# Lecture and Exercises: Stochastic Filtering (SS 2016)

**Lecturer:**JProf. Dr. Philipp Harms**Lecture:**Thursdays 12-14 in HS II**Exercises:**Wednesdays 16-18 in SR 218**Intended Audience:**MSc and PhD students with the required prerequisites**Language:**English

## Contents

We start in discrete time with an overview of the relevant questions, concepts, and results. This allows us to quickly and easily gather intuition and get started with numerical experiments. We then proceed to continuous time, where we focus on the derivation of the filtering equations. We present a new and unified approach to deriving the filtering equations. That is, we start in a general semimartingale setting and deduce various well-known results as special cases. Numerics and applications will be discussed on the way.

- Discrete time:
- State and parameter estimation
- Bayes' formula and changes of probability measures
- Filtering, smoothing, and prediction

- Continuous time:
- Optional and predictable projections
- Construction of the filter as a measure-valued process
- Right-continuity of the observation filtration
- Change of measure versus innovations approach
- Kushner-Stratonovich and Zakai equation

- Numerics:
- Baum-Welch algorithm
- Viterbi algorithm
- Kalman filter
- Particle filter
- Optionally: Feedback particle filter
- Optionally: Galerkin approximations

- Applications:
- Asset price modeling
- Credit risk modeling
- Robotics
- DNA sequencing
- Signal processing

## Prerequisites

- Measure and probability theory
- Stochastic integration with respect to Brownian motion
- Ito calculus
- Coding in R, MATLAB, or your favorite programming language

Some basic semimartingale theory will be helpful, but is not required.

## Literature

- Kallenberg, Olaf. 2010.
*Foundations of Modern Probability Theory, Second Edition.*Springer-Verlag New York. - Cappé, Moulines, Ryden. 2005.
*Inference in Hidden Markov Models.*Springer-Verlag New York. - Ramon van Handel. 2008.
*Hidden Markov Models.*Lecture Notes, Princeton University. - Bain, Crisan. 2009.
*Fundamentals of Stochastic Filtering.*Springer-Verlag New York. - Liptser, Shiryaev. 2001.
*Statistics of random processes, Volumes I and II.*Springer-Verlag Berlin Heidelberg. - Grigelionis, Mikulevicius. 2011.
*Nonlinear filtering equations for stochastic processes with jumps.*The Oxford handbook of nonlinear filtering, pp.95-128. Oxford University Press.

## Exercises

Solving 60% of the exercises is required for a grade or certificate of participation.

- Exercise Sheet 1, due on April 27
- Exercise Sheet 2, due on May 4
- Exercise Sheet 3, due on May 11
- Exercise Sheet 4 and Code Template 4, due on May 25. Code solution 4.
- Exercise Sheet 5, due on June 1
- Exercise Sheet 6, due on June 8. Code solution 6.
- Exercise Sheet 7, due on June 15.
- Exercise Sheet 8, due on June 22.
- Exercise Sheet 9, due on July 1.
- Exercise Sheet 10, due on July 8.
- Exercise Sheet 11, due on July 13.
- Exercise Sheet 12, due on July 20.

## Announcements

- The date of the first lecture is April 21 and of the first exercise class April 27.
- Here is a summary of the recursions for filtering, smoothing, and prediction covered in the lecture: pdf
- The exercises on Wednesday June 29 and July 6 will be shifted to Friday 10-12, SR 218.