Mathematics 16:642:611 Selected Topics in Applied Mathematics: Variational Inequalities, Obstacle and Free Boundary Problems in Mathematical Finance

Instructor

Paul Feehan

Questions about the course?

Please feel welcome to contact me!

Schedule

The course will be offered during the Fall 2011 semester (Tuesdays and Fridays 10:20-11:40 in Hill 425, Wednesdays 10:20-11:40 in SEC 204). The Friday meeting times are reserved for make-up classes, optional problem sessions, and guest lectures. The Tuesday meeting time may be moved slightly earlier (10:00-11:20) to avoid conflict with a seminar later that morning.

Course summary

The goal of the course is to introduce graduate students to the theory required to do research on variational inequalities, obstacle problems, and free boundary problems and their applications to pure and applied mathematics. To accomplish this goal, we plan to cover the following topics:
  • Existence, uniqueness, and global regularity results for solutions to elliptic obstacle problems (stationary variational inequalities).
  • Optimal regularity of solutions to elliptic obstacle problems near the free boundary.
  • Regularity of the free boundary for solutions to elliptic obstacle problems.
  • Existence, uniqueness, and global regularity results for solutions to parabolic obstacle problems (evolutionary variational inequalities).
  • Optimal regularity of solutions to parabolic obstacle problems near the free boundary.
  • Regularity of the free boundary for solutions to parabolic obstacle problems.
  • Weighted Sobolev spaces and applications to degenerate elliptic and parabolic partial differential equations and obstacle problems.
  • Applications: sample problems will be drawn from mathematical finance, through additional problems for other applications will be considered depending on the interests of course participants.

Optional third weekly meeting

The optional third weekly meeting times are reserved for make-up classes, problem sessions, and guest lectures. Examples of guest lectures may include
  • Presentations by other faculty members on topics related to the course. Examples include numerical solution of variational inequalities, degenerate partial differential equations and weighted Sobolev spaces, specific obstacle problems in applied mathematics, relationship with probability theory and stochastic processes, more applications to mathematical finance, among others.
  • Presentations by students on topics related to the course. Examples include thesis research problems or solutions to homework problems (all homework assignments are optional).

Audience

Second and higher-year doctoral students; second-year master's students with required mathematics background; junior faculty members.

Pre-requisites

A one-semester graduate level course on real analysis (for example, Math 16:640:501) covering measure theory, Hilbert spaces, and Banach spaces. An undergraduate course on real analysis (for example, Math 01:640:411-412) may suffice for motivated graduate students.

The following short text is recommended reading prior to the course for anyone who has not taken a graduate level class on partial differential equations:
  • Q. Han and F. Lin, Elliptic partial differential equations, Courant Lecture Notes, American Mathematical Society, Providence, RI, 2011.

Co-requisites

A one-semester graduate level course on partial differential equations (for example, Math 16:640:517 or a similar course based on the text by Evans) is recommended, but not required. The course will be self-contained in order to accommodate beginning second-year students who wish to explore potential research topics in partial differential equations and their applications.

Non-requisites

  • Do I need to know anything about probability theory or stochastic processes? No! I will provide references and additional problems to students interested in applications to probability theory and Markov processes, but probability theory and stochastic processes will not be covered in the course. Students interested in probability theory and stochastic processes should consider also taking Math 16:642:591 Topics in Probability and Ergodic Theory I.
  • Do I need to know anything about mathematical finance? No! I will provide references and additional problems to students interested in applications to mathematical finance (and other applications), but mathematical finance itself will not be covered in the course.
  • Should I have taken an undergraduate level course on partial differential equations? While prior exposure to partial differential equations through a course such as Math 01:640:423 has some marginal benefit, that is not required.

Textbooks

  1. A. Petrosyan, H. Shahgholian, N. Ural'tseva, Regularity of free boundaries in obstacle type problems, publication expected in 2011.
  2. J-F. Rodrigues, Obstacle problems in mathematical physics , North-Holland, New York, 1987.

Primary reference texts

  1. A. Bensoussan and J. L. Lions, Applications of variational inequalities in stochastic control, North-Holland, New York, 1982.
  2. L. Caffarelli and S. Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68, American Mathematical Society, Providence, RI, 2005.
  3. A. Friedman, Variational principles and free boundary problems, Dover, New York, 2010.
  4. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic, New York, 1980.
  5. V. I. Maz'ya, Sobolev spaces: with applications to elliptic partial differential equations, Springer, New York, 2011.
  6. G. M. Troianiello, Elliptic differential equations and obstacle problems, Plenum, New York, 1987.

Supplementary reference texts

  1. L. C. Evans, Partial differential equations, second edition, American Mathematical Society, Providence, RI, 2010.
  2. A. Friedman, Partial differential equations, Dover, New York, 2008.
  3. A. Friedman, Partial differential equations of parabolic type, Dover, New York, 2008.
  4. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, New York, 2001.
  5. Q. Han and F. Lin, Elliptic partial differential equations, Courant Lecture Notes, American Mathematical Society, Providence, RI, 2011.
  6. N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996.
  7. N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008.
  8. G. M. Lieberman, Second order parabolic differential equations, World Scientific, River Edge, NJ, 1996.
  9. R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, American Mathematical Society, Providence, RI, 1996. (Free e-copy from AMS).

Reference articles

Research articles relevant to the course will be provided, including those by the following authors:
  • L. Caffarelli and collaborators.
  • P. Daskalopoulos and collaborators.
  • P. Feehan and collaborators.
  • P. Laurence and collaborators.
  • H. Shahgholian and collaborators.

Grading

The course is intended for second and higher-year graduate students primarily interested in research in partial differential equations and applications, so there will be no exams or required homework. Optional weekly written homework and reading assignments will be provided and homework assignments may be discussed during the optional third weekly meeting periods.

Syllabus, lecture notes, homework problems, and reading assignments

The weekly syllabus, readings, homework problems, lecture notes and research articles will be provided on Sakai to registered participants.