ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 1, pp. 53–71.
Pleiades Publishing, Inc., 2008.
Original Russian Text
A.I. Ovseevich, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 1, pp. 59–73.
Kalman Filter and Quantization
A. I. Ovseevich
Institute for Problems in Mechanics, RAS, Moscow
Received October 15, 2007
Abstract—We give an interpretation of the problem of ﬁltering of diﬀusion processes as a
quantization problem. Based on this, we show that the classical Kalman–Bucy linear ﬁlter
describes a ﬂow of automorphisms of the Heisenberg algebra. We obtain new formulas for the
unnormalized conditional density in the linear case, a new interpretation of the Mehler formula
for the fundamental solution of the Schr¨odinger operator for a harmonic oscillator, and formulas
for a regularized determinant of a Sturm–Liouville operator.
The paper is aimed at a conceptually simple and intuitive approach to the Kalman ﬁlter. We
deal with both the linear and nonlinear ﬁltering, but ﬁnal results concern the linear case.
The structure of these notes and our arguments are roughly as follows.
First we state the problem of ﬁltering of a partially observable diﬀusion process, i.e., that of op-
timal estimation of an unobserved component based on observations. To avoid geometric problems
irrelevant to our main topic, we conﬁne ourselves with diﬀusions in a Euclidean space R
By using the Girsanov theorem, the problem of nonlinear ﬁltering can be reduced to that of
computing a path integral, which is related to quantization of a classical Hamiltonian system. The
corresponding Hamiltonian system arises via a deterministic optimal control problem, which might
be interpreted as that of ﬁnding a “most probable” unobservable trajectory compatible with obser-
vations. In a most favorable situation, computation of a path integral can be performed by using
Generally, computation of the above-mentioned path integral is performed by using several basic
tools from quantum mechanics. The ﬁrst of them is the Hamiltonian (or Schr¨odinger) approach,
which is equivalent to the Zakai equation of nonlinear ﬁltering. This is a stochastic partial diﬀer-
ential equation for the conditional distribution density of an unobservable component of a diﬀusion
process when the observable component is known. In fact, the Zakai equation does not exactly
describe this conditional density but rather its projective class, i.e., the density considered up to
multiplication by a constant. This, of course, complies with the spirit of quantum mechanics since
quantum states are projective classes of vectors.
The Zakai equation is a generalization of the Kolmogorov equation for the (unconditional) dis-
tribution density of a diﬀusion process and reduces to the latter when the observation is absent or
The second quantum tool is the Heisenberg (or Lax) equation for observed values. It can be
eﬃciently applied in the linear situation where we take linear combinations of coordinates and
Supported in part by the Russian Foundation for Basic Research, project nos. 05-08-50226 and 06-01-