Here is material I wrote for a course on stochastic analysis at UW-Madison in Fall 2003. The intention is to provide a stepping stone to deeper books such as Protter's monograph. Hopefully this text is accessible to students who do not have an ideal background in analysis and probability theory, and useful for instructors who (like me) are not experts on stochastic analysis. The PDF file of the text is here (currently almost 400 pages, last updated Fall semester 2014). Comments are welcome.
The text covers the development of the stochastic integral of predictable processes with respect to cadlag semimartingale integrators, Itô's formula in an open domain in R^n, and an existence and uniqueness theorem for an equation of the type dX=dH+F(t,X)dY where Y is a cadlag semimartingale.
The text is self-contained except for some basics of analysis
and probability that are explained but not proved. Also,
the reader needs to accept without proof these
two martingale theorems:
(1) the existence of quadratic variation for a cadlag local martingale;
(2) the so-called fundamental theorem of local martingales that states the following: given a cadlag local martingale M and a positive constant c, M can be decomposed as N+A where N and A are cadlag local martingales, jumps of N are bounded by c, and A has paths of finite variation.
The author has been partially supported by the National Science Foundation and by the Wisconsin Alumni Research Foundation during the preparation of this manuscript.