Existence, uniqueness and tail behavior of solutions to Zakai equations with unbounded coefficients

Conditions are given to guarantee the existence and uniqueness of solutions to the Duncan-Mortensen-Zakai equation for nonlinear filtering of multivariable diffusions with unbounded coefficients. Sharp upper and lower bounds on the tail of conditional densities are also obtained. A methodology is described to treat these problems using classical p.d.e. methods applied to the "robust" version of the DMZ equation. Several examples are included.