In this article we consider a linear regression model of the type
equation(1)
View the MathML sourcedx(t)=ϑ′a(t)dt+dW(t),t≥0
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with the initial condition x(0)=x0x(0)=x0. Here we assume that (W(t),t≥0)(W(t),t≥0) is an adapted one-dimensional standard Wiener process on a filtered probability space (Ω,F,(Ft)t≥0,P),ϑ(Ω,F,(Ft)t≥0,P),ϑ an unknown parameter from some subset ΘΘ of Rp+1,(a(t),t≥0)Rp+1,(a(t),t≥0) an observable adapted (p+1)(p+1)-dimensional cadlag process and x=(x(t),t≥0)x=(x(t),t≥0) solves Eq. (1). We assume that p>1p>1.
The model described includes several more concrete cases like linear stochastic differential equations (SDE’s) of first or of higher-order (CARMA-processes) linear stochastic delay differential equations (SDDE’s). They can be found e.g. in Brockwell (2001), Galtchouk and Konev (2001), Konev and Pergamenshchikov, 1985 and Konev and Pergamenshchikov, 1992, Küchler and Vasiliev, 2001, Küchler and Vasiliev, 2003, Küchler and Vasiliev, 2005, Küchler and Vasiliev, 2006 and Küchler and Vasiliev, 2008 and Liptzer and Shiryaev (1977).
In what follows we will study the problem of sequential estimating the parameter ϑϑ from ΘΘ based on the observation of (x(t),a(t))t≥0(x(t),a(t))t≥0.
We shall construct for every ε>0ε>0 and arbitrary but fixed q≥2q≥2 a sequential procedure ϑ(ε)ϑ(ε) to estimate ϑϑ with εε-accuracy in the sense
equation(2)
View the MathML source‖ϑ(ε)−ϑ‖q2≤ε.
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Here the LqLq-norm is defined as View the MathML source‖⋅‖q=(Eϑ‖⋅‖q)1q, where View the MathML source‖a‖=(∑i=0mai2)12 and EϑEϑ denotes the expectation under PϑPϑ for ϑ∈Θϑ∈Θ (the number q≥2q≥2 is fixed in what follows).
Moreover, we shall determine the rate of convergence of the duration of observations T(ε)T(ε) to infinity and almost surely convergence of ϑ(ε)ϑ(ε) if ε→0ε→0.
The new results presented here consist in the greater generality of the conditions on (a(t))(a(t)) than in previous papers of Küchler and Vasiliev, 2001, Küchler and Vasiliev, 2003, Küchler and Vasiliev, 2005 and Küchler and Vasiliev, 2006. A similar estimation problem for a more general model was investigated in Galtchouk and Konev (2001). The authors have considered the problem of sequential estimation of parameters in multivariate stochastic regression models with martingale noise and an arbitrary finite number of unknown parameters. The estimation procedure in Galtchouk and Konev (2001) is based on the least squares method with a special choice of weight matrices. The proposed procedure enables them to estimate the parameters with any prescribed mean square accuracy under appropriate conditions on the regressors (a(t))(a(t)). Among conditions on the regressors there is one limiting the growth of the maximum eigenvalue of the symmetric design matrix with respect to its minimal eigenvalue. This condition is slightly stronger than those usually imposed in asymptotic investigations and it is not possible to apply this estimation procedure to continuous-time models with essentially different behaviour of the eigenvalues (if, for example, the smallest eigenvalue growth linearly and the largest one — exponentially with the observation time).
The paper (Galtchouk & Konev, 2001) also includes extended hints to earlier works of different authors on sequential estimations for parameters of both continuous as well as discrete-time processes.
The methods applied in this paper to (1) were inspired by the following basic examples for (1):
I. SDE’s of autoregressive type given by
equation(3)
View the MathML sourcedxt(p)=∑i=0pϑixt(p−i)dt+dW(t),t≥0.
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II. SDDE’s given by
equation(4)
View the MathML sourcedX(t)=∑i=0pϑiX(t−ri)dt+dW(t),t≥0.
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The sequential parameter estimation problem of the process (3) was solved in Konev and Pergamenshchikov (1992) under some additional condition on the roots of its characteristic equation (and as follows, on the corresponding parameters). Similar to Galtchouk and Konev (2001), in Konev and Pergamenshchikov (1992) obtained the sequential estimators of the parameter ϑϑ with given accuracy in the mean square sense.
Our paper considers the sequential parameter estimation problem of the process (3) with p=1p=1 as an example of the general estimation procedure, elaborated for linear regression model (1).
It is shown, that the presented sequential estimation procedure works for all parameters View the MathML sourceΘ̃={ϑ∈R2:ϑ1≠0}. As usual, the condition ϑ1≠0ϑ1≠0 means the knowledge of the order (p=1)(p=1) of the process (3). It should be noted that the problem of sequential estimation for the case View the MathML sourceΘ̃∖{ϑ∈R2:ϑ0=0} has been solved in Konev and Pergamenshchikov, 1985 and Konev and Pergamenshchikov, 1992.
The problem of sequential parameter estimation for the process (4) was considered in Küchler and Vasiliev, 2001, Küchler and Vasiliev, 2003, Küchler and Vasiliev, 2005 and Küchler and Vasiliev, 2006 under some additional conditions on the underlying parameters. The general estimation procedure, presented in this paper, works under the most weakest possible assumptions on the parameters. Thus it is shown, that in the case p=1p=1 in the model (4) the constructed general estimation procedure gives the possibility to solve the parameter estimation problem with guaranteed accuracy for all parameter points ϑ∈R2ϑ∈R2 except for some curves Lebesgue of measure zero.
The estimators with such property may be used in various adaptive procedures (control, prediction, filtration).
