For Tyrone Duncan

Tyrone Duncan has made seminal contributions in the field of Filtering, Stochastic Control and the interface of probability theory and geometry. In his pioneering thesis, he presented a fundamental theory of nonlinear filtering and was instrumental in introducing the modern theory of martingales as developed by Doob, Meyer, Kunita-Watanabe and others in the study of nonlinear filtering. As he said to one of us, he was fortunate to learn the subject from Watanabe in the course on Martingale theory which Watanabe gave at Stanford when Tyrone was a graduate student. The celebrated stochastic partial differential equation describing the evolution of the unnormalized conditional density of the state given the observations was introduced in his thesis (independently by Mortensen and Zakai). As an outgrowth of the ideas presented in his thesis, he was the first to give a causal representation for the Radon-Nikodym derivative of the probability measure representing the signal plus noise with respect-to the probability measure representing noise only-the celebrated likelihood ratio formula obtained via the Girsanov Theorem. In important work, he also gave the formula for mutual information between a stochastic signal (the state) and an observation where the state is observed in additive white noise.