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

Authors: Xiu Kan; Huisheng Shu; Yan Che
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameterθ:dXt=(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.


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## Body

## 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. MT 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 {ℱt}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:(2.1)Xt=X0+∫0t(α(s)+β(s)Xs)ds+∫0tσ(s)dWs
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 θ∈R as follows:
(2.2)dXt=(θα(t)+β(t)Xt)dt+σ(t)dWtt≥0,
and we observe the process Xt by the following filtering observations:
(2.3)dYt=μ(t)Xtdt+γ(t)dVtt≥0,
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 estimateθ 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,t≥0). Then, we can rewrite (2.2) as a two-component system for (Xt,θt) as follows:
(2.4)(dXtdθt)=(β(t)α(t)00)(Xtθt)dt+(σ(t)0)dWtt≥0.
Similarly, filtering observations system (2.3) can be expressed as follows:
(2.5)dYt=(μ(t)0)(Xtθt)dt+γ(t)dVtt≥0.
Therefore, we can use the Kalman-Bucy linear filtering theory to estimate θ0 as follows:
(2.6)θ^t=E[θ0∣Ys,0≤s≤t],
and moreover, we also have X^t=𝔼[Xt|Ys,0≤s≤t].For given Gaussian initial conditionsX0 and θ0, it is well known from Kalman-Bucy linear filtering theory that error covariance matrix S(t) satisfies the following Riccati equation:
(2.7)S˙(t)=AS+SAT-SCT(DDT)-1CS+BBT,
where A=(β(t)α(t)00),B=(σ(t)0),C=(μ(t)0),D=γ(t), and as we all know the error covariance matrix S(t) is defined as follows:
(2.8)S(t)=(Sxx(t)Sxθ(t)Sθx(t)Sθθ(t))=(E[(Xt-X^t)2]E[(Xt-X^t)(θ0-θ^t)]E[(Xt-X^t)(θ0-θ^t)]E[(θ0-θ^t)2]).
Set  a=Sxx,b=Sxθ=Sθx, and c=Sθθ. From Riccati equation (2.7), one can get the following system:
(2.9)a˙=2βa+2αb+σ2-μ2γ2a2,b˙=βb+αc-μ2γ2ab,c˙=-μ2γ2b2.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 equationc˙=-(μ2/γ2)b2, we can see that the error variance 𝔼[(θ0-θ^t)2] is monotonically decreasing.


## 3. Asymptotic Convergence Analysis

Assume that the initial conditionsX0 and θ0 are independent and have nonvariances, so that b(0)=0 and a(0)=𝔼[X02]>0,c(0)=𝔼[θ02]>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 conditionsX0 and θ0 for system (2.2) are independent and have nonvanishing variances.
(a2) Letα(t),β(t),σ(t),μ(t),γ(t)∈C(ℝ+).
Then, the error covariance matrixS(t) satisfies det(S(t))>0 for all t≥0, and
(3.1)Sxx(t)>0,Sθθ(t)>0∀t≥0.Proof.
By Kalman-Bucy linear filtering theory, we know thatdet(S(t))>0 for all t≥0. Furthermore, it is not difficult to show that (3.1) holds for all t≥0.
Sincedet(S(t))>0, it follows that S-1(t)  exists. Set
(3.2)R(t)=S-1(t)=(e(t)f(t)f(t)g(t)).
As we know that R=1/S implies that R˙=-(1/S2)S˙, one can easily have that
(3.3)R˙=-RS˙R.
It follows readily form (2.9) and (3.3) that
(3.4)R˙=-RA-ATR+CT(DDT)-1C-RBBTR.
Using a similar computation as (2.9), we can get
(3.5)e˙=μ2γ2-2βe-σ2e2,f˙=-αe-βf-σ2ef,g˙=-2αf-σ2f2.
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 e˙=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,
(3.6)e˙(t)=lim⁡Δt→0e(t-)-e(t--Δt)Δt<0,
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 ase˙(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.
Defineh(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, Sθθ>0 for all t≥0. Finally, we assume that there exists t0 such that, Sxx(t0)=0, then g(t0)=Sxx(t0)h(t0)=0, so that h(t0)=e(t0)g(t0)-f2(t0)≤0,  and this contradicts h(t0)>0. Hence, Sxx>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 e˙=μ2/γ2-2βe-σ2e2 when β,σ,μ,γ are equal to constants.In the casee(0)≠l2, we get
(3.7)e(t)=l1+l2Lexp⁡[(l1+l2)σ2t]Lexp⁡[(l1+l2)σ2t]-1,
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 casee(0)=l2, the solution shows that e(t)=l2 for all t≥0.Thus, for eachα>0, β>0, σ>0, μ>0, γ>0, the solution e(t) obviously satisfies
(3.8)e(t)⟶l2ast⟶+∞.The convergence rate of the estimator is given by following theorem.Theorem 3.2.
Assume thatα,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+σ12l22)>σ22l21 where l2i=(-2βi/σi2+(4βi2)/(σi4)+(4μi2)/(σi2γi2))/2,i=1,2.
Then, for arbitraryϵ>0 and T>0, we have
(3.9)P(|θ0-θ^t|>ϵ)≤1ϵ2CT-1,
where C is a positive constant independent of ϵ and T.Proof.
Letei be the solution to e˙i=μi2/γi2-2βiei-σi2ei2,i=1,2, and ei(t0)=e(t0).
Sinceμ22/γ22-2β2e-σ22e2≤e˙=μ2/γ2-2βe-σ2e2≤μ12/γ12-2β1e-σ12e2  for all t≥t0, by the comparison theorem [2, 36], we obtain that
(3.10)e2(t)≤e(t)≤e1(t)∀t≥t0.
It follows from (3.7) that e is bounded, and for any given δ∈(0,1), there is a t1≥t0 such that
(3.11)0<l22(1-δ)≤e(r)≤l21(1+δ)∀r≥t1.
For t≥t1, we can obtain from (3.5) and f(0)=0 that
(3.12)f(t)=-∫0texp⁡[-∫st‍(β(r)+σ2(r)e(r))dr]α(s)e(s)ds=-exp⁡[-∫0t‍(β(r)+σ2(r)e(r))dr]∫0t1‍exp⁡[∫0s‍(β(r)+σ2(r)e(r))dr]α(s)e(s)ds-∫t1t‍exp⁡[-∫st‍(β(r)+σ2(r)e(r))dr]α(s)e(s)ds.
As β(r)+σ2(r)e(r)≥β1+σ12l22(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
(3.13)|∫t1texp⁡⁡[-∫st(β(r)+σ2(r)e(r))dr]α(s)e(s)ds|(1-δ)≤∫0texp⁡⁡[-(β1+σ12l22(1-δ))(t-s)]l21(1+δ)ds(1-δ)=l21(1+δ)β1+σ12l22(1-δ)∫0texp⁡⁡[-(β1+σ12l22(1-δ))(t-s)]d(β1+σ12l22(1-δ))s(1-δ)=l21(1+δ)β1+σ12l22(1-δ)(1-exp⁡⁡[-(β1+σ12l22(1-δ))t])(1-δ)≤l21(1+δ)β1+σ12l22(1-δ).
By similar arguments, we obtain that
(3.14)|∫t1t‍exp⁡[-∫st(β(r)+σ2(r)e(r))dr]α(s)e(s)ds|≥l22(1-δ)β2+σ22l21(1+δ).
Therefore, for any ξ>0, there exists t(ξ)>0 such that
(3.15)l22(1-δ)β2+σ22l21(1+δ)≤|f(t)|≤l21(1+δ)β1+σ12l22(1-δ)∀t≥t(ξ).
For all t≥t(ξ), we can get from (3.5) that
(3.16)g˙=(2|α|-σ2|f|)|f|≥(2α1-σ22l21(1+δ)β1+σ12l22(1-δ))l22(1-δ)β2+σ22l21(1+δ)=(2α1(β1+σ12l22)-σ22(l21(1+δ))β1+σ12l22(1-δ))l22(1-δ)β2+σ22l21(1+δ).
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
(3.17)E(θ0-θ^t)2=Sθθ=eh≤Ct-1.
Hence, for arbitrary ϵ>0 and all T>0, it follows from Chebyshev’s inequality that
(3.18)P(|θ0-θ^t|>ϵ)≤1ϵ2CT-1.
The proof is complete.Remark 3.3.
From the proof of Theorem3.2, we can see that θ0-θ^t goes to 0 in L2-sense under the given conditions. In other words, θ^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 (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 parameterθ 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 θ^t as the following form.The Kalman-Bucy linear filtering theory shows us(4.1)(dXtdθt)=(A(t)-CT(t)C(t)D2(t)S(t))(Xtθt)dt+C(t)D2(t)S(t)dYt=(β(t)-μ2(t)γ2(t)Sxx(t)α(t)-μ2(t)γ2(t)Sθx(t)0)(Xtθt)dt+μ2(t)γ2(t)(Sxx(t)Sθx(t))dYt
with initial conditions X^0=𝔼[X0] and θ^0=𝔼[θ0]. If we denote that Φ(t) is the matrix fundamental solution of the deterministic linear system
(4.2)(x˙ty˙t)=(β(t)-μ2(t)γ2(t)Sxx(t)α(t)-μ2(t)γ2(t)Sθx(t)0)(x(t)y(t)),
then the solution to (4.1) is given by
(4.3)(X^tθ^t)=Φ(t)Φ-1(0)(E[X0]E[θ0])+∫0t‍Φ(t)Φ-1(s)(Sxx(t)Sθx(t))dYs.
And for every particular experiment ω, the quantity (θ0(ω)-θ^t(ω))2 would be the squared estimation error.But in this paperθ 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-θ^t, not θ0-θ^t. It is therefore desirable that estimator θ^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 (Wt,Vt)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 Theorem3.2, we know that e,f is bonded and g is monotonically increasing, moreover, Sxx(t)=a=g/h=g/(eg-f2)=(g-f2/e+f2/e)/(eg-f2)=1/e+f2/e(eg-f2) and Sθx(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)
:θ^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 fixedυ∈ℝ, we have
(4.4)θ^t(υ,⋅)⟶υ,P-a.s.,ast⟶∞.Proof.
We will show that (4.4) holds for all υ∈Nc, where η(N)=0.
By Kalman-Bucy linear filtering theory, we know(4.5)(dXtdθt)=(A(t)-CT(t)C(t)D2(t)S(t))(Xtθt)dt+C(t)D2(t)S(t)dYt=(β(t)-μ2(t)γ2(t)Sxx(t)α(t)-μ2(t)γ2(t)Sθx(t)0)(Xtθt)dt+μ2(t)γ2(t)(Sxx(t)Sθx(t))dYt
with initial conditions X^0=𝔼[X0]andθ^0=𝔼[θ0]=𝔼[υ]=υ.
Since the following linear equations:(4.6)(x˙ty˙t)=(β(t)-μ2(t)γ2(t)Sxx(t)α(t)-μ2(t)γ2(t)Sθx(t)0)(x(t)y(t))
equal to
(4.7)x˙t=[β(t)-μ2(t)γ2(t)Sxx(t)]x(t)+α(t)Y(t),y˙t=-μ2(t)γ2(t)Sθx(t)x(t),
it follows from (c1)–(c3) that
(4.8)β1-μ12γ12a1≤β(t)-μ2(t)γ2(t)Sxx(t)≤β2-μ22γ22a2<0,α1≤α(t)≤α2,-μ12γ12b1≤-μ2(t)γ2(t)Sθx(t)≤-μ22γ22b2.
For linear equations:
(4.9)(x˙ty˙t)=(β1-μ12γ12a1α1-μ12γ12b10)(x(t)y(t)),(x˙ty˙t)=(β2-μ22γ22a2α2-μ22γ22b20)(x(t)y(t)),
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
(4.10)Φ1(t)≤Φ(t)≤Φ2(t).
It is not difficult to explore (4.9), and get
(4.11)Φ1(t)=(-λ1′N21eλ1′t-λ2′N21eλ2′teλ1′teλ2′t),Φ2(t)=(-λ1M21eλ1t-λ2M21eλ2teλ1teλ2t),Φ1-1(t)=(-N21λ1′-λ2e-λ1′t-λ2′λ1′-λ2′e-λ1′tN21λ1′-λ2′e-λ2′tλ1′λ1′-λ2′e-λ2′t),Φ2-1(t)(-M21λ1-λ2e-λ1t-λ2λ1-λ2e-λ1tM21λ1-λ2e-λ2tλ1λ1-λ2e-λ2t),
whereN11=β1-(μ12/γ12)a1,N12=α1,N21=(μ12/γ12)b1,λ1′=(N11+N112-4N12N21)/2,λ2′=(N11-N112-4N12N21)/2, M11=β2-(μ22/γ22)a2,M12=α2,M21=(μ22/γ22)b2,λ1=(M11+M112-4M12M21)/2,λ2=(M11-M112-4M12M21)/2.
By assumption (c2) and (c3), we know thatλ1′<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
(4.12)(X^tθ^t)=Φ(t)Φ-1(0)(E[X0]E[θ0])+∫0t‍Φ(t)Φ-1(s)(Sxx(t)Sθx(t))dYs.
Using the similar method, we can also obtain the solutions for the following two equations:
(4.13)(dX^tdθ^t)=(β1-μ12γ12a1α1-μ12γ12b10)(X^tθ^t)dt+μ1γ1(a1b1)dYt,(4.14)(dX^tdθ^t)=(β2-μ22γ22a2α2-μ22γ22b20)(X^tθ^t)dt+μ2γ2(a2b2)dYt,
where X^0=𝔼[X0] and θ^0=𝔼[θ0]=𝔼[υ]=υ.
The solutions of the two equations are explored as the following form:(4.15)(X^tθ^t)=Φ1(t)Φ1-1(0)(E[X0]E[θ0])+∫0t‍Φ1(t)Φ1-1(s)(a1b1)dYs,(X^tθ^t)=Φ2(t)Φ2-1(0)(E[X0]E[θ0])+∫0t‍Φ2(t)Φ2-1(s)(a2b2)dYs.
For (4.14), we have that
(4.16)(X^tθ^t)=Φ2(t)Φ2-1(0)(E[X0]E[θ0])+∫0t‍Φ2(t)Φ2-1(s)(a2b2)dYs
yields that
(4.17)θ^t=∫0t‍[a2(M21λ1-λ2e-λ2(t-s)-M21λ1-λ2e-λ2(t-s))+b2(λ1λ1-λ2e-λ2(t-s)-λ2λ1-λ2e-λ2(t-s))]dYs+(M21λ1-λ2e-λ2t-M21λ1-λ2e-λ2t)X0+(λ1λ1-λ2e-λ2t-λ2λ1-λ2e-λ2t)θ0.
Since λ1<0 and λ2<0, it is easy to get
(4.18)θ^t(υ,⋅)⟶υ,P-a.s.,ast⟶∞.
For (4.13), we can also get
(4.19)θ^t(υ,⋅)⟶υ,P-a.s.,ast⟶∞.
Hence, for (4.1), we can get the following result:
(4.20)θ^t(υ,⋅)⟶υ,P-a.s.,ast⟶∞.
The proof is complete.Remark 4.2.
Under the probability space used in this paper, we can see that Theorem3.2 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 θ^t is a martingale, while, in our result, θ^t can be not a martingale. Furthermore, when θ^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.



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