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The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameter

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 [

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 [

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 [

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)

The notation used here is fairly standard except where otherwise stated.

Hull-White model is a continuous-time, real stochastic process as follows:

Now, our aim is to estimate

For given Gaussian initial conditions

Equation (

From the equation

Assume that the initial conditions

(a1) Assume the initial conditions

(a2) Let

Then, the error covariance matrix

By Kalman-Bucy linear filtering theory, we know that

Since

As long as

Define

The proof is complete.

In order to obtain the convergence rate, the Riccati equation must be solved, and we just need the solution of (

In the case

In the other case

Thus, for each

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

Assume that

:

:

:

:

:

:

Then, for arbitrary

Let

Since

The proof is complete.

From the proof of Theorem

It is well known that Kalman-Bucy linear filtering theory remains valid if one replaces the Brownian motion (

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

The Kalman-Bucy linear filtering theory shows us

But in this paper

In the proof of Theorem

Assume that the following two conditions are satisfied:

:

:

:

Then, for all fixed

We will show that (

By Kalman-Bucy linear filtering theory, we know

Since the following linear equations:

It is not difficult to explore (

By assumption (c2) and (c3), we know that

By the ODE theory [

The solutions of the two equations are explored as the following form:

For (

Under the probability space used in this paper, we can see that Theorem

The strong consistency in Deck [

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.

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.