# Stochastic Differential Equations

### Welcome to AMS216 Winter 2015!!

General Class Information

Instructor: Tatiana Xifara (xifara@soe.ucsc.edu)

Office Hours: Wednesday 10:00-11:00am or by appointment

Lectures: Monday, Wednesday 5:00pm-6:45pm @ Kresge Clrm 319

Course ObjectivesThis is an introduction to modeling and inference with Stochastic differential equations (SDEs) that arise in many branches of science and engineering. This is a graduate level course that requires only upper division probability and differential equations, since we will approach the analysis of questions about SDE through the associated differential equations and the inference through the normal (gaussian) distribution.

Textbook:  B. Oksendal (2013). Stochastic Differential Equations: An Introduction with Applications (sixth edition, sixth corrected printing. Springer.

Further Reading: R. Durrett (1996). Stochastic Calculus: A Practical Introduction. CRC Press.

### Syllabus

Course topics (tentative list, check calendar for most updated mterial) include

•  Brief review of probability theory (conditional probability, conditional average, Gaussian distribution, characteristic function, Bayes theorem) and ordinary differential equations
• Brownian motion, White noise, Itô's Formula, Gambler's ruin problem
• Geometric Brownian motion, Brownian bridge
• Ornstein-Uhlenbeck Process, Maxwell-Boltzmann distribution, Cox-Ingersoll-Ross process
• Stochastic integrals and stochastic differential equations, Itô interpretation, Stratonovich interpretation, relation between Itô interpretation and Stratonovich interpretation, Stochastic integrals based on axioms
• Kolmogorov backward equation of a stochastic differential equation, Fokker-Planck equation (forward equation) of a stochastic differential equation, meaning of the forward equation, meaning of the backward equation, adjoint differential operators, relation between the forward equation and the backward equation
• Escape problem, differential equation for the average exit time, Kramers approximation, several special cases
• Going backward in time in Ornstein-Uhlenbeck process, time reversibility of equilibrium
• Feynman-Kac formula for the backward equation, Feynman-Kac formula for the forward equation, application of Feynman-Kac formula in reconstructing molecular bond potential
• Numerical methods for SDEs
• Generalized Ornstein-Uhlenbeck process, Diffusion processes
• Likelihood inference for stochastic differential equations
• Inference for the Onrstein-Uhlenbeck Process
• Bayesian Inference for Diffusions
• The Metropolis-adjusted Langevin algorithm (if there is time)

## Announcements Archive

• 1/2:  Get ready ... !  First class Monday Jan 5th ...
• 1/6: Class wil be moved to Kresge Clrm 319 - effective Wednesday 1/7.
• 1/12: HM1 is up and is due to the 21st.
• 1/14:  Lecture notes on Brwonian motion are up!  See "Homeworks and handouts".
• 1/21:
• Extra reading for Gambler's ruin problem (Senior thesis by Daniel Ladd)
• HW1 due by today. (typo at 2(b), the denomiantor should be 2^k, not 2k)
• 1/29:
• HM2 is up and is due to Feb 11th.
• Our Midterm will be on Feb 18th (Remember February 16th is President's day so the class will not meet.
• 2/16: R code is added in the attachments for simulation of Brownian motion and more.
• 2/23: Homework 3 is due to Wednesday March 4th.
• 2/25: Start thinking about your final project. Feel free to come and find me for ideas and guidance.
• 3/9: Final Project is due to finals week.

DRC ACCOMMODATIONS

If you qualify for classroom accommodations because of a disability, please get an Accommodation Authorization from the Disability Resource Center (DRC) and submit it to me in person outside of class (e.g., office hours) within the first two weeks of the quarter. Contact DRC at 459-2089 (voice), 459-4806 (TTY), or http://drc.ucsc.edu for more information on the requirements and/or process.