Moshe Zakai

On the optimal filtering of diffusion processes

SummaryLet x(t) be a diffusion process satisfying a stochastic differential equation and let the observed process y(t) be related to x(t) by dy(t) = g(x(t)) + dw(t) where w(t) is a Brownian motion. The problem considered is that of finding the conditional probability of x(t) conditioned on the observed path y(s), 0≦s≦t. Results on the Radon-Nikodym derivative of measures induced by diffusions processes are applied to derive equations which determine the required conditional probabilities.

Some lower bounds on signal parameter estimation

New bounds are presented for the maximum accuracy with which parameters of signals imbedded in white noise can be estimated. The bounds are derived by comparing the estimation problem with related optimal detection problems. They are, with few exceptions, independent of the bias and include explicitly the dependence on the a priori interval. The new results are compared with previously known results.

Improved Lower Bounds on Signal Parameter Estimation

An improved technique for bounding the mean-square error of signal parameter estimates is presented. The resulting bounds are independent of the bias and stronger than previously known bounds.

Mutual information of the white Gaussian channel with and without feedback

The following model for the white Gaussian channel with or without feedback is considered: \begin{equation} Y(t) = \int_o ^{t} \phi (s, Y_o ^{s} ,m) ds + W(t) \end{equation} where m denotes the message, Y(t) denotes the channel output at time t , Y_o ^ {t} denotes the sample path Y(\theta), 0 \leq \theta \leq t. W(t) is the Brownian motion representing noise, and \phi(s, y_o ^ {s} ,m) is the channel input (modulator output). It is shown that, under some general assumptions, the amount of mutual information I(Y_o ^{T} ,m) between the message m and the output path Y_o ^ {T} is directly related to the mean-square causal filtering error of estimating \phi (t, Y_o ^{t} ,m) from the received data Y_o ^{T} , 0 \leq t \leq T . It follows, as a corollary to the result for I(Y_o ^ {T} ,m) , that feedback can not increase the capacity of the nonband-limited additive white Gaussian noise channel.

A lower bound on the estimation error for certain diffusion processes

A lower bound on the minimal mean-square error in estimating nonlinear diffusion processes is derived. The bound holds for causal and noncausal filtering.

Riemann-Stieltjes approximations of stochastic integrals

SummaryWe consider the space C[0, 1] together with its Borel σ-algebra A and a Wiener measure P. Let Ω denote a point in C[0, 1] and let x(Ω, t) denote the coordinate process. Then, {x(Ω, t), tε[0, 1]} is a Wiener process, and stochastic integrals of the form

The Malliavin calculus

\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)

Lower and upper bounds on the optimal filtering error of certain diffusion processes

can be defined for a suitable class of ϕ. In this paper we consider a sequence of Stieltjes integrals of the form