ECE 784 - Winter 2018

ECE 784 - STATS 902: Introduction to Stochastic Calculus

Instructor: Andrew Heunis (E&CE and STATS)

Email: heunis@uwaterloo.ca

Phone: 519-888-4567 x32083

Location: EIT 3115

The stochastic calculus finds ready application in a variety of areas including electrical engineering (nonlinear filtering stochastic optimal control, stochastic adaptive control), physics (quantum and stochastic mechanics) and mathematical finance (pricing of derivative securities etc.).

The goal of this course is to establish the main principles of stochastic calculus within the simplest setting of stochastic integration with respect to continuous semimartingales. Our emphasis will be on the basic principles and theorems of stochastic calculus rather than on specific applications.

The main prerequisites for enrolling in this course are competence in basic measure theory (Lebesgue integral and its properties, convergence theorems, Radon-Nikodym theorem, Fubini-Tonelli theorem, extension theorem for measures) and elementary probability theory (independence, expectations, conditional expectations). These prerequisites are provided by STATS 901.

Contents:

  1. Preliminaries: brief overview of prerequisites, introduction to monotone and Dynkin classes of sets, monotone and Dynkin class theorems.
  2. Discrete-parameter martingales: discrete-parameter filtrations and stopping times, optional sampling theorem, supermartingale inequalities, supermartingale convergence theorem, uniform integrability and uniformly integrable supermartingales.
  3. Elements of continuous-parameter stochastic processes: processes with independent increments, the Wiener process, continuous-parameter filtrations and standard filtrations, continuous-parameter stopping times, corlol processes, progressively measurable processes.
  4. Continuous-parameter martingales: structure of (super)-martingale sample paths, continuous- parameter analogs of the main results for discrete-parameter martingales, continuous local martin- gales, quadratic variation process of a continuous local martingale.
  5. Stochastic integration of progressively measurable integrands: sample-path integrals with respect to processes of locally-bounded variation, Kunita-Watanabe inequalities and Ito integrals with respect to continuous local martingales and continuous semimartingales, Ito’s formula, exponential local martingales, Novikov theorem, martingale characterization of the Wiener process, changes of measure and the Girsanov theorem.
  6. Representation of local martingales as stochastic integrals.

Course Materials:

Course notes, problem sets and solutions will be available on UW-LEARN at the start of the Winter 2018 term.