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## Stochastic Processes

### Summary

EC | 8 |

Location | Vrije Universiteit |

Weeks | 6 - 21 |

Lecture | Wednesday, 14:00 - 16:45 |

Provider | Stochastics (Star) |

Links | Course page (requires login) |

**Prerequisites**

The Mastermath course "Measure-Theoretic Probability" is sufficient. Alternatively: basic knowledge of Probability (equivalent to Chapters 1-8 of "A First Course in Probability" by S. Ross, 9th Edition, or Chapters 1-5 of "Statistical Inference" by G. Casella and R. Berger, 2nd Edition), and of Measure and Integration (equivalent to Chapters 1-5 of "Measure Theory" by D. Cohn, 2nd Edition).

**Aim of the course**

The aim of this course is to cover the elementary theory of stochastic processes by discussing some of the fundamental classes of processes, namely Brownian motion, continuous-time martingales and Markov and Feller processes. At the end of the course the student: - Is able to recognize the measure-theoretic aspects of the construction of stochastic processes, including the canonical space, the distribution and trajectory of a stochastic process, filtrations and stopping times. - Is able to state Kolmogorov's Extension Theorem and use it to show the existence of stochastic processes with a prescribed distribution. - Can describe the abstract construction of Brownian motion and prove some of the elementary properties of this process, such as continuity of trajectories and the distribution of the maximum of a trajectory. - Can define continuous-time martingales (and super- and sub-martingales) and prove some of the fundamental result concerning them, including the optional sampling theorem and Doob's inequality. - Can define Markov processes and continuous-time Markov chains; for the latter, can explain the relation between the transition function and the infinitesimal generator, and give a construction involving Poisson processes. - Can define Feller processes and prove the fundamental Hile-Yosida theorem relating generators and operator semigroups.

**Lecturers**Daniel Valesin (RuG), Christian Hirsch (RuG)