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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 342705, 15 pages
http://dx.doi.org/10.1155/2012/342705
Research Article

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

1School of Information Science and Technology, Donghua University, Shanghai 200051, China
2School of Science, Donghua University, Shanghai 200051, China

Received 21 June 2012; Accepted 21 July 2012

Academic Editor: Jun Hu

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.

Abstract

The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameter 𝜃𝑑𝑋𝑡=(𝜃𝛼(𝑡)+𝛽(𝑡)𝑋𝑡)𝑑𝑡+𝜎(𝑡)𝑑𝑊𝑡. Continuous-time Kalman-Bucy linear filtering theory is first used to estimate the unknown parameter θ 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 [18]. 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 [1015] 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, 1626]. 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 Itô 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, 3135]. 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 authors’ 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. =(,+) and +=[0,+). For a vector 𝑥=, |𝑥| is the Euclidean norm (or 𝐿2 norm) with |𝑥|=𝑥𝑥. 𝑀𝑇 and 𝑀1 represent the transpose and inverse of the matrix 𝑀. det(𝑀) denotes the determinant of the matrix 𝑀. 𝐼 denotes the identity matrix of compatible dimension. Moreover, let (Ω,,𝐏) be a complete probability space with a natural filtration {𝑡}𝑡0 satisfying the usual conditions (i.e., it is right continuous, and 0 contains all 𝐏-null sets). 𝔼[𝑥] stands for the expectation of the stochastic variable 𝑥 with respect to the given probability measure 𝐏. 𝐶(+) denotes the class of all continuous time on 𝑡+.

2. Problem Statement

Hull-White model is a continuous-time, real stochastic process as follows: 𝑋𝑡=𝑋0+𝑡0𝛼(𝑠)+𝛽(𝑠)𝑋𝑠𝑑𝑠+𝑡0𝜎(𝑠)𝑑𝑊𝑠(2.1) with initial value 𝑋0 as a Gaussian random variable, where 𝛼,𝛽,𝜎 are deterministic continuous functions on time 𝑡, 𝑊𝑡 is a Brownian motion independent of the initial value 𝑋0. Obviously, Hull-White model (2.1) is a general continuous-time linear SDE for 𝑋𝑡, and we assume that the coefficient 𝛼 contains an unknown parameter 𝜃𝑅 as follows: 𝑑𝑋𝑡=𝜃𝛼(𝑡)+𝛽(𝑡)𝑋𝑡𝑑𝑡+𝜎(𝑡)𝑑𝑊𝑡𝑡0,(2.2) and we observe the process 𝑋𝑡 by the following filtering observations: 𝑑𝑌𝑡=𝜇(𝑡)𝑋𝑡𝑑𝑡+𝛾(𝑡)𝑑𝑉𝑡𝑡0,(2.3) where 𝜇,𝛾 are deterministic bounded continuous functions on time 𝑡, and 𝑉𝑡 is a Brownian motion independent of 𝑊𝑡.

Now, our aim is to estimate 𝜃 in (2.2) based on the observation of (2.3). First, we can use Bayesian analysis to deal with the unknown parameter 𝜃. We model 𝜃 as a random variable and denoted it as 𝜃0. We assume 𝜃0 normally distributed and independent of 𝜎(𝑊𝑡,𝑉𝑡,𝑡0). Then, we can rewrite (2.2) as a two-component system for (𝑋𝑡,𝜃𝑡) as follows: 𝑑𝑋𝑡𝑑𝜃𝑡=𝑋𝛽(𝑡)𝛼(𝑡)00𝑡𝜃𝑡0𝑑𝑡+𝜎(𝑡)𝑑𝑊𝑡𝑡0.(2.4) Similarly, filtering observations system (2.3) can be expressed as follows: 𝑑𝑌𝑡=𝑋𝜇(𝑡)0𝑡𝜃𝑡𝑑𝑡+𝛾(𝑡)𝑑𝑉𝑡𝑡0.(2.5) Therefore, we can use the Kalman-Bucy linear filtering theory to estimate 𝜃0 as follows: ̂𝜃𝑡𝜃=𝔼0𝑌𝑠,,0𝑠𝑡(2.6) and moreover, we also have 𝑋𝑡=𝔼[𝑋𝑡|𝑌𝑠,0𝑠𝑡].

For given Gaussian initial conditions 𝑋0 and 𝜃0, it is well known from Kalman-Bucy linear filtering theory that error covariance matrix 𝑆(𝑡) satisfies the following Riccati equation: ̇𝑆(𝑡)=𝐴𝑆+𝑆𝐴𝑇𝑆𝐶𝑇𝐷𝐷𝑇1𝐶𝑆+𝐵𝐵𝑇,(2.7) where 𝐴=𝛽(𝑡)𝛼(𝑡)00,𝐵=0𝜎(𝑡),𝐶=(𝜇(𝑡)0),𝐷=𝛾(𝑡), and as we all know the error covariance matrix 𝑆(𝑡) is defined as follows: 𝑆𝑆(𝑡)=𝑥𝑥(𝑡)𝑆𝑥𝜃𝑆(𝑡)𝜃𝑥(𝑡)𝑆𝜃𝜃=𝔼𝑋(𝑡)𝑡𝑋𝑡2𝔼𝑋𝑡𝑋𝑡𝜃0̂𝜃𝑡𝔼𝑋𝑡𝑋𝑡𝜃0̂𝜃𝑡𝔼𝜃0̂𝜃𝑡2.(2.8) Set 𝑎=𝑆𝑥𝑥,𝑏=𝑆𝑥𝜃=𝑆𝜃𝑥, and 𝑐=𝑆𝜃𝜃. From Riccati equation (2.7), one can get the following system: ̇𝑎=2𝛽𝑎+2𝛼𝑏+𝜎2𝜇2𝛾2𝑎2,̇𝜇𝑏=𝛽𝑏+𝛼𝑐2𝛾2𝜇𝑎𝑏,̇𝑐=2𝛾2𝑏2.(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 𝑡+.

Remark 2.2. From the equation ̇𝑐=(𝜇2/𝛾2)𝑏2, we can see that the error variance 𝔼[(𝜃0̂𝜃𝑡)2] is monotonically decreasing.

3. Asymptotic Convergence Analysis

Assume that the initial conditions 𝑋0 and 𝜃0 are independent and have nonvariances, so that 𝑏(0)=0 and 𝑎(0)=𝔼[𝑋20]>0,𝑐(0)=𝔼[𝜃20]>0; thus, 𝑆(0) is a regular matrix. For the property of continuity of 𝑆(𝑡), 𝑆1(𝑡) exists at least for small times. In order to obtain the rate of convergence of the estimator, 𝑆(𝑡) should satisfy the regularity conditions. The following Theorem certifies the regularity of 𝑆(𝑡).

Theorem 3.1.  (a1) Assume the initial conditions 𝑋0 and 𝜃0 for system (2.2) are independent and have nonvanishing variances.
  (a2) Let 𝛼(𝑡),𝛽(𝑡),𝜎(𝑡),𝜇(𝑡),𝛾(𝑡)𝐶(+).
 Then, the error covariance matrix 𝑆(𝑡) satisfies det(S(t))>0 for all 𝑡0, and 𝑆𝑥𝑥(𝑡)>0,𝑆𝜃𝜃(𝑡)>0𝑡0.(3.1)

Proof. By Kalman-Bucy linear filtering theory, we know that det(𝑆(𝑡))>0 for all 𝑡0. Furthermore, it is not difficult to show that (3.1) holds for all 𝑡0.
Since det(𝑆(𝑡))>0, it follows that 𝑆1(𝑡) exists. Set 𝑅(𝑡)=𝑆1(𝑡)=𝑒(𝑡)𝑓(𝑡)𝑓(𝑡)𝑔(𝑡).(3.2) As we know that 𝑅=1/𝑆 implies that ̇𝑅=(1/𝑆2)̇𝑆, one can easily have that ̇̇𝑅=𝑅𝑆𝑅.(3.3) It follows readily form (2.9) and (3.3) that ̇𝑅=𝑅𝐴𝐴𝑇𝑅+𝐶𝑇𝐷𝐷𝑇1𝐶𝑅𝐵𝐵𝑇𝑅.(3.4) Using a similar computation as (2.9), we can get 𝜇̇𝑒=2𝛾22𝛽𝑒𝜎2𝑒2,̇𝑓=𝛼𝑒𝛽𝑓𝜎2𝑒𝑓,̇𝑔=2𝛼𝑓𝜎2𝑓2.(3.5) The condition (a1) shows that 𝑎(0)>0,𝑏(0)=0, and 𝑐(0)>0, which implies that 𝑒(0)>0,𝑓(0)=0, and 𝑔(0)>0. Since the Riccati equations (2.9) have unique solutions on 𝑅+, thus the nonlinear system (3.5) has a unique solution on +. Furthermore, the first equation ̇𝑒=𝜇2/𝛾22𝛽𝑒𝜎2𝑒2 with initial condition 𝑒(0)>0 has a unique solution on a maximal time interval [0,𝑇), where 𝑇+. Assume that there exists a smallest time 𝑡(0,𝑇) such that 𝑒(𝑡)=0. By the property of continuity of 𝑒(𝑡), we have 𝑒(𝑡)>0, for 0𝑡<𝑡. Thus, ̇𝑒(𝑡)=limΔ𝑡0𝑒𝑡𝑒𝑡Δ𝑡Δ𝑡<0,(3.6) this contradicts with ̇𝑒(𝑡)=𝜇2(𝑡)/𝛾2(𝑡)2𝛽(𝑡)𝑒(𝑡)𝜎2(𝑡)𝑒2(𝑡)𝜇2(𝑡)/𝛾2(𝑡) for all 𝑡[0,𝑇). Therefore, 𝑒(𝑡)>0, for 𝑡[0,𝑇).
As long as ̇𝑒(𝑡)=𝜇2(𝑡)/𝛾2(𝑡)2𝛽(𝑡)𝑒(𝑡)𝜎2(𝑡)𝑒2(𝑡)𝜇2(𝑡)/𝛾2(𝑡) for all 𝑡[0,𝑇) and 𝜇(𝑡),𝛾(𝑡) are bounded, we have ̇𝑒(𝑡)𝐶, where 𝐶 is a constant. So that 𝑒(𝑡) is bounded from below by 0 and from above by 𝑒(0)+𝑡, which implies that 𝑒(𝑡) cannot explode in finite time, thus 𝑇=+. This shows that system (3.5) has a unique solution on + because the second equation is a linear equation for 𝑓 which can be solved analytically on +, and 𝑔 can get by integration.
Define (𝑡)=det(𝑅(𝑡))=𝑒(𝑡)𝑔(𝑡)𝑓2(𝑡). Since det(𝑆(𝑡))>0 for all 𝑡0, thus (𝑡)=det(𝑅(𝑡))=1/det(𝑆(𝑡))>0 for all 𝑡0, moreover, 𝑆𝜃𝜃>0 for all 𝑡0. Finally, we assume that there exists 𝑡0 such that, 𝑆𝑥𝑥(𝑡0)=0, then 𝑔(𝑡0)=𝑆𝑥𝑥(𝑡0)(𝑡0)=0, so that (𝑡0)=𝑒(𝑡0)𝑔(𝑡0)𝑓2(𝑡0)0, and this contradicts (𝑡0)>0. Hence, 𝑆𝑥𝑥>0 for all 𝑡0.
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 ̇𝑒=𝜇2/𝛾22𝛽𝑒𝜎2𝑒2 when 𝛽,𝜎,𝜇,𝛾 are equal to constants.

In the case 𝑒(0)𝑙2, we get 𝑒𝑙(𝑡)=1+𝑙2𝑙𝐿exp1+𝑙2𝜎2𝑡𝑙𝐿exp1+𝑙2𝜎2𝑡,1(3.7) where 𝐿=(𝑒(0)+𝑙1)/(𝑒(0)𝑙2), 𝑙1=(2𝛽/𝜎2+4𝛽2/𝜎4+4𝜇2/𝜎2𝛾2)/2, 𝑙2=((2𝛽/𝜎2)+4𝛽2/𝜎4+4𝜇2/𝜎2𝛾2)/2.

In the other case 𝑒(0)=𝑙2, the solution shows that 𝑒(𝑡)=𝑙2 for all 𝑡0.

Thus, for each 𝛼>0, 𝛽>0, 𝜎>0, 𝜇>0, 𝛾>0, the solution 𝑒(𝑡) obviously satisfies 𝑒(𝑡)𝑙2as𝑡+.(3.8)

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

Theorem 3.2. Assume that 𝛼,𝛽,𝜎,𝜇,𝛾𝐶(+), are all bounded, and there are constants 𝛼1, 𝛼2, 𝛽1, 𝛽2, 𝜎1, 𝜎2, 𝜇1, 𝜇2, 𝛾1, 𝛾2, and 𝑡0, such that(b1): 0<𝛼1|𝛼(𝑡)|𝛼2 for all 𝑡𝑡0;(b2): 0<𝛽1|𝛽(𝑡)|𝛽2 for all 𝑡𝑡0;(b3): 0<𝜎1|𝜎(𝑡)|𝜎2 for all 𝑡𝑡0;(b4): 0<𝜇2|𝜇(𝑡)|𝜇1 for all 𝑡𝑡0;(b5): 0<𝛾1|𝛾(𝑡)|𝛾2 for all 𝑡𝑡0;(b6): 2𝛼1(𝛽1+𝜎21𝑙22)>𝜎22𝑙21 where 𝑙2𝑖=(2𝛽𝑖/𝜎2𝑖+(4𝛽2𝑖)/(𝜎4𝑖)+(4𝜇2𝑖)/(𝜎2𝑖𝛾2𝑖))/2,𝑖=1,2.
Then, for arbitrary 𝜖>0 and 𝑇>0, we have 𝑃||𝜃0̂𝜃𝑡||1>𝜖𝜖2𝐶𝑇1,(3.9) where 𝐶 is a positive constant independent of 𝜖 and 𝑇.

Proof. Let 𝑒𝑖 be the solution to ̇𝑒𝑖=𝜇2𝑖/𝛾2𝑖2𝛽𝑖𝑒𝑖𝜎2𝑖𝑒2𝑖,𝑖=1,2, and 𝑒𝑖(𝑡0)=𝑒(𝑡0).
Since 𝜇22/𝛾222𝛽2𝑒𝜎22𝑒2̇𝑒=𝜇2/𝛾22𝛽𝑒𝜎2𝑒2𝜇21/𝛾212𝛽1𝑒𝜎21𝑒2 for all 𝑡𝑡0, by the comparison theorem [2, 36], we obtain that 𝑒2(𝑡)𝑒(𝑡)𝑒1(𝑡)𝑡𝑡0.(3.10) It follows from (3.7) that 𝑒 is bounded, and for any given 𝛿(0,1), there is a 𝑡1𝑡0 such that 0<𝑙22(1𝛿)𝑒(𝑟)𝑙21(1+𝛿)𝑟𝑡1.(3.11) For 𝑡𝑡1, we can obtain from (3.5) and 𝑓(0)=0 that 𝑓(𝑡)=𝑡0exp𝑡𝑠𝛽(𝑟)+𝜎2(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠=exp𝑡0𝛽(𝑟)+𝜎2(𝑟)𝑒(𝑟)𝑑𝑟𝑡10exp𝑠0𝛽(𝑟)+𝜎2(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠𝑡𝑡1exp𝑡𝑠𝛽(𝑟)+𝜎2(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠.(3.12) As 𝛽(𝑟)+𝜎2(𝑟)𝑒(𝑟)𝛽1+𝜎21𝑙22(1𝛿) holds for all 𝑡𝑡1, thus, the first term in (3.12) goes to 0 as 𝑡. For the second term in (3.12), we have ||||𝑡𝑡1exp𝑡𝑠𝛽(𝑟)+𝜎2||||(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠𝑡0𝛽exp1+𝜎21𝑙22𝑙(1𝛿)(𝑡𝑠)21=𝑙(1+𝛿)𝑑𝑠21(1+𝛿)𝛽1+𝜎21𝑙22(1𝛿)𝑡0𝛽exp1+𝜎21𝑙22𝑑𝛽(1𝛿)(𝑡𝑠)1+𝜎21𝑙22𝑠=𝑙(1𝛿)21(1+𝛿)𝛽1+𝜎21𝑙22𝛽(1𝛿)1exp1+𝜎21𝑙22𝑡𝑙(1𝛿)21(1+𝛿)𝛽1+𝜎21𝑙22.(1𝛿)(3.13) By similar arguments, we obtain that ||||𝑡𝑡1exp𝑡𝑠𝛽(𝑟)+𝜎2||||𝑙(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠22(1𝛿)𝛽2+𝜎22𝑙21.(1+𝛿)(3.14) Therefore, for any 𝜉>0, there exists 𝑡(𝜉)>0 such that 𝑙22(1𝛿)𝛽2+𝜎22𝑙21||||𝑙(1+𝛿)𝑓(𝑡)21(1+𝛿)𝛽1+𝜎21𝑙22(1𝛿)𝑡𝑡(𝜉).(3.15) For all 𝑡𝑡(𝜉), we can get from (3.5) that ̇𝑔=2|𝛼|𝜎2||𝑓||||𝑓||2𝛼1𝜎22𝑙21(1+𝛿)𝛽1+𝜎21𝑙22𝑙(1𝛿)22(1𝛿)𝛽2+𝜎22𝑙21=(1+𝛿)2𝛼1𝛽1+𝜎21𝑙22𝜎22𝑙21(1+𝛿)𝛽1+𝜎21𝑙22(𝑙1𝛿)22(1𝛿)𝛽2+𝜎22𝑙21(.1+𝛿)(3.16) By assumption (b6), we get ̇𝑔>0 for a sufficiently small 𝜉>0. This implies that 𝑔(𝑡) goes to infinity at least as a linear function. Thus, there exists a constant 𝐶>0, such that 𝔼𝜃0̂𝜃𝑡2=𝑆𝜃𝜃=𝑒𝐶𝑡1.(3.17) Hence, for arbitrary 𝜖>0 and all 𝑇>0, it follows from Chebyshev’s inequality that 𝑃||𝜃0̂𝜃𝑡||1>𝜖𝜖2𝐶𝑇1.(3.18)
The proof is complete.

Remark 3.3. From the proof of Theorem 3.2, we can see that 𝜃0̂𝜃𝑡 goes to 0 in 𝐿2-sense under the given conditions. In other words, ̂𝜃𝑡 is asymptotically unbiased.

Remark 3.4. It is well known that Kalman-Bucy linear filtering theory remains valid if one replaces the Brownian motion (𝑊𝑡,𝑉𝑡) 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 𝜃 is, a genuine Gaussian random variable, then we can have a clear statistical interpretation for the convergence rate. Firstly, we pick 𝜃0 at random; secondly, let system (2.2) run up to time 𝑡 and simultaneously observe 𝑌 by system (2.3); finally, compute ̂𝜃𝑡 as the following form.

The Kalman-Bucy linear filtering theory shows us 𝑑𝑋𝑡𝑑𝜃𝑡=𝐶𝐴(𝑡)𝑇(𝑡)𝐶(𝑡)𝐷2𝑋(𝑡)𝑆(𝑡)𝑡𝜃𝑡𝑑𝑡+𝐶(𝑡)𝐷2(𝑡)𝑆(𝑡)𝑑𝑌𝑡=𝜇𝛽(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥𝜇(𝑡)𝛼(𝑡)2(𝑡)𝛾2(𝑆𝑡)𝜃𝑥𝑋(𝑡)0𝑡𝜃𝑡𝜇𝑑𝑡+2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥𝑆(𝑡)𝜃𝑥(𝑡)𝑑𝑌𝑡(4.1) with initial conditions 𝑋0=𝔼[𝑋0] and ̂𝜃0=𝔼[𝜃0]. If we denote that Φ(𝑡) is the matrix fundamental solution of the deterministic linear system ̇𝑥𝑡̇𝑦𝑡=𝜇𝛽(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥𝜇(𝑡)𝛼(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝜃𝑥(𝑡)0𝑥(𝑡)𝑦(𝑡),(4.2) then the solution to (4.1) is given by 𝑋𝑡̂𝜃𝑡=Φ(𝑡)Φ1𝔼𝑋(0)0𝔼𝜃0+𝑡0Φ(𝑡)Φ1𝑆(𝑠)𝑥𝑥𝑆(𝑡)𝜃𝑥(𝑡)𝑑𝑌𝑠.(4.3) And for every particular experiment 𝜔, the quantity (𝜃0̂𝜃(𝜔)𝑡(𝜔))2 would be the squared estimation error.

But in this paper 𝜃 is a fixed parameter, so we can only choose 𝜃0(𝜔)=𝜃, and then the statistical mean over different values of 𝜃0(𝜔) has no experimental meaning. The true estimation error is given by ̂𝜃𝜃𝑡, not 𝜃0̂𝜃𝑡. It is therefore desirable that estimator ̂𝜃𝑡 converges to 𝜃0 for “all fixed values 𝜐=𝜃0" a.s. To establish such an assertion we work with a product space (𝑅×Ω,(𝑅),𝜂𝑃), where 𝜂 denotes the law of 𝜃0, and (Ω,,𝑃) is the underlying probability space for Brownian motion (𝑊𝑡,𝑉𝑡)𝑡0. This space is most appropriate because one can make 𝑃 a.s. statements for fixed 𝜐. Notice that in this representation we have 𝜃0(𝜐,𝜔)=𝜐 for all (𝜐,𝜔)×Ω. 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 𝑒,𝑓 is bonded and 𝑔 is monotonically increasing, moreover, 𝑆𝑥𝑥(𝑡)=𝑎=𝑔/=𝑔/(𝑒𝑔𝑓2)=(𝑔𝑓2/𝑒+𝑓2/𝑒)/(𝑒𝑔𝑓2)=1/𝑒+𝑓2/𝑒(𝑒𝑔𝑓2) and 𝑆𝜃𝑥(𝑡)=𝑏=𝑓/=𝑓/(𝑒𝑔𝑓2). Thus, there exist positive constants 𝑎1,𝑎2,𝑏1, and 𝑏2 such that 𝑎1𝑎𝑎2 and 𝑏1𝑏𝑏2.

Theorem 4.1. Assume that the following two conditions are satisfied:(c1): ̂𝜃𝑡 converges to 𝜃0 in 𝐿2(𝜂𝑃);(c2): 𝛽2𝜇22/𝛾22<0;(c3): (𝛽2(𝜇22/𝛾22)𝑎2)24𝛼2(𝜇22/𝛾22)𝑏2<0.
Then, for all fixed 𝜐, we have ̂𝜃𝑡(𝜐,)𝜐,𝑃-𝑎.𝑠.,𝑎𝑠𝑡.(4.4)

Proof. We will show that (4.4) holds for all 𝜐𝑁𝑐, where 𝜂(𝑁)=0.
By Kalman-Bucy linear filtering theory, we know 𝑑𝑋𝑡𝑑𝜃𝑡=𝐶𝐴(𝑡)𝑇(𝑡)𝐶(𝑡)𝐷2𝑋(𝑡)𝑆(𝑡)𝑡𝜃𝑡𝑑𝑡+𝐶(𝑡)𝐷2(𝑡)𝑆(𝑡)𝑑𝑌𝑡=𝜇𝛽(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥𝜇(𝑡)𝛼(𝑡)2(𝑡)𝛾2(𝑆𝑡)𝜃𝑥𝑋(𝑡)0𝑡𝜃𝑡𝜇𝑑𝑡+2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥𝑆(𝑡)𝜃𝑥(𝑡)𝑑𝑌𝑡(4.5) with initial conditions 𝑋0=𝔼[𝑋0̂𝜃]and0=𝔼[𝜃0]=𝔼[𝜐]=𝜐.
Since the following linear equations: ̇𝑥𝑡̇𝑦𝑡=𝜇𝛽(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥𝜇(𝑡)𝛼(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝜃𝑥(𝑡)0𝑥(𝑡)𝑦(𝑡)(4.6) equal to ̇𝑥𝑡=𝜇𝛽(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥(𝑡)𝑥(𝑡)+𝛼(𝑡)𝑌(𝑡),̇𝑦𝑡𝜇=2(𝑡)𝛾2𝑆(𝑡)𝜃𝑥(𝑡)𝑥(𝑡),(4.7) it follows from (c1)–(c3) that 𝛽1𝜇21𝛾21𝑎1𝜇𝛽(𝑡)2(𝑡)𝛾2𝑆(𝑡)𝑥𝑥(𝑡)𝛽2𝜇22𝛾22𝑎2𝛼<0,1𝛼(𝑡)𝛼2,𝜇21𝛾21𝑏1𝜇2(𝑡)𝛾2𝑆(𝑡)𝜃𝑥𝜇(𝑡)22𝛾22𝑏2.(4.8) For linear equations: ̇𝑥𝑡̇𝑦𝑡=𝛽1𝜇21𝛾21𝑎1𝛼1𝜇21𝛾21𝑏10,𝑥(𝑡)𝑦(𝑡)̇𝑥𝑡̇𝑦𝑡=𝛽2𝜇22𝛾22𝑎2𝛼2𝜇22𝛾22𝑏20𝑥,(𝑡)𝑦(𝑡)(4.9) if we set Φ1(𝑡) and Φ2(𝑡) that are the matrix fundamental solution of (4.9), we can obtain from the comparison theorem that Φ1(𝑡)Φ(𝑡)Φ2(𝑡).(4.10)
It is not difficult to explore (4.9), and get Φ1𝜆(𝑡)=1𝑁21𝑒𝜆1𝑡𝜆2𝑁21𝑒𝜆2𝑡𝑒𝜆1𝑡𝑒𝜆2𝑡,Φ2𝜆(𝑡)=1𝑀21𝑒𝜆1𝑡𝜆2𝑀21𝑒𝜆2𝑡𝑒𝜆1𝑡𝑒𝜆2𝑡,Φ11(𝑁𝑡)=21𝜆1𝜆2𝑒𝜆1𝑡𝜆2𝜆1𝜆2𝑒𝜆1𝑡𝑁21𝜆1𝜆2𝑒𝜆2𝑡𝜆1𝜆1𝜆2𝑒𝜆2𝑡,Φ21(𝑀𝑡)21𝜆1𝜆2𝑒𝜆1𝑡𝜆2𝜆1𝜆2𝑒𝜆1𝑡𝑀21𝜆1𝜆2𝑒𝜆2𝑡𝜆1𝜆1𝜆2𝑒𝜆2𝑡,(4.11) where𝑁11=𝛽1(𝜇21/𝛾21)𝑎1,𝑁12=𝛼1,𝑁21=(𝜇21/𝛾21)𝑏1,𝜆1=(𝑁11+𝑁2114𝑁12𝑁21)/2,𝜆2=(𝑁11𝑁2114𝑁12𝑁21)/2, 𝑀11=𝛽2(𝜇22/𝛾22)𝑎2,𝑀12=𝛼2,𝑀21=(𝜇22/𝛾22)𝑏2,𝜆1=(𝑀11+𝑀2114𝑀12𝑀21)/2,𝜆2=(𝑀11𝑀2114𝑀12𝑀21)/2.
By assumption (c2) and (c3), we know that 𝜆1<0,𝜆2<0,𝜆1<0, and 𝜆2<0.
By the ODE theory [37, 38] and above discussion, we know that the solution of (4.1) is given by 𝑋𝑡̂𝜃𝑡=Φ(𝑡)Φ1𝔼𝑋(0)0𝔼𝜃0+𝑡0Φ(𝑡)Φ1𝑆(𝑠)𝑥𝑥𝑆(𝑡)𝜃𝑥(𝑡)𝑑𝑌𝑠.(4.12) Using the similar method, we can also obtain the solutions for the following two equations: 𝑑𝑋𝑡𝑑̂𝜃𝑡=𝛽1𝜇21𝛾21𝑎1𝛼1𝜇21𝛾21𝑏10𝑋𝑡̂𝜃𝑡𝜇𝑑𝑡+1𝛾1𝑎1𝑏1𝑑𝑌𝑡𝑑𝑋,(4.13)𝑡𝑑̂𝜃𝑡=𝛽2𝜇22𝛾22𝑎2𝛼2𝜇22𝛾22𝑏20𝑋𝑡̂𝜃𝑡𝜇𝑑𝑡+2𝛾2𝑎2𝑏2𝑑𝑌𝑡,(4.14) where 𝑋0=𝔼[𝑋0] and ̂𝜃0=𝔼[𝜃0]=𝔼[𝜐]=𝜐.
The solutions of the two equations are explored as the following form: 𝑋𝑡̂𝜃𝑡=Φ1(𝑡)Φ11𝔼𝑋(0)0𝔼𝜃0+𝑡0Φ1(𝑡)Φ11𝑎(𝑠)1𝑏1𝑑𝑌𝑠,𝑋𝑡̂𝜃𝑡=Φ2(𝑡)Φ21(𝔼𝑋0)0𝔼𝜃0+𝑡0Φ2(𝑡)Φ21(𝑎𝑠)2𝑏2𝑑𝑌𝑠.(4.15)
For (4.14), we have that 𝑋𝑡̂𝜃𝑡=Φ2(𝑡)Φ21𝔼𝑋(0)0𝔼𝜃0+𝑡0Φ2(𝑡)Φ21𝑎(𝑠)2𝑏2𝑑𝑌𝑠(4.16) yields that ̂𝜃𝑡=𝑡0𝑎2𝑀21𝜆1𝜆2𝑒𝜆2(𝑡𝑠)𝑀21𝜆1𝜆2𝑒𝜆2(𝑡𝑠)+𝑏2𝜆1𝜆1𝜆2𝑒𝜆2(𝑡𝑠)𝜆2𝜆1𝜆2𝑒𝜆2(𝑡𝑠)𝑑𝑌𝑠+𝑀21𝜆1𝜆2𝑒𝜆2𝑡𝑀21𝜆1𝜆2𝑒𝜆2𝑡𝑋0+𝜆1𝜆1𝜆2𝑒𝜆2𝑡𝜆2𝜆1𝜆2𝑒𝜆2𝑡𝜃0.(4.17) Since 𝜆1<0 and 𝜆2<0, it is easy to get ̂𝜃𝑡(𝜐,)𝜐,𝑃-a.s.,as𝑡.(4.18) For (4.13), we can also get ̂𝜃𝑡(𝜐,)𝜐,𝑃-a.s.,as𝑡.(4.19) Hence, for (4.1), we can get the following result: ̂𝜃𝑡(𝜐,)𝜐,𝑃-a.s.,as𝑡.(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 Chebyshev’s inequality on the result of Theorem 4.1.

Remark 4.3. The strong consistency in Deck [30] requires that ̂𝜃𝑡 is a martingale, while, in our result, ̂𝜃𝑡 can be not a martingale. Furthermore, when ̂𝜃𝑡 is a martingale, our result is more strong than Deck’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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