### Source materials

See the Resources page.

### Weekly schedule (tentative)

week | topics |
---|---|

1 | Review of multivariate normal random variables via linear algebra. Discrete time Gaussian processes. Stability, transience, and cointegration. Gaussian nature of paths. |

2 | Discrete Markov chains, path space, transtion probabilities, evolution of probability, value functions, backward and forward equations. Various random walks as examples. |

3 | Continuous time Gaussian Markov processes. Brownian motion and Ornstein-Uhleneck. Continuous paths as Gaussian random objects. Convergence of discrete time Gaussian processes to continuous time processes. |

4 | The heat equation, diffusion, and the relation to Brownian motion. Green's functions as transition probability densities, the semi-group property and the tower property. |

5 | Integrals with respect to Brownian motion, the Ito integral. Filtrations and non-anticipating functions. Convergence of the Riemann sum approximations. |

6 | Functions of Brownian motion, the Ito isometry formula, Ito's lemma. The relation between Ito's lemma and backward equations. Geometric Brownian motion. |

7 | General diffusion processes, infinitesimal mean and variance, quadratic variation, a more general version of Ito's lemma. |

8 | Hiting times, stopping times, boundary conditions. Martingales, Doob stopping time theorem. |

9 | Stochastic differential equations, writing diffusions as functions of Brownian motion. |

10 | The relation between diffusion processes and partial differential equations (PDEs) of diffusion type. Backward equations, the Feynman Kac formula. |

11 | Finite difference approximate solution of PDEs of diffusion type. Monte Carlo and simulation of diffusions. |

12 | Modeling and approximation with diffusions. Continuous time approximations of large but discrete processes, small but finite time intervals. Diffusive scaling. |

13 | Change of measure and Girsanov transformations. |

14 | Steady states, equilibrium, recurrence and transience. Review and summary for the final exam. |

15 | Final exam, same time, same room |