Speaker
Description
This talk deals with the so-called filtering problem, the optimal estimation of a hidden state given partial and noisy observations. In the seminal paper by Kushner \cite{KUSHNER1967179}, a nonlinear stochastic partial differential equation for the conditional density of continuous processes with continuous observations has been derived. In a subsequent paper, Zakai \cite{zakai1969optimal} was able to obtain a linear stochastic partial differential equation for the unnormalized density using Girsanov transformation. In the general case, these filtering equations are infinite dimensional. Only in very special cases, arguably the most commonly known of them, the Kalman-Bucy filter, one obtains finite dimensional filtering equations. Another case in which the filtering equations are finite dimensional is when the so called Bene\v{s} condition is fulfilled (\cite{benevs1981exact}). We consider an instance in which this Bene\v{s} condition is satisfied and for which we can explicitly formulate filter and smoother density and statistics, relying on the work in \cite{ocone1982explicit}. Interestingly, the filtering density turn out to be a mixture of Gaussians.
This is joint work with Aleksandar Arandjelović (ETH Zürich) and David Hirnschall (WU Wien).
Affiliation
WU Wien / UNIQA Insurance Group