On the optimal filtering problem for the cubic sensor

We consider the problem of optimal filtering of a scalar diffusion process measured by a monotone nonlinear sensor in a low-noise channel. The specific sensors considered are of the form h_n(x)=|x|ⁿ sgn(x). This case represents a wide class of sensors with a critical inflection point, since it is the leading term in Taylor's expansion of the measurement function in the critical region. We give for the first time a formal asymptotic approximation of the conditional and of the mean square estimation errors of the optimal filter as interpolation formulas. We also construct an asymptotic approximation to the optimal filter and compare its performance with that of a constant gain filter.