Fall 2012 Math136/Stats219 Stochastic Processes

MWF 12:15-1:05 pm. Room 200-205.

Instructor:

Isabelle Camilier

Room 383 FF (math building)

Email: camilier (at) stanford (dot) edu

Office hours:

Mondays 2:30-4:30 pm and Wednesdays 11-12, or by appointment (email me).

Course Assistant:

Ka Wai Tsang

Email: ktsang@stanford.edu

Office hours: Tuesday 9-11 am and 4-5 pm

Room: Sequoia 220

Assignments

Homework (20 %) 7 HW assignments (lowest grade dropped).

Midterm exam (30%) in class. Friday, October 26, in class.

Final exam (50%) in class on Wed, December 12, 8:30 am. Room TBA.

At least 60% required for Credit grade.

Syllabus

Course goal: This course prepares students for a rigorous study of Stochastic Differential Equations

Prerequisites: Probability at the level of Stat116/Math105/Math151 and real analysis at the level of Math115.

Material:

Lectures notes: Download the course lecture notes

Optional textbooks:

• Grimmett and Stirzaker, Probability and Random Processes (remark: detailed probability review in chapters 1-5)
• Rosenthal, A first look at rigorous probability theory (remark: review of prerequisite material from real analysis in Appendix A)

Coursework: HW, solutions, and announcements will be on the Coursework page.

Schedule

Preliminary schedule:

M Sept 24 Probability spaces and sigma-algebras (1.1)

W Sept 26 Indicators, simple functions. Random variables (1.2.1, 1.2.2)

F Sept 28 Expectations, motonicity and linearity. Jensen's and Markov's inequalities.(1.2.3)

M Oct 1 Convergence of random variables. Distribution, density and characteristic function. Convergence almost surely, in probability, and in distribution. (1.3.1, 1.4.1)

W Oct 3 Dominated convergence, L^q space (1.3.2,1.4.2) HW1 due

F Oct 5 Independence (1.4.3).

M Oct 8 Conditional expectation and Hilbert space (2.1, 2.2)

W Oct 10 Properties of conditional expectation (2.3) HW2 due

F Oct 12 Properties of conditional expectation (2.3)

M Oct 15 Stochastic processes: definition.

W Oct 17 Stochastic processes and characteristic functions (3.1, 3.2.1) HW3 due

F Oct 19 Gaussian variables, vectors and processes (3.2.2)

M Oct 22 Stationary processes and sample path continuity, right-continuous with left-limits processes. (3.2.3, 3.3)

W Oct 24 Review

F Oct 26 Midterm exam in class.

M Oct 29 Discrete time martingales and filtrations (4.1.1, 4.1.2)

W Oct 31 Sub and supermartingales (4.1.3) HW4 due

F Nov 2 Continuous time martingales (4.2)

M Nov 5 Stopping times for discrete parameter filtrations (4.3.1)

W Nov 7 Stopping times for discrete parameter filtrations (4.3.1) HW5 due

F Nov 9 Martingale representation and inequalities. ( 4.4)

M Nov 12 Martingale convergence theorems (4.5, 4.6)

W Nov 14 Martingales HW6 due

F Nov 16 Brownian motion: definition, Gaussian construction, independence of increments.

Thanksgiving break

M Nov 26 Brownian motion: law of its maximum in an interval and first hitting time of positive levels.

W Nov 28 Brownian bridge and the reflection principle. (5.1, 5.2) HW7 due

F Nov 30 Geometric Brownian motion, Brownian bridge and Ornstein-Uhlenbeck process.

M Dec 3 Markov chain and processes (6.1)

W Dec 5 Poisson distribution, approximation, and process: definition, rate, construction, independence of increments.

F Dec 7 Review

Final exam