On girsanov linearization of stochastic systems and application to filtering

The problem of extending state-feedback linearization methods of deterministic Control theory to stochastic systems is addressed. For Stratonovitch stochastic differential equations with smooth vector fields, necessary and sufficient geometric conditions for local and global linearization by diffeomorphism and absolutely continuous change of probability law are obtained, using the interpretation of Girsanov transformations as state-feedback on brownian motions. For stochastic systems with single input (or one-dimensional brownian motion) and single output (or one-dimensional observation process), necessary and sufficient geometric conditions to transform the Duncan-Mortensen-Zakai (DMZ) equation of filtering into that of a linear prime system are obtained, as well as an interpretation of gauge transformation as Girsanov change of probability law.