# 401-3642-00L Brownian Motion and Stochastic Calculus

Semester | Spring Semester 2015 |

Lecturers | J. Teichmann |

Periodicity | yearly course |

Language of instruction | English |

Abstract | This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic calculus. The following topics will be discussed: - Brownian motion - Markov processes - Stochastic calculus - Levy processes |

Objective | This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic calculus. The following topics will be discussed: - Brownian motion: definition, construction, some important properties - Markov processes: basics, strong Markov property, generators and martingale problems - Stochastic calculus: semimartingales, stochastic integrals, Ito formula, Girsanov transformation, stochastic differential equations - Levy processes: basic notions, some important properties |

Content | This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic calculus. The following topics will be discussed: - Brownian motion: definition, construction, some important properties - Markov processes: basics, strong Markov property, generators and martingale problems - Stochastic calculus: semimartingales, stochastic integrals, Ito formula, Girsanov transformation, stochastic differential equations - Levy processes: basic notions, some important properties |

Lecture notes | will be available for purchase |

Literature | - Durrett, R., "Stochastic Calculus. A Practical Introduction", CRC Press, 1996. - Ikeda, N. and Watanabe, S., "Stochastic Differential Equations and Diffusion Processes", second edition, North Holland, Amsterdam, 1979. - Karatzas, I. and Shreve, S., "Brownian Motion and Stochastic Calculus", second edition, Springer, Berlin, 1991. - Revuz, D. and Yor, M., "Continuous Martingales and Brownian Motion", second edition, Springer, Berlin, 1994. - Rogers, L.C.G. and Williams, D., "Diffusions, Markov Processes, and Martingales", vol. 1 and 2, Wiley, Chichester, 2000, 1994. - Sato, K., "Levy Processes and Infinitely Divisible Distributions", Cambridge University Press, 1999. |

Prerequisites / Notice | Familiarity with measure-theoretic probability as in the standard D-MATH course "Wahrscheinlichkeitstheorie" will be assumed. Textbook accounts can be found for example in - Jacod, J. and Protter, P., "Probability Essentials", second edition, Springer, 2004 or - Durrett, R., "Probability: Theory and Examples", second edition, Duxbury Press, 1996 (Chapters 1-4 and Appendix) |