Topics in Probability: Stochastic Calculus Math 7880-1, Spring 2008 University of Utah

______________________________________________________________________ Time & Place: W 3:00-5:00 p.m. JWB 308 Instructor: Davar Khoshnevisan JWB 102 ______________________________________________________________________ Course Outline: This is a first course in stochastic calculus and, more generally, infinite-dimensional analysis. Basic concepts: Brownian motion and stochastic integrals; martingale calculus; Itô's formula; stochastic differential equations. Advanced topics (several or possibly all of the following): One-dimensional SDEs; diffusions and their connections to PDEs; examples from interacting particle systems; stochastic differential equations in Hilbert spaces (aka stoch. PDEs). This is a vast subject of great mathematical beauty. Also, it has deep and diverse applications throughout mathematics, as well as other theoretical sciences. Throughout the course we shall draw a few central examples from real analysis (potential theory and PDEs); mathematical finance (option pricing); evolutionary biology (population dynamics); and statistical mechanics (random matrices). ______________________________________________________________________ Suggested Reading: -Chung, K.L. and Williams R.J. (*) Introduction to Stochastic Integration, Birkhäuser (1990). -Itô, K. (*) Stochastic Processes (Aarhus university lectures), Springer Verlag (2004). -Karatzas, I. & Shreve, S.E. (*) Brownian Motion and Stochastic Calculus, Springer Verlag (2004). -McKean, H.P. (*) Stochastic Integrals, Reprinted by AMS Chelsea publishing (2005). -Øksendal, B. (*) Stochastic Differential Equations, Springer Verlag (2007). -Revuz, D. & Yor, M. (*) Continuous Martingales and Brownian Motion, Springer Verlag (2004). Basic Probability Texts (background): -Durrett, R. (*) Probability: Theory and Examples, Duxbury Press (2004). -Karlin, S. & Taylor, H.M. (*) A First Course in Stochastic Processes, Academic Press (1975). -Khoshnevisan, D. (*) Probability, American Mathematical Society (2007). -Williams, D. (*) Probability with Martingales, Cambridge University Press (1991). Lecture Notes: Made available as we proceed. Reading for the advanced topics: -Bass, R.F.: (*) Probabilistic Techniques in Analysis, Springer Verlag (1994). (*) Diffusions and Elliptic Operators, Springer Verlag (1997). -Da Prato, G. & Zabczyk, J. (*) Stochastic Equations in Infinite Dimensions, Cambridge University Press (1992). -Karlin, S. & Taylor, H.M. (*) A Second Course in Stochastic Processes, Academic Press (1981). -Kurtz, T.G. & Ethier, S.N. (*) Markov Processes, Wiley (1986). -Walsh, J.B. (*) An Introduction to Stochastic Partial Differential Equations, Springer Verlag (1984). ______________________________________________________________________ Recommended Prerequisites: Math 6040 and Math 6210. Some knowledge of elementary stochastic processes is helpful. ______________________________________________________________________ Grading: Based on weekly assignments. ______________________________________________________________________ Announcements: The whole ball of wax (Last update: April 9, 2008; warning: Ch. 4 is highly incomplete, & will remain that way)