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

______________________________________________________________________Time & Place:W 3:00-5:00 p.m. JWB 308Instructor: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)