Classification and equivalence in estimation theory

We give a rapid review of some recent developments in nonlinear filtering and suggest a method of classification of estimation problems based on the Lie algebra generated by the operators which appear in the conditional density equation. This is, to some extent implicit in [1] and [2]. We then go on to study a natural class of automorphisms of this algebra thus giving a systematic method of generating problems of equivalent difficulty. Finally, we give here a new class of nonlinear filtering problems which have filtering equations which are themselves nonlinear in an essential way.