Topics in Probability: Stochastic Calculus
Math 7880-1, Spring 2008
University of Utah
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Time & Place:	W 3:00-5:00 p.m.	JWB 308
Instructor:	Davar Khoshnevisan	JWB 102
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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).
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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).
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Recommended Prerequisites:
Math 6040 and Math 6210. Some knowledge of elementary stochastic processes is helpful.
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Grading:
Based on weekly assignments.
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Announcements:
The whole ball of wax (Last update: April 9, 2008;
	warning: Ch. 4 is highly incomplete, & will remain that way)