There is no one text book for the course. Lectures will be self-contained, and handouts will be available on some parts of the course. However, the course will be based mainly on the following texts, and we will refer to them regularly. These books are on reserve in the library. The second book mentioned (Probability: Theory and Examples by Durrett) was used as a textbook in Math850 and will be used as a reference for the broader background material.


• Main literature.
  "Stochastic Differential Equations: An Introduction with Applications", B. Oksendal, Universitext, Springer.
  "Probability: Theory and Examples", R. Durrett, Duxbury Press.
  "Introduction to stochastic integration", K.L. Chung and R.J. Williams, Birkhäuser.
  "Brownian Motion and Stochastic Calculus", I. Karatzas and S. E. Shreve, Springer.
  "Continuous martingales and Brownian motion", D. Revuz and M. Yor, Springer.
  


• Additional reading.
  "Introduction to Stochastic Calculus with Applications", F. Klebaner, Imperial College press.
  "Stochastic Integration and Differential Equations", P. E. Protter, Springer.
  "Stochastic Calculus and Financial Applications", J.M. Steele, Springer.
  "Stochastic Calculus - A practical introduction", R. Durrett, CRC Press.
  "Diffusions, Markov Processes, and Martingales", Volume I, L. C. G. Rogers and D. Williams Cambridge University Press.
  "Diffusions, Markov Processes, and Martingales", Volume II, L. C. G. Rogers and D. Williams Cambridge University Press.
  


• Academic Honesty.   All students must be honest and forthright in their academic studies. Students are expected to do their own work and neither give nor receive unauthorized assistance. Any violation of this standard must be reported to the Office of Judicial Affairs. Know the University’s Policy on Academic Dishonesty found in the Student Guide to University Policies. Copies of it may be obtained on the Web at http://www.udel.edu/stuguide/07-08, in the University Advisement Center (102 Elliott Hall ), or in the Office of Judicial Affairs (218 Hullihen Hall). For all other administrative questions/problems see your home department or the Office of Campus Life.

Syllabus

Discrete and continuous martingales, stopping times, martingale inequalities, martingale convergence theorems, optional stopping theorem, ergodic theory, Brownian motion, construction, strong Markov property, sample path properties, stochastic (Ito) integration with respect to Brownian motion and martingales, It\^o formula, martingale representation theorem, Girsanov theorem, stochastic differential equations, diffusion processes, applications, for example to analysis (PDE's), engineering (filtering) and finance (Black-Scholes).

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Prerequisites

The course will focus on the probabilistic theory but is intended to appeal to a broad audience: Mathematics graduate students with a pure and applied focus as well as engineering and finance graduate students. The optimal preparation for the course is Math850 (Theory of Probability), an introductory measure theoretic course in probability theory. But the course could also be followed by students who have taken Math630 and Math631. Some elementary knowledge of measure theory (that could also be acquired just prior or parallel to the beginning of the course) would be necessary.

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Grading

There will be one midterm and one final exam (which may be take-home exams or projects). The midterm exam is tentatively scheduled for Tuesday, March 27 (Class 14). If you miss the midterm exam due to legal reasons (you will need proof for this) the final exam will count for the midterm as well. The date of the final exam will be announced later in the semester. You will need to write the final exam for successful completion of the course. Problems on written exams will be on a similar level as the simper examples done in class and homework problems while take home exams will contain some problems on the level of the more challenging examples and homework problems. As for the solutions to your problem sets, clear explanations of your steps and a consistent argument will be essential for full credit.

Grading: Tentatively, graded homework 50%, midterm 20%, final exam 30% of the course grade.

Assignments

Homework will be assigned Thursdays of most weeks. Your solutions are due the following Thursday. A subset of the problems will be corrected and graded. Full credit will only be given for solutions that include clear explanations of your steps and a consistent argument. I will require you to write the solutions out neatly and formally as well as notationally correct. Barring some exceptions like serious illness late homework will not be accepted. While I encourage you to work together and discuss the material and homework, the solutions to the problem sets that you hand in will have to be your own, in your own words.

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