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

Abstract

The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameter 𝜃∶𝑑𝑋_𝑡=(𝜃𝛼(𝑡)+𝛽(𝑡)𝑋_𝑡)𝑑𝑡+𝜎(𝑡)𝑑𝑊_𝑡. 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 [1–8]. 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 [10–15] 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, 16–26]. 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 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, 31–35]. 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 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. ℝ=(−∞,+∞) and ℝ₊=[0,+∞). For a vector 𝑥=∈ℝ, |𝑥| is the Euclidean norm (or 𝐿² norm) with |𝑥|=√𝑥⋅𝑥. 𝑀^𝑇 and 𝑀⁻¹ 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,ℱ,𝐏) be a complete probability space with a natural filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous, and ℱ₀ 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(ℝ₊) denotes the class of all continuous time on 𝑡∈ℝ₊.

2. Problem Statement

Hull-White model is a continuous-time, real stochastic process as follows: 𝑋_𝑡=𝑋₀+^𝑡0𝛼(𝑠)+𝛽(𝑠)𝑋_𝑠𝑑𝑠+^𝑡0𝜎(𝑠)𝑑𝑊_𝑠(2.1) with initial value 𝑋₀ as a Gaussian random variable, where 𝛼,𝛽,𝜎 are deterministic continuous functions on time 𝑡, 𝑊_𝑡 is a Brownian motion independent of the initial value 𝑋₀. Obviously, Hull-White model (2.1) is a general continuous-time linear SDE for 𝑋_𝑡, and we assume that the coefficient 𝛼 contains an unknown parameter 𝜃∈𝑅 as follows: 𝑑𝑋_𝑡=𝜃𝛼(𝑡)+𝛽(𝑡)𝑋_𝑡𝑑𝑡+𝜎(𝑡)𝑑𝑊_𝑡𝑡≥0,(2.2) and we observe the process 𝑋_𝑡 by the following filtering observations: 𝑑𝑌_𝑡=𝜇(𝑡)𝑋_𝑡𝑑𝑡+𝛾(𝑡)𝑑𝑉_𝑡𝑡≥0,(2.3) where 𝜇,𝛾 are deterministic bounded continuous functions on time 𝑡, and 𝑉_𝑡 is a Brownian motion independent of 𝑊_𝑡.

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(𝑊_𝑡,𝑉_𝑡,𝑡≥0). Then, we can rewrite (2.2) as a two-component system for (𝑋_𝑡,𝜃_𝑡) as follows: ⎛⎜⎜⎝𝑑𝑋_𝑡𝑑𝜃_𝑡⎞⎟⎟⎠=⎛⎜⎜⎝𝛽0(𝑡)𝛼0(𝑡)⎞⎟⎟⎠⎛⎜⎜⎝𝑋_𝑡𝜃_𝑡⎞⎟⎟⎠𝑑𝑡+⎛⎜⎜⎝𝜎0(𝑡)⎞⎟⎟⎠𝑑𝑊_𝑡𝑡≥0.(2.4) Similarly, filtering observations system (2.3) can be expressed as follows: 𝑑𝑌_𝑡=𝜇(𝑡)0⎛⎜⎜⎝𝑋^𝑡𝜃_𝑡⎞⎟⎟⎠𝑑𝑡+𝛾(𝑡)𝑑𝑉_𝑡𝑡≥0.(2.5) Therefore, we can use the Kalman-Bucy linear filtering theory to estimate 𝜃₀ as follows: 𝜃̂_𝑡=𝔼𝜃₀∣𝑌_𝑠,0≤𝑠≤𝑡,(2.6) and moreover, we also have 𝑋_𝑡=𝔼[𝑋_𝑡|𝑌_𝑠,0≤𝑠≤𝑡].

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: 𝑆̇(𝑡)=𝐴𝑆+𝑆𝐴^𝑇−𝑆𝐶^𝑇𝐷𝐷^𝑇⁻¹𝐶𝑆+𝐵𝐵^𝑇,(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: 𝑆(𝑡)=⎛⎜⎜⎝𝑆^𝑥𝑥𝑆(𝑡)𝑆_𝑥𝜃(𝑡)_𝜃𝑥(𝑡)𝑆_𝜃𝜃(𝑡)⎞⎟⎟⎠=⎛⎜⎜⎜⎝𝔼𝑋^𝑡−𝑋_𝑡²𝔼𝑋^𝑡𝔼𝑋_𝑡−𝑋_𝑡𝜃⁰−𝜃̂_𝑡−𝑋_𝑡𝜃⁰−𝜃̂_𝑡𝔼𝜃₀−𝜃̂_𝑡²⎞⎟⎟⎟⎠.(2.8) Set 𝑎=𝑆_𝑥𝑥,𝑏=𝑆_𝑥𝜃=𝑆_𝜃𝑥, and 𝑐=𝑆_𝜃𝜃. From Riccati equation (2.7), one can get the following system: 𝑎̇=2𝛽𝑎+2𝛼𝑏+𝜎²−𝜇²𝛾²𝑏̇=𝛽𝑏+𝛼𝑐−𝜇₂𝑎²,𝛾²𝑐̇=−𝜇₂𝑎𝑏,𝛾²𝑏².(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∈ℝ₊.

Remark 2.2. From the equation 𝑐̇=−(𝜇²/𝛾²)𝑏², we can see that the error variance 𝔼[(𝜃₀−𝜃̂_𝑡)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), 𝑆⁻¹(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(ℝ₊).
 Then, the error covariance matrix 𝑆(𝑡) satisfies det(S(t))>0 for all 𝑡≥0, and 𝑆_𝑥𝑥(𝑡)>0,𝑆_𝜃𝜃(𝑡)>0∀𝑡≥0.(3.1)

Proof. By Kalman-Bucy linear filtering theory, we know that det(𝑆(𝑡))>0 for all 𝑡≥0. Furthermore, it is not difficult to show that (3.1) holds for all 𝑡≥0.
Since det(𝑆(𝑡))>0, it follows that 𝑆⁻¹(𝑡) exists. Set 𝑅(𝑡)=𝑆⁻¹(𝑡)=⎛⎜⎜⎝𝑓𝑒((𝑡𝑡))𝑓𝑔((𝑡𝑡))⎞⎟⎟⎠.(3.2) As we know that 𝑅=1/𝑆 implies that 𝑅̇=−(1/𝑆²)𝑆̇, one can easily have that 𝑅̇=−𝑅𝑆̇𝑅.(3.3) It follows readily form (2.9) and (3.3) that 𝑅̇=−𝑅𝐴−𝐴^𝑇𝑅+𝐶^𝑇𝐷𝐷^𝑇⁻¹𝐶−𝑅𝐵𝐵^𝑇𝑅.(3.4) Using a similar computation as (2.9), we can get 𝑒̇=𝜇²𝛾²𝑓̇=−𝛼−𝑒2−𝛽𝛽𝑒𝑓−−𝜎𝜎²𝑒²₂,𝑔̇=−2𝛼𝑓−𝜎₂𝑒𝑓,𝑓².(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 ℝ₊. Furthermore, the first equation 𝑒̇=𝜇²/𝛾²−2𝛽𝑒−𝜎²𝑒² with initial condition 𝑒(0)>0 has a unique solution on a maximal time interval [0,𝑇), where 𝑇∈ℝ₊. 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 0≤𝑡<⁻𝑡. Thus, 𝑒̇(𝑡)=lim_Δ𝑡→0𝑒⁻𝑡−𝑒⁻𝑡−Δ𝑡Δ𝑡<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 ℝ₊ because the second equation is a linear equation for 𝑓 which can be solved analytically on ℝ₊, and 𝑔 can get by integration.
Define ℎ(𝑡)∶=det(𝑅(𝑡))=𝑒(𝑡)𝑔(𝑡)−𝑓²(𝑡). Since det(𝑆(𝑡))>0 for all 𝑡≥0, thus ℎ(𝑡)=det(𝑅(𝑡))=1/det(𝑆(𝑡))>0 for all 𝑡≥0, moreover, 𝑆_𝜃𝜃>0 for all 𝑡≥0. Finally, we assume that there exists 𝑡₀ such that, 𝑆_𝑥𝑥(𝑡₀)=0, then 𝑔(𝑡₀)=𝑆_𝑥𝑥(𝑡₀)ℎ(𝑡₀)=0, so that ℎ(𝑡₀)=𝑒(𝑡₀)𝑔(𝑡₀)−𝑓²(𝑡₀)≤0, and this contradicts ℎ(𝑡₀)>0. Hence, 𝑆_𝑥𝑥>0 for all 𝑡≥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 𝑒̇=𝜇²/𝛾²−2𝛽𝑒−𝜎²𝑒² when 𝛽,𝜎,𝜇,𝛾 are equal to constants.

In the case 𝑒(0)≠𝑙₂, we get 𝑒(𝑡)=𝑙¹+𝑙₂𝐿exp𝑙₁+𝑙₂𝜎²𝑡𝐿exp𝑙₁+𝑙₂𝜎²𝑡−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 𝑡≥0.

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,𝛾∈𝐶(ℝ₊), 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 𝑡≥𝑡₀;(b2): 0<𝛽₁≤|𝛽(𝑡)|≤𝛽₂ for all 𝑡≥𝑡₀;(b3): 0<𝜎₁≤|𝜎(𝑡)|≤𝜎₂ for all 𝑡≥𝑡₀;(b4): 0<𝜇₂≤|𝜇(𝑡)|≤𝜇₁ for all 𝑡≥𝑡₀;(b5): 0<𝛾₁≤|𝛾(𝑡)|≤𝛾₂ for all 𝑡≥𝑡₀;(b6): 2𝛼₁(𝛽₁+𝜎²¹𝑙₂₂)>𝜎²²𝑙₂₁ where 𝑙_2𝑖=(−2𝛽_𝑖/𝜎^2𝑖+(4𝛽^2𝑖)/(𝜎^4𝑖)+(4𝜇^2𝑖)/(𝜎^2𝑖𝛾^2𝑖))/2,𝑖=1,2.
Then, for arbitrary 𝜖>0 and 𝑇>0, we have 𝑃||𝜃₀−𝜃̂_𝑡||>𝜖≤1𝜖²𝐶𝑇⁻¹,(3.9) where 𝐶 is a positive constant independent of 𝜖 and 𝑇.

Proof. Let 𝑒_𝑖 be the solution to 𝑒_𝑖̇=𝜇^2𝑖/𝛾^2𝑖−2𝛽_𝑖𝑒_𝑖−𝜎^2𝑖𝑒^2𝑖,𝑖=1,2, and 𝑒_𝑖(𝑡₀)=𝑒(𝑡₀).
Since 𝜇²²/𝛾²²−2𝛽₂𝑒−𝜎²²𝑒²≤𝑒̇=𝜇²/𝛾²−2𝛽𝑒−𝜎²𝑒²≤𝜇²¹/𝛾²¹−2𝛽₁𝑒−𝜎²¹𝑒² for all 𝑡≥𝑡₀, by the comparison theorem [2, 36], we obtain that 𝑒₂(𝑡)≤𝑒(𝑡)≤𝑒₁(𝑡)∀𝑡≥𝑡₀.(3.10) It follows from (3.7) that 𝑒 is bounded, and for any given 𝛿∈(0,1), there is a 𝑡₁≥𝑡₀ such that 0<𝑙₂₂(1−𝛿)≤𝑒(𝑟)≤𝑙₂₁(1+𝛿)∀𝑟≥𝑡₁.(3.11) For 𝑡≥𝑡₁, we can obtain from (3.5) and 𝑓(0)=0 that 𝑓(𝑡)==−−ex^𝑡0pexp−−_𝑡0^𝑡𝑠𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠−𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)𝑑𝑟^𝑡1₀exp^𝑠0𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠_𝑡𝑡1exp−^𝑡𝑠𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠.(3.12) As 𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)≥𝛽₁+𝜎²¹𝑙₂₂(1−𝛿) holds for all 𝑡≥𝑡₁, thus, the first term in (3.12) goes to 0 as 𝑡→∞. For the second term in (3.12), we have ||||^𝑡𝑡₁exp≤−^𝑡𝑠𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠_𝑡0||||=e𝑙xp−𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)(𝑡−𝑠)𝑙₂₁(1+𝛿)𝑑𝑠₂₁(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)^𝑡0=𝑙exp−𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)(𝑡−𝑠)𝑑𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)₂₁𝑠(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)1−exp−𝛽¹≤𝑙+𝜎²¹𝑙₂₂(1−𝛿)𝑡₂₁(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂(1−𝛿).(3.13) By similar arguments, we obtain that ||||^𝑡𝑡₁exp−^𝑡𝑠𝛽(𝑟)+𝜎²(𝑟)𝑒(𝑟)𝑑𝑟𝛼(𝑠)𝑒(𝑠)𝑑𝑠||||≥𝑙²²(1−𝛿)𝛽₂+𝜎²²𝑙₂₁(1+𝛿).(3.14) Therefore, for any 𝜉>0, there exists 𝑡(𝜉)>0 such that 𝑙₂₂(1−𝛿)𝛽₂+𝜎²²𝑙₂₁(1+𝛿)≤||𝑓(𝑡)||≤𝑙²¹(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)∀𝑡≥𝑡(𝜉).(3.15) For all 𝑡≥𝑡(𝜉), we can get from (3.5) that 𝑔̇=≥22|𝛼𝛼|₁−𝜎²||𝑓||||𝑓||−𝜎²²𝑙₂₁(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)𝑙²²(1−𝛿)=2𝛼₁𝛽₂+𝜎²²𝑙₂₁(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂−𝜎²²𝑙₂₁(1+𝛿)𝛽₁+𝜎²¹𝑙₂₂(1−𝛿)𝑙²²(1−𝛿)𝛽₂+𝜎²²𝑙₂₁(1+𝛿).(3.16) By assumption (b6), we get 𝑔̇>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 𝔼𝜃₀−𝜃̂_𝑡²=𝑆_𝜃𝜃=𝑒ℎ≤𝐶𝑡⁻¹.(3.17) Hence, for arbitrary 𝜖>0 and all 𝑇>0, it follows from Chebyshev’s inequality that 𝑃||𝜃₀−𝜃̂_𝑡||>𝜖≤1𝜖²𝐶𝑇⁻¹.(3.18)
The proof is complete.

Remark 3.3. From the proof of Theorem 3.2, we can see that 𝜃₀−𝜃̂_𝑡 goes to 0 in 𝐿²-sense under the given conditions. In other words, 𝜃̂_𝑡 is asymptotically unbiased.

Remark 3.4. It is well known that Kalman-Bucy linear filtering theory remains valid if one replaces the Brownian motion (𝑊_𝑡,𝑉_𝑡) 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 𝜃̂_𝑡 as the following form.

The Kalman-Bucy linear filtering theory shows us ⎛⎜⎜⎝𝑑𝑋_𝑡𝑑𝜃_𝑡⎞⎟⎟⎠=𝐴(𝑡)−𝐶^𝑇(𝑡)𝐶(𝑡)𝐷²(𝑡)𝑆(𝑡)⎛⎜⎜⎝𝑋𝜃^𝑡_𝑡⎞⎟⎟⎠𝑑𝑡+𝐶(𝑡)𝐷²(𝑡)𝑆(𝑡)𝑑𝑌^𝑡=⎛⎜⎜⎜⎝𝛽(𝑡)−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝑥𝑥−𝜇(𝑡)𝛼(𝑡)₂(𝑡)𝛾²(𝑡)𝑆^𝜃𝑥(𝑡)0⎞⎟⎟⎟⎠⎛⎜⎜⎝𝑋𝜃^𝑡_𝑡⎞⎟⎟⎠𝑑𝑡+𝜇²(𝑡)𝛾²(𝑡)⎛⎜⎜⎝𝑆^𝑥𝑥𝑆(𝑡)_𝜃𝑥(𝑡)⎞⎟⎟⎠𝑑𝑌^𝑡(4.1) with initial conditions 𝑋₀=𝔼[𝑋₀] and 𝜃̂₀=𝔼[𝜃₀]. If we denote that Φ(𝑡) is the matrix fundamental solution of the deterministic linear system ⎛⎜⎜⎝𝑥̇_𝑡𝑦̇_𝑡⎞⎟⎟⎠=⎛⎜⎜⎜⎝𝛽(𝑡)−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝑥𝑥−𝜇(𝑡)𝛼(𝑡)₂(𝑡)𝛾²(𝑡)𝑆^𝜃𝑥(𝑡)0⎞⎟⎟⎟⎠⎛⎜⎜⎝𝑥𝑦((𝑡𝑡))⎞⎟⎟⎠,(4.2) then the solution to (4.1) is given by ⎛⎜⎜⎝𝑋_𝑡𝜃̂_𝑡⎞⎟⎟⎠=Φ(𝑡)Φ₋₁(0)⎛⎜⎜⎝𝔼𝑋⁰𝔼𝜃₀⎞⎟⎟⎠+^𝑡0Φ(𝑡)Φ⁻¹(𝑠)⎛⎜⎜⎝𝑆^𝑥𝑥𝑆(𝑡)_𝜃𝑥(𝑡)⎞⎟⎟⎠𝑑𝑌^𝑠.(4.3) And for every particular experiment 𝜔, the quantity (𝜃₀(𝜔)−𝜃̂_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 𝜃−𝜃̂_𝑡, not 𝜃₀−𝜃̂_𝑡. It is therefore desirable that estimator 𝜃̂_𝑡 converges to 𝜃₀ for “all fixed values 𝜐=𝜃₀" a.s. To establish such an assertion we work with a product space (𝑅×Ω,ℬ(𝑅)⊗ℱ,𝜂⊗𝑃), where 𝜂 denotes the law of 𝜃₀, and (Ω,ℱ,𝑃) is the underlying probability space for Brownian motion (𝑊_𝑡,𝑉_𝑡)_𝑡≥0. This space is most appropriate because one can make 𝑃 a.s. statements for fixed 𝜐∈ℝ. Notice that in this representation we have 𝜃₀(𝜐,𝜔)=𝜐 for all (𝜐,𝜔)∈ℝ×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, 𝑆_𝑥𝑥(𝑡)=𝑎=𝑔/ℎ=𝑔/(𝑒𝑔−𝑓²)=(𝑔−𝑓²/𝑒+𝑓²/𝑒)/(𝑒𝑔−𝑓²)=1/𝑒+𝑓²/𝑒(𝑒𝑔−𝑓²) and 𝑆_𝜃𝑥(𝑡)=𝑏=𝑓/ℎ=𝑓/(𝑒𝑔−𝑓²). Thus, there exist positive constants 𝑎₁,𝑎₂,𝑏₁, and 𝑏₂ such that 𝑎₁≤𝑎≤𝑎₂ and 𝑏₁≤𝑏≤𝑏₂.

Theorem 4.1. Assume that the following two conditions are satisfied:(c1): 𝜃̂_𝑡 converges to 𝜃₀ in 𝐿²(𝜂⊗𝑃);(c2): 𝛽₂−𝜇²²/𝛾²²<0;(c3): (𝛽₂−(𝜇²²/𝛾²²)𝑎₂)²−4𝛼₂(𝜇²²/𝛾²²)𝑏₂<0.
Then, for all fixed 𝜐∈ℝ, we have 𝜃̂_𝑡(𝜐,⋅)⟶𝜐,𝑃-𝑎.𝑠.,𝑎𝑠𝑡⟶∞.(4.4)

Proof. We will show that (4.4) holds for all 𝜐∈𝑁^𝑐, where 𝜂(𝑁)=0.
By Kalman-Bucy linear filtering theory, we know ⎛⎜⎜⎝𝑑𝑋_𝑡𝑑𝜃_𝑡⎞⎟⎟⎠=𝐴(𝑡)−𝐶^𝑇(𝑡)𝐶(𝑡)𝐷²(𝑡)𝑆(𝑡)⎛⎜⎜⎝𝑋𝜃^𝑡_𝑡⎞⎟⎟⎠𝑑𝑡+𝐶(𝑡)𝐷²(𝑡)𝑆(𝑡)𝑑𝑌^𝑡=⎛⎜⎜⎜⎝𝛽(𝑡)−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝑥𝑥−𝜇(𝑡)𝛼(𝑡)₂(𝑡)𝛾²(𝑡)𝑆^𝜃𝑥(𝑡)0⎞⎟⎟⎟⎠⎛⎜⎜⎝𝑋𝜃^𝑡_𝑡⎞⎟⎟⎠𝑑𝑡+𝜇²(𝑡)𝛾²(𝑡)⎛⎜⎜⎝𝑆^𝑥𝑥𝑆(𝑡)_𝜃𝑥(𝑡)⎞⎟⎟⎠𝑑𝑌^𝑡(4.5) with initial conditions 𝑋₀=𝔼[𝑋₀]and𝜃̂₀=𝔼[𝜃₀]=𝔼[𝜐]=𝜐.
Since the following linear equations: ⎛⎜⎜⎝𝑥̇_𝑡𝑦̇_𝑡⎞⎟⎟⎠=⎛⎜⎜⎜⎝𝛽(𝑡)−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝑥𝑥−𝜇(𝑡)𝛼(𝑡)₂(𝑡)𝛾²(𝑡)𝑆^𝜃𝑥(𝑡)0⎞⎟⎟⎟⎠⎛⎜⎜⎝𝑥𝑦((𝑡𝑡))⎞⎟⎟⎠(4.6) equal to 𝑥̇_𝑡=𝛽(𝑡)−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝑥𝑥𝑦̇_𝑡(𝑡)𝑥(𝑡)+𝛼(𝑡)𝑌(𝑡),=−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝜃𝑥(𝑡)𝑥(𝑡),(4.7) it follows from (c1)–(c3) that 𝛽₁−𝜇²¹𝛾²¹𝑎₁≤𝛽(𝑡)−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝑥𝑥(𝑡)≤𝛽₂−𝜇²²𝛾²²𝛼𝑎₂<0,₁−𝜇₂₁≤𝛼(𝑡)≤𝛼₂,𝛾²¹𝑏₁≤−𝜇²(𝑡)𝛾²(𝑡)𝑆^𝜃𝑥(𝑡)≤−𝜇²²𝛾²²𝑏₂.(4.8) For linear equations: ⎛⎜⎜⎝𝑥̇_𝑡𝑦̇_𝑡⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝛽¹−𝜇𝛾²¹²¹𝑎₁𝛼₁−𝜇²¹𝛾²¹⎛⎜⎜⎝𝑥̇_𝑡𝑏₁0⎞⎟⎟⎟⎟⎠⎛⎜⎜⎝𝑥𝑦((𝑡𝑡))⎞⎟⎟⎠,𝑦̇_𝑡⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝛽²−𝜇𝛾²²²²𝑎₂𝛼₂−𝜇²²𝛾²²𝑏₂0⎞⎟⎟⎟⎟⎠⎛⎜⎜⎝𝑥𝑦((𝑡𝑡))⎞⎟⎟⎠,(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 Φ₁(𝑡)=⎛⎜⎜⎜⎜⎝−𝜆^1𝑁₂₁𝑒^𝜆′1𝑡−𝜆^2𝑁₂₁𝑒^𝜆′2𝑡𝑒^𝜆′1𝑡𝑒^𝜆′2𝑡⎞⎟⎟⎟⎟⎠,Φ₂(𝑡)=⎛⎜⎜⎜⎜⎝−𝜆¹𝑀₂₁𝑒^𝜆1𝑡−𝜆²𝑀₂₁𝑒^𝜆2𝑡𝑒^𝜆1𝑡𝑒^𝜆2𝑡Φ₁₋₁⎞⎟⎟⎟⎟⎠,(𝑡)=⎛⎜⎜⎜⎜⎜⎝−𝑁²¹𝜆^1−𝜆₂𝑒^{−𝜆′1𝑡}−𝜆^2𝜆^1−𝜆^2𝑒^{−𝜆′1𝑡}𝑁₂₁𝜆^1−𝜆^2𝑒^{−𝜆′2𝑡}𝜆^1𝜆^1−𝜆^2𝑒^{−𝜆′2𝑡}⎞⎟⎟⎟⎟⎟⎠,Φ₂₋₁(𝑡)⎛⎜⎜⎜⎜⎜⎝−𝑀²¹𝜆₁−𝜆₂𝑒^−𝜆1𝑡−𝜆²𝜆₁−𝜆₂𝑒^−𝜆1𝑡𝑀₂₁𝜆₁−𝜆₂𝑒^−𝜆2𝑡𝜆₁𝜆₁−𝜆₂𝑒^−𝜆2𝑡⎞⎟⎟⎟⎟⎟⎠,(4.11) where𝑁₁₁=𝛽₁−(𝜇²¹/𝛾²¹)𝑎₁,𝑁₁₂=𝛼₁,𝑁₂₁=(𝜇²¹/𝛾²¹)𝑏₁,𝜆^1=(𝑁₁₁+𝑁²¹¹−4𝑁₁₂𝑁₂₁)/2,𝜆^2=(𝑁₁₁−𝑁²¹¹−4𝑁₁₂𝑁₂₁)/2, 𝑀₁₁=𝛽₂−(𝜇²²/𝛾²²)𝑎₂,𝑀₁₂=𝛼₂,𝑀₂₁=(𝜇²²/𝛾²²)𝑏₂,𝜆₁=(𝑀₁₁+𝑀²¹¹−4𝑀₁₂𝑀₂₁)/2,𝜆₂=(𝑀₁₁−𝑀²¹¹−4𝑀₁₂𝑀₂₁)/2.
By assumption (c2) and (c3), we know that 𝜆^1<0,𝜆^2<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 ⎛⎜⎜⎝𝑋_𝑡𝜃̂_𝑡⎞⎟⎟⎠=Φ(𝑡)Φ₋₁(0)⎛⎜⎜⎝𝔼𝑋⁰𝔼𝜃₀⎞⎟⎟⎠+^𝑡0Φ(𝑡)Φ⁻¹(𝑠)⎛⎜⎜⎝𝑆^𝑥𝑥𝑆(𝑡)_𝜃𝑥(𝑡)⎞⎟⎟⎠𝑑𝑌^𝑠.(4.12) Using the similar method, we can also obtain the solutions for the following two equations: ⎛⎜⎜⎝𝑑𝑋_𝑡𝑑𝜃̂_𝑡⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝛽¹−𝜇𝛾²¹²¹𝑎₁𝛼₁−𝜇²¹𝛾²¹𝑏₁0⎞⎟⎟⎟⎟⎠⎛⎜⎜⎝𝑋^𝑡𝜃̂_𝑡⎞⎟⎟⎠𝑑𝑡+𝜇₁𝛾₁⎛⎜⎜⎝𝑎₁𝑏₁⎞⎟⎟⎠𝑑𝑌_𝑡⎛⎜⎜⎝𝑑𝑋,(4.13)_𝑡𝑑𝜃̂_𝑡⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝛽²−𝜇𝛾²²²²𝑎₂𝛼₂−𝜇²²𝛾²²𝑏₂0⎞⎟⎟⎟⎟⎠⎛⎜⎜⎝𝑋^𝑡𝜃̂_𝑡⎞⎟⎟⎠𝑑𝑡+𝜇₂𝛾₂⎛⎜⎜⎝𝑎₂𝑏₂⎞⎟⎟⎠𝑑𝑌_𝑡,(4.14) where 𝑋₀=𝔼[𝑋₀] and 𝜃̂₀=𝔼[𝜃₀]=𝔼[𝜐]=𝜐.
The solutions of the two equations are explored as the following form: ⎛⎜⎜⎝𝑋_𝑡𝜃̂_𝑡⎞⎟⎟⎠=Φ₁(𝑡)Φ¹⁻¹(0)⎛⎜⎜⎝𝔼𝑋⁰𝔼𝜃₀⎞⎟⎟⎠+^𝑡0Φ₁(𝑡)Φ¹⁻¹(𝑠)⎛⎜⎜⎝𝑎¹𝑏₁⎞⎟⎟⎠𝑑𝑌_𝑠⎛⎜⎜⎝𝑋_𝑡,𝜃̂_𝑡⎞⎟⎟⎠=Φ₂(𝑡)Φ²⁻¹(0)⎛⎜⎜⎝𝔼𝑋⁰𝔼𝜃₀⎞⎟⎟⎠+^𝑡0Φ₂(𝑡)Φ²⁻¹(𝑠)⎛⎜⎜⎝𝑎²𝑏₂⎞⎟⎟⎠𝑑𝑌_𝑠.(4.15)
For (4.14), we have that ⎛⎜⎜⎝𝑋_𝑡𝜃̂_𝑡⎞⎟⎟⎠=Φ₂(𝑡)Φ²⁻¹(0)⎛⎜⎜⎝𝔼𝑋⁰𝔼𝜃₀⎞⎟⎟⎠+^𝑡0Φ₂(𝑡)Φ²⁻¹(𝑠)⎛⎜⎜⎝𝑎²𝑏₂⎞⎟⎟⎠𝑑𝑌_𝑠(4.16) yields that 𝜃̂_𝑡=^𝑡0𝑎²𝜆₁𝑀−²¹𝜆₂𝑒^{−𝜆2(𝑡−𝑠)}−𝑀²¹𝜆₁−𝜆₂𝑒^{−𝜆2(𝑡−𝑠)}+𝑏²𝜆₁𝜆−¹𝜆₂𝑒^{−𝜆2(𝑡−𝑠)}−𝜆²𝜆₁−𝜆₂𝑒^{−𝜆2(𝑡−𝑠)}𝑑𝑌^𝑠+𝑀²¹𝜆₁−𝜆₂𝑒^−𝜆2𝑡−𝑀²¹𝜆₁−𝜆₂𝑒^−𝜆2𝑡𝑋⁰+𝜆¹𝜆₁−𝜆₂𝑒^−𝜆2𝑡−𝜆²𝜆₁−𝜆₂𝑒^−𝜆2𝑡𝜃⁰.(4.17) Since 𝜆₁<0 and 𝜆₂<0, it is easy to get 𝜃̂_𝑡(𝜐,⋅)⟶𝜐,𝑃-a.s.,as𝑡⟶∞.(4.18) For (4.13), we can also get 𝜃̂_𝑡(𝜐,⋅)⟶𝜐,𝑃-a.s.,as𝑡⟶∞.(4.19) Hence, for (4.1), we can get the following result: 𝜃̂_𝑡(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 𝜃̂_𝑡 is a martingale, while, in our result, 𝜃̂_𝑡 can be not a martingale. Furthermore, when 𝜃̂_𝑡 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.