Deterministic feedback linearization, Girsanov transformations and finite-dimensional filters

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 geffucient 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-with single-input (or onde-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 an affine prime system are obtained, as well as interpretation of gauge transformation as Girsanov change of probability law.