Stochastic calculus, Fall 2012

Course Details and Syllabus

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