# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 236A: Stochastic Dynamics and Applications

**Approved:**2009-03-01, Albert Fannjiang

**Units/Lecture:**

Fall, alternate years; 4 units; lecture/term paper or discussion

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

**Prerequisites:**

MAT/STA 235B, or consent of instructor. A working knowledge of the material in the entire probability sequence MAT/STA 235ABC is recommended. MAT/STA 235AB may be taken concurrently.

**Course Description:**

Stochastic processes, Brownian motion, Stochastic integration, martingales, stochastic differential equations. Diffusions, connections with partial differential equations, mathematical finance.

**Suggested Schedule:**

Department Syllabus

MAT 236A: Stochastic Dynamics and Applications

When taught: | Fall, alternate years |

Suggested text ($): | B. Oksendal, Stochastic Differential Equations, ISBN-10: 038715292X ($50) |

Units/lectures: | 4 units; lecture/term paper or discussion |

Prerequisites: | MAT/STA 235 |

Lectures | Sections | Topics/Comments |
---|---|---|

1 | 1.1 - 1.6 | Motivation, examples |

1 | 2.1 - 2.2 | Review: probability space, stochastic processes |

3 | 3.1 - 3.3 | Ito integrals, chaos expansion |

2 | 4.1 - 4.2 | Ito formula: 1-d, multi-d |

1 | 4.3 | Martingale representation theorem |

1 | 5.1 | Examples and solution methods for SDE's |

1 | 5.2 | Existence and uniqueness |

1 | 5.3 | Weak and strong solutions |

2 | 7.1 - 7.2 | Diffusions: Markov property |

2 | 7.3 - 7.4 | Generator, Dynkin formula |

1 | 8.1 | Kolmogorov's backward equations |

1 | 8.2 | Feynman-Kac formula |

1 | 8.3 | Martingale problem |

2 | 8.5 | Random time change |

2 | 8.6 | Girsanov theorem |

2 | 9.1 - 9.3 | Dirchlet problem, Poisson problem |

2 | 6.1 - 6.3 | Linear filtering |

2 | 10.1 - 10.4 | Optimal stopping (shortened and simplified) |