Description

Many applications, from guidance and navigation to time series analysis and credit risk estimation, require one to estimate the current state of a system which cannot be directly observed. Stochastic filtering is the mathematical procedure of determining the posterior distribution of a hidden state from noisy observations, as the system evolves randomly in time.

Some of the early applications of stochastic filtering were in aerospace navigation, for instance by NASA in the Apollo missions to the moon. It is now a well-established branch of stochastic analysis, with widespread use, particularly in engineering and financial mathematics.

The primary goal of filtering is to derive the stochastic differential equation satisfied by the posterior distribution. In some special but important cases, such as finite-state Markov chains and Gaussian processes with linear dynamics, this equation is finite-dimensional, and provides a tractable solution to the filtering problem. In more general settings, the resulting infinite-dimensional equation is difficult to solve, and various numerical procedures have thus been proposed to approximate the solution.

In this project, you would learn some of the basics of filtering theory, and then explore and potentially try out some of the various approaches to approximate the solution of the filtering equations. This could involve in particular particle filters, in which a cloud of simulated particles is propagated in such a way that their joint distribution converges to the posterior distribution of the hidden state as the number of particles tends to infinity.

There is scope for either a somewhat pure project on the theoretical aspects of the filtering equations, or a slightly more applied project involving the derivation, implementation and comparison of various filtering algorithms. You could also investigate applications of filtering, for instance to finance, or to problems of stochastic control in partially observable systems.

Prerequisites

Probability II and Mathematical Finance III would be suitable prerequisites. This project would be particularly suitable for students also intending to take the course Stochastic Analysis IV.

Resources

There are a number of books on the subject, such as:

• Alan Bain and Dan Crisan, Fundamentals of Stochastic Filtering, Springer, 2009.
• Lakhdar Aggoun and Robert J. Elliott, Measure Theory and Filtering, Cambridge University Press, 2004.

You can also find various sets of lecture notes online, such as:

• Ramon van Handel, Stochastic Calculus, Filtering, and Stochastic Control, Lecture notes, 2007.