Course Description:

Continuation of MA/STAT 638. This course studies Ito's Stochastic Calculus and the solutions of Stochastic Differential Equations. They will be used to provide examples of stochastic processes for the investigation of the concept of Markov and Diffusion Processes, Gaussian and Stationary Processes, Ergodic Theory and many others. The emphasis will be on the applications of probability theory in mathematics, engineering and physical sciences.

Instructor:

Aaron Nung Kwan Yip

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 432

Office phone: (765) 494-1941

Fax: (765) 496-1551

Email: click here

Mailing address:

Department of Mathematics

Purdue University

150 N. University Street

West Lafayette, IN 47907-1395

Lecture Time and Place:

MWF 12:30pm-1:20pm, CL50 125

Office Hours:

MWF: 9:15am-10:15am or by appointment

Textbook:

[KS] Brownian Motion and Stochastic Calculus (2nd Ed.), by I. Karatzas, S. E. Shreve

[O] Stochastic Differential Equations (6th Ed.), by B. Oksendal

Reference Books (on reserve in math libary):

[Ev] An introduction to stochastic differential equations(pdf file), L. C. Evans

[RY] Continuous Martingales and Brownian Motion by D. Revuz and M. Yor

[Bi] Probability and Measure (3rd Ed.), by P. Billingsley

[Br] Probability, by L. Brieman

[Fe] An introduction to probability theory and its applications, v.1 and 2, by W. Feller

Prerequisites:

A good knowledge of basic properties of Ito's Calculus and Stochastic Differential Equation: Ito's Formula, Quadratic Variations, Existence and Uniqueness of SDEs. The Lecture Note [Ev] listed above gives a very comprehensive review of the necessary background needed.

Homework:

Homeworks will be assigned occasionally. Please refer to the course announcement below.

• No late homeworks are accepted (in principle).
  
• You are encouraged to discuss the homework problems with your classmates but all your handed-in homeworks must be your own work.

Examinations:

Presentation of some research paper/topic.

Grading Policy:

Homeworks (50%)

Final Exam (50%)

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade, plus any other disciplinary action, deemed appropriate.

Course Outline:

Core Topics:

Markov and Diffusion Process: Feller Process, generator, invariant measure and ergordic property

Relation between SDE and PDE: Representation of solutions of PDEs using SDEs, Feynman-Kac Formulas, Potential Theory, Martingale Representation, Girsanov Transformation

Stability of SDE

Exit Time Problems

Freidlin-Wentzell Large Deviation Theory

Additional Topics:

Averaging Principle

Monte-Carlo, Simulated Annealing, Stochastic Optimization

Convergence of Markov Chain to Diffusion Processes

Levy Process

Fractional Brownian Motion

Course Progress and Announcement:

(You should consult this section regularly, for homework assignments, additional materials and announcements.)

Some materials on physical consideration of Brownian Motion:
Stochastic Problems in Physics and Astronomy, S. Chandrasekhar, Rev. of Mod. Phys., Vol 15(1943), pp 1-89
The Mathematics of Brownian Motion and Johnson Noise D. T. Gillespie, Am. J. Phys., Vol 64(1996), pp 225-240
Fluctuation and Dissipation in Brownian Motion D. T. Gillespie, Am. J. Phys., Vol 61(1993), pp 1077-1083
Dynamical Theories of Brownian Motion E. Nelson, Princeton University Notes

Hw 1: [KS: 5.2.26], due Feb 1, Friday, 3pm (slide under my office door if I am not in the office.)

Some materials on the foundation of Markov Processes:
Semi-Group for Markov Processes (Dynkin, 1956)
Infinitesimal Generators of Markov Processes (Dynkin, 1956)
Strong Markov Processes (Dynkin & Yushkevich, 1956)

Some materials on the Martingale Formulation of Diffusion Processes:
A very brief introduction(Stroock-Varadhan, 1970)
Chapter 0 of Stroock-Varadhan Book(Stroock-Varadhan, 1979)
Summary of Stroock-Varadhan I, II(Stroock-Varadhan, 1970)
Part I(Stroock-Varadhan, 1969)
Part II(Stroock-Varadhan, 1969)

Hw 2: Due Mar 7, Friday, 3pm

Some materials on the Cameron-Martin-Girsanov Transformation:
Transformation under translation(Cameron-Martin, 1944)
Girsanov Transformation (Girsanov, 1960)

Some materials on the evalutaion of Wiener Functional and Differential Equations
Evaluation of Wiener Functionals and Sturm-Liouville Differential Equations (Cameron-Martin, 1944)
Connections between Probability and Differential Equations (Kac, 1950)

Some materials on the Clark's Representation Formula
Representation of Wiener Functionals Using Stochastic Integrals (Clack, 1970)
Representation of Functionals of Diffusion Processes and Malliavin's Calculus (Ocone, 1984)
Generalized Clark Representation with Applications to Optimal Portfolios (Ocone, Karatzas, 1991)

Some materials on the theory and applications of boundary crossing of Wiener Process
Boundary Crossing Probabilities for Wiener Processes (H. Robbins, D. Siegmund, 1970)
Statistical Tests and PDEs (H. Robbins, D. Siegmund, 1973)

Some materials on fluctuation Theory
Occupancy Times for Markov Processes(D. A. Darling, M. Kac, 1956)
Fluctuation Theory of Recurrent Events(W. Feller, 1949)

A Survey on Wiener Integral and the Relationship between Wiener Functional and Differential Equations
Koval'chik (1963)

Some materials on Large Deviations
Chernoff: Meaaure of Asymptotic Efficiency (1952)

Some materials on Large Deviations and Small Perturbations of Dynamical Systems
Small Perturbations of Random Dynamics Systesm Friedlin and Wentzell
Excerpt from "Large Deviations and Metastability", by E. Olivier and M. E. Vares
Excerpt from "Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences", by C. W. Gardiner

Information about Final Exam Presentation

PRESETANTATION SCHEDULE

Location: MATH 817

Monday, Apr 28:
9-10:30am (Shan Yang: Introduction to Levy process and its application on the optimal stopping time problem);
10:30am-12:00pm (Shuhao Cao: Criteria for Recurrence and Existence of Invariance Measures for Multidimensional Diffusions);
1:00-2:30pm (Joseph Zedah: Parabolic Anderson Problem and Intermittency);

Tuesday, Apr 29:
9-10:30am;
10:30am-12:00pm (Jung-Pin Chen: Some Remarks on the Smoluchowski-Kramers Approximation);
1:00-2:30pm (Liang Cheng: Levy Processes);