ģ‚šœƒš‘”(šœ,ā‹…)āŸ¶šœ,š‘ƒ-š‘Ž.š‘ .,š‘Žš‘ š‘”āŸ¶āˆž.(4.4)

Proof. We will show that (4.4) holds for all šœāˆˆš‘š‘, where šœ‚(š‘)=0.
By Kalman-Bucy linear filtering theory, we know āŽ›āŽœāŽœāŽš‘‘š‘‹š‘”š‘‘šœƒš‘”āŽžāŽŸāŽŸāŽ =ī‚µš¶š“(š‘”)āˆ’š‘‡(š‘”)š¶(š‘”)š·2ī‚¶āŽ›āŽœāŽœāŽš‘‹(š‘”)š‘†(š‘”)š‘”šœƒš‘”āŽžāŽŸāŽŸāŽ š‘‘š‘”+š¶(š‘”)š·2(š‘”)š‘†(š‘”)š‘‘š‘Œš‘”=āŽ›āŽœāŽœāŽœāŽšœ‡š›½(š‘”)āˆ’2(š‘”)š›¾2š‘†(š‘”)š‘„š‘„āˆ’šœ‡(š‘”)š›¼(š‘”)2(š‘”)š›¾2(š‘†š‘”)šœƒš‘„āŽžāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽš‘‹(š‘”)0š‘”šœƒš‘”āŽžāŽŸāŽŸāŽ šœ‡š‘‘š‘”+2(š‘”)š›¾2āŽ›āŽœāŽœāŽš‘†(š‘”)š‘„š‘„š‘†(š‘”)šœƒš‘„āŽžāŽŸāŽŸāŽ (š‘”)š‘‘š‘Œš‘”(4.5) with initial conditions īš‘‹0=š”¼[š‘‹0Ģ‚šœƒ]and0=š”¼[šœƒ0]=š”¼[šœ]=šœ.
Since the following linear equations: āŽ›āŽœāŽœāŽĢ‡š‘„š‘”Ģ‡š‘¦š‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽšœ‡š›½(š‘”)āˆ’2(š‘”)š›¾2š‘†(š‘”)š‘„š‘„āˆ’šœ‡(š‘”)š›¼(š‘”)2(š‘”)š›¾2š‘†(š‘”)šœƒš‘„āŽžāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ (š‘”)0š‘„(š‘”)š‘¦(š‘”)(4.6) equal to Ģ‡š‘„š‘”=ī‚øšœ‡š›½(š‘”)āˆ’2(š‘”)š›¾2š‘†(š‘”)š‘„š‘„ī‚¹(š‘”)š‘„(š‘”)+š›¼(š‘”)š‘Œ(š‘”),Ģ‡š‘¦š‘”šœ‡=āˆ’2(š‘”)š›¾2š‘†(š‘”)šœƒš‘„(š‘”)š‘„(š‘”),(4.7) it follows from (c1)ā€“(c3) that š›½1āˆ’šœ‡21š›¾21š‘Ž1šœ‡ā‰¤š›½(š‘”)āˆ’2(š‘”)š›¾2š‘†(š‘”)š‘„š‘„(š‘”)ā‰¤š›½2āˆ’šœ‡22š›¾22š‘Ž2š›¼<0,1ā‰¤š›¼(š‘”)ā‰¤š›¼2,āˆ’šœ‡21š›¾21š‘1šœ‡ā‰¤āˆ’2(š‘”)š›¾2š‘†(š‘”)šœƒš‘„šœ‡(š‘”)ā‰¤āˆ’22š›¾22š‘2.(4.8) For linear equations: āŽ›āŽœāŽœāŽĢ‡š‘„š‘”Ģ‡š‘¦š‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽš›½1āˆ’šœ‡21š›¾21š‘Ž1š›¼1āˆ’šœ‡21š›¾21š‘10āŽžāŽŸāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ ,āŽ›āŽœāŽœāŽš‘„(š‘”)š‘¦(š‘”)Ģ‡š‘„š‘”Ģ‡š‘¦š‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽš›½2āˆ’šœ‡22š›¾22š‘Ž2š›¼2āˆ’šœ‡22š›¾22š‘20āŽžāŽŸāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽš‘„āŽžāŽŸāŽŸāŽ ,(š‘”)š‘¦(š‘”)(4.9) if we set Ī¦1(š‘”) and Ī¦2(š‘”) that are the matrix fundamental solution of (4.9), we can obtain from the comparison theorem that Ī¦1(š‘”)ā‰¤Ī¦(š‘”)ā‰¤Ī¦2(š‘”).(4.10)
It is not difficult to explore (4.9), and get Ī¦1āŽ›āŽœāŽœāŽœāŽœāŽāˆ’šœ†(š‘”)=ī…ž1š‘21š‘’šœ†ā€²1š‘”āˆ’šœ†ī…ž2š‘21š‘’šœ†ā€²2š‘”š‘’šœ†ā€²1š‘”š‘’šœ†ā€²2š‘”āŽžāŽŸāŽŸāŽŸāŽŸāŽ ,Ī¦2āŽ›āŽœāŽœāŽœāŽœāŽāˆ’šœ†(š‘”)=1š‘€21š‘’šœ†1š‘”āˆ’šœ†2š‘€21š‘’šœ†2š‘”š‘’šœ†1š‘”š‘’šœ†2š‘”āŽžāŽŸāŽŸāŽŸāŽŸāŽ ,Ī¦1āˆ’1(āŽ›āŽœāŽœāŽœāŽœāŽœāŽāˆ’š‘š‘”)=21šœ†ī…ž1āˆ’šœ†2š‘’āˆ’šœ†ā€²1š‘”āˆ’šœ†ī…ž2šœ†ī…ž1āˆ’šœ†ī…ž2š‘’āˆ’šœ†ā€²1š‘”š‘21šœ†ī…ž1āˆ’šœ†ī…ž2š‘’āˆ’šœ†ā€²2š‘”šœ†ī…ž1šœ†ī…ž1āˆ’šœ†ī…ž2š‘’āˆ’šœ†ā€²2š‘”āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,Ī¦2āˆ’1(āŽ›āŽœāŽœāŽœāŽœāŽœāŽāˆ’š‘€š‘”)21šœ†1āˆ’šœ†2š‘’āˆ’šœ†1š‘”āˆ’šœ†2šœ†1āˆ’šœ†2š‘’āˆ’šœ†1š‘”š‘€21šœ†1āˆ’šœ†2š‘’āˆ’šœ†2š‘”šœ†1šœ†1āˆ’šœ†2š‘’āˆ’šœ†2š‘”āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,(4.11) whereš‘11=š›½1āˆ’(šœ‡21/š›¾21)š‘Ž1,š‘12=š›¼1,š‘21=(šœ‡21/š›¾21)š‘1,šœ†ī…ž1=(š‘11+ī”š‘211āˆ’4š‘12š‘21)/2,šœ†ī…ž2=(š‘11āˆ’ī”š‘211āˆ’4š‘12š‘21)/2, š‘€11=š›½2āˆ’(šœ‡22/š›¾22)š‘Ž2,š‘€12=š›¼2,š‘€21=(šœ‡22/š›¾22)š‘2,šœ†1=(š‘€11+ī”š‘€211āˆ’4š‘€12š‘€21)/2,šœ†2=(š‘€11āˆ’ī”š‘€211āˆ’4š‘€12š‘€21)/2.
By assumption (c2) and (c3), we know that šœ†ī…ž1<0,šœ†ī…ž2<0,šœ†1<0, and šœ†2<0.
By the ODE theory [37, 38] and above discussion, we know that the solution of (4.1) is given by āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =Ī¦(š‘”)Ī¦āˆ’1āŽ›āŽœāŽœāŽš”¼ī€ŗš‘‹(0)0ī€»š”¼ī€ŗšœƒ0ī€»āŽžāŽŸāŽŸāŽ +ī€œš‘”0Ī¦(š‘”)Ī¦āˆ’1āŽ›āŽœāŽœāŽš‘†(š‘ )š‘„š‘„š‘†(š‘”)šœƒš‘„(āŽžāŽŸāŽŸāŽ š‘”)š‘‘š‘Œš‘ .(4.12) Using the similar method, we can also obtain the solutions for the following two equations: āŽ›āŽœāŽœāŽš‘‘īš‘‹š‘”š‘‘Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽš›½1āˆ’šœ‡21š›¾21š‘Ž1š›¼1āˆ’šœ‡21š›¾21š‘10āŽžāŽŸāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ šœ‡š‘‘š‘”+1š›¾1āŽ›āŽœāŽœāŽš‘Ž1š‘1āŽžāŽŸāŽŸāŽ š‘‘š‘Œš‘”āŽ›āŽœāŽœāŽš‘‘īš‘‹,(4.13)š‘”š‘‘Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽš›½2āˆ’šœ‡22š›¾22š‘Ž2š›¼2āˆ’šœ‡22š›¾22š‘20āŽžāŽŸāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ šœ‡š‘‘š‘”+2š›¾2āŽ›āŽœāŽœāŽš‘Ž2š‘2āŽžāŽŸāŽŸāŽ š‘‘š‘Œš‘”,(4.14) where īš‘‹0=š”¼[š‘‹0] and Ģ‚šœƒ0=š”¼[šœƒ0]=š”¼[šœ]=šœ.
The solutions of the two equations are explored as the following form: āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =Ī¦1(š‘”)Ī¦1āˆ’1āŽ›āŽœāŽœāŽš”¼ī€ŗš‘‹(0)0ī€»š”¼ī€ŗšœƒ0ī€»āŽžāŽŸāŽŸāŽ +ī€œš‘”0Ī¦1(š‘”)Ī¦1āˆ’1āŽ›āŽœāŽœāŽš‘Ž(š‘ )1š‘1āŽžāŽŸāŽŸāŽ š‘‘š‘Œš‘ ,āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =Ī¦2(š‘”)Ī¦2āˆ’1(āŽ›āŽœāŽœāŽš”¼ī€ŗš‘‹0)0ī€»š”¼ī€ŗšœƒ0ī€»āŽžāŽŸāŽŸāŽ +ī€œš‘”0Ī¦2(š‘”)Ī¦2āˆ’1(āŽ›āŽœāŽœāŽš‘Žš‘ )2š‘2āŽžāŽŸāŽŸāŽ š‘‘š‘Œš‘ .(4.15)
For (4.14), we have that āŽ›āŽœāŽœāŽīš‘‹š‘”Ģ‚šœƒš‘”āŽžāŽŸāŽŸāŽ =Ī¦2(š‘”)Ī¦2āˆ’1āŽ›āŽœāŽœāŽš”¼ī€ŗš‘‹(0)0ī€»š”¼ī€ŗšœƒ0ī€»āŽžāŽŸāŽŸāŽ +ī€œš‘”0Ī¦2(š‘”)Ī¦2āˆ’1āŽ›āŽœāŽœāŽš‘Ž(š‘ )2š‘2āŽžāŽŸāŽŸāŽ š‘‘š‘Œš‘ (4.16) yields that Ģ‚šœƒš‘”=ī€œš‘”0ī‚øš‘Ž2ī‚µš‘€21šœ†1āˆ’šœ†2š‘’āˆ’šœ†2(š‘”āˆ’š‘ )āˆ’š‘€21šœ†1āˆ’šœ†2š‘’āˆ’šœ†2(š‘”āˆ’š‘ )ī‚¶+š‘2ī‚µšœ†1šœ†1āˆ’šœ†2š‘’āˆ’šœ†2(š‘”āˆ’š‘ )āˆ’šœ†2šœ†1āˆ’šœ†2š‘’āˆ’šœ†2(š‘”āˆ’š‘ )ī‚¶ī‚¹š‘‘š‘Œš‘ +ī‚µš‘€21šœ†1āˆ’šœ†2š‘’āˆ’šœ†2š‘”āˆ’š‘€21šœ†1āˆ’šœ†2š‘’āˆ’šœ†2š‘”ī‚¶š‘‹0+ī‚µšœ†1šœ†1āˆ’šœ†2š‘’āˆ’šœ†2š‘”āˆ’šœ†2šœ†1āˆ’šœ†2š‘’āˆ’šœ†2š‘”ī‚¶šœƒ0.(4.17) Since šœ†1<0 and šœ†2<0, it is easy to get Ģ‚šœƒš‘”(šœ,ā‹…)āŸ¶šœ,š‘ƒ-a.s.,asš‘”āŸ¶āˆž.(4.18) For (4.13), we can also get Ģ‚šœƒš‘”(šœ,ā‹…)āŸ¶šœ,š‘ƒ-a.s.,asš‘”āŸ¶āˆž.(4.19) Hence, for (4.1), we can get the following result: Ģ‚šœƒš‘”(šœ,ā‹…)āŸ¶šœ,š‘ƒ-a.s.,asš‘”āŸ¶āˆž.(4.20) The proof is complete.

Remark 4.2. Under the probability space used in this paper, we can see that Theorem 3.2 is the particular form of Theorem 4.1 if we use Chebyshevā€™s inequality on the result of Theorem 4.1.

Remark 4.3. The strong consistency in Deck [30] requires that Ģ‚šœƒš‘” is a martingale, while, in our result, Ģ‚šœƒš‘” can be not a martingale. Furthermore, when Ģ‚šœƒš‘” is a martingale, our result is more strong than Deckā€™s, so in that case we can relax the conditions as Deck.

5. Conclusions

In this paper, we have investigated the parameter estimation problem for a class of linear stochastic systems called Hull-White stochastic differential equations which are important models in finance. Firstly, Bayesian viewpoint is first chosen to analyze the parameter estimation problem based on Kalman-Bucy linear filtering theory. Secondly, some sufficient conditions on coefficients are given to study the asymptotic convergence problem. Finally, the strong consistent property of estimator is discussed by Kalman-Bucy linear filtering theory and comparison theorem.

Acknowledgments

This work was supported by the National Nature Science Foundation of China under Grant no. 60974030 and the Science and Technology Project of Education Department in Fujian Province JA11211.

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Copyright © 2012 Xiu Kan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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