Optimal Linear Nonstationary Filtering
On the probability space (Ω, F, P) with a distinguished family of the σ-algebras (F t ), t ≤ T, we shall consider the two-dimensional Gaussian random process (θ t , F t ), 0 ≤ t ≤ T, satisfying the stochastic differential equations $$d{\theta _t}\, = \,a(
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10.1 The Kalman-Buey Method 10.1.1. On the probability space (n,:F, P) with a distinguished family of the u-algebras (:Ft ), t :$ T, we shall consider the two-dimensional Gaussian random process (Bt,:Ft ), 0 :$ t :$ T, satisfying the stochastic differential equations
dBt = a(t)Btdt + b(t)dWI(t), det = A(t)Btdt + B(t)dW2(t),
(10.1)
(10.2)
where WI = (WI(t),:Ft ) and W2 = (W2(t),:Ft ) are two independent Wiener processes and Bo, eo are :Fo-measurable. It will be assumed that the measurable functions a(t), b(t), A(t), B(t) are such that T 1T \a(t)\dt < 00, b2(t)dt < 00, (10.3)
1
1T \A(t)\dt < 00,
1T B2(t)dt < 00.
(10.4)
From Theorem 4.10 it follows that the linear equation given by (10.1) has a unique, continuous solution, given by the formula t
Bt = exp [1 a(U)dU] [Bo +
1t
exp {_1B a(U)dU} b(S)dW1 (S)].
(10.5)
The problem of optimal linear nonstationary filtering (Bt on et,) examined by Kalman and Bucy consists of the following. Suppose the process Bt , 0 ::; t ::; T, is inaccessible for observation, and one can observe only the values et, 0 :$ t :$ T, containing incomplete (due to the availability in (10.2) of the multiplier A(t) and the noise B(s)dW2(S» information on the values Bt . It is required at each moment t to estimate (to filter) in the 'optimal' way the values Bt on the basis of the observed process: eb = {es, 0:$ S :$ t}. If we take the optimality of estimation in the mean square sense, then the optimal (at t) estimate for Bt given eb = {eB' 0 :$ S :$ t} coincides with the conditional expectation I
J:
1
Henceforth only the measurable modifications of conditional expectations will be taken.
R. S. Liptser et al., Statistics of Random Processes © Springer-Verlag Berlin Heidelberg 2001
376
10. Optimal Linear Nonstationary Filtering
(10.6) (in the notation of Chapter 8, mt we denote by
= lI't{O)). An error of estimation {of filtering) (1O.7)
The method employed by Kalman and Bucy to find mt and 'Yt yields a closed system of dynamic equations (see (10.1O)-{10.11)) for the estimate in a form convenient for instrumentation of an optimal 'filter'. The process (Ot,et), 0 ~ t ~ T, studied by Kalman and Bucy is Gaussian. As a consequence, the optimal estimate mt = M{Otl.r;) turns out to be linear (see Lemma 10.1). The next chapter contains an essential generalization of the Kalman-Bucy scheme. It will be shown there that in the so-called conditionally Gaussian case for mt = M{Otl.r;) and 'Yt = M[{Ot-mt)21.r;] a closed system of equations can also be obtained (see (12.29), (12.30)), although the estimate mt will then be, generally speaking, nonlinear. In the case of (1O.1) and (1O.2) the equations for mt and 'Yt can be easily deduced from the general equations of filtering obtained in Chapter 8. This will be done in Sections 10.2 and 10.3. In Subsections 10.1.2-10.1.4 the filtering equations for mt and 'Yt will be deduced (with some modifications and refinements) following the scheme originally suggested by Kalman and Bucy. As noted in the introduction, (10.24) is the basis of this deduction (
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