Optimal filtering in the presence of multiplicative noise

Optimal filtering for a specific class of nonlinear, discrete dynamical systems having multiplicative input noise and multiplicative measurement noise is studied. The most general solution is the probability density function of the state conditioned on the measurements. Fundamental to the recursive solution derived in this study are the lognormal probability laws assumed for the multiplicative input noise and the multiplicative measurement noise. The basic theory is developed for a scalar system with positive state. The conditions for stability of the optimal estimate (i.e., conditional expectation) are also derived. The theory is then extended to a scalar system where the state can be either positive or negative.