Stochastic filtering: a functional and probabilistic approach.

In this thesis I study the stochastic filtering of a Stochastic Differential Equation (in continuous time), both in a linear and non-linear setting. This means, essentially, finding the best possible estimate (in a certain sense) of an unobservable stochastic process given the information contained on the observations of another process (which is a function of the unobservable process plus some noise). After an introduction on stochastic calculus and SDE, I will present the linear case. I will prove the derivation of the well-known “Kalman Bucy Filter” using a functional approach based on projection on Hilbert spaces. I will provide also some simple but interesting application with Matlab. Then, I will present the non-linear case using, instead, an heavy probabilist approach based on change of measure and Girsanov Theorem.