On Some Approximation Techniques in Non Linear Filtering

Abstract

The following problem has been considered by J. Picard [3]. Let a signal be governed by the equation

$$ ^{\begin{array}{*{20}{c}} {dx = b\left( x \right)dt + {\varepsilon ^r}\sigma \left( x \right)d{w^1}} \\ {x\left( 0 \right) = \xi } \end{array}} $$

(1.1)

This signal is “observed” by the process

$$ \begin{array}{*{20}{c}} {dy = h\left( x \right)dt + {\varepsilon ^{1 - }}{}^rd{w^2}} \\ {y\left( 0 \right) = 0} \end{array} $$

(1.2)

where w^l, w² are two independent Wiener processes, and ξ is a random variable independent of w^l, w². The parameter ε is small and 0 < γ < ½. This assumption means that there is a high signal to noise ratio. Let

$$ \phi \left( t \right) = E\left[ {\phi \left( {{x_t}} \right)\left| { y\left( s \right), 0 \leqslant s \leqslant t} \right.} \right] $$

(1.3)