Statistics 603b
Advanced Stochastic processes

I find Chung & Williams (1990) a good overall reference to stochastic calculus, with the more thorough Metivier (1982) better on predictability and Doob-Meyer.

Dellacherie & Meyer (1982) is the ultimate guide to any technical quesion about stochastic integrals and the ``theory of processes", but it makes very difficult reading. I have often discovered the answers to puzzling questions buried somewhere in D&M.

Andersen, P. K., Borgan, Ø, Gill, R. D. & Keiding, N. (1993), Statistical Models Based on Counting Processes, Springer-Verlag, New York.
Billingsley, P. (1999), Convergence of Probability Measures, Wiley.
I own the first edition, from 1968, but I am tempted to get 1999 edition as well. The original was the classic, and best, reference for the theory of convergence in distribution in C[0,1] and D[0,1]. A great book.
Chung, K. L. & Williams, R. J. (1990), Introduction to Stochastic Integration, Birkhäuser, Boston.
Crisp mathematical style. Similar coverage to 603 except for the restriction to continuous sample paths. Lots of mathematical detail, but not as overwhelming as some books at the same technical level.
Dellacherie, C. & Meyer, P. A. (1978), Probabilities and Potential A, North-Holland, Amsterdam.
Covers the hard modern theory of stochastic processes. Analytic sets, section theorems, Choquet capacity, and much more.
Dellacherie, C. & Meyer, P. A. (1982), Probabilities and Potential B: Theory of Martingales, North-Holland, Amsterdam.
Understand this book and you know it all. Difficult. I treat it as the ultimate source.
Durrett, R. (1984), Brownian Motion and Martingales in Analysis, Wadsworth, Belmont CA.
A good quick introduction to stochastic integrals, with a few details omitted. Earlier incarnation of Stochastic Calculus: A Practical Introduction (CRC Press 1996).
Hall, P. & Heyde, C. C. (1980), Martingale Limit Theory and Its Application, Academic Press, New York, NY.
Limit theory for martingales in discrete time. Good source for martingale CLT.
Harrison, J. M. & Kreps, D. M. (1979), `Martingales and arbitrage in multiperiod securities markets', Journal of Economic Theory 20, 381--408.
Harrison, J. M. & Pliska, S. R. (1981), `Martingales and stochastic integrals in the theory of continuous trading', Stochastic Processes and their Applications 11, 215--260.
Readable account explaining why folks interested in finance should know about semimartingales.
Métivier, M. (1982), Semimartingales: A Course on Stochastic Processes, De Gruyter, Berlin.
More thorough than Chung & Williams, and a lot easier than Dellacherie & Meyer B. Assumes the reader is very comfortable with measure theory. One of my favourites.
Neveu, J. (1975), Discrete-Parameter Martingales, North-Holland, Amsterdam.
An excellent reference for martingales in discrete time.
Øksendal (1998), Stochastic Differential Equations: An Introduction with Applications, Springer.
Covers a lot of ground by skipping over the harder mathematical details. Now into a sixth edition, I believe.
Pollard (1984), Convergence of Stochastic Processes, Springer.
Slightly outdated, but it's free. Look at: Chapter 4 for convergence in distribution in C[0,1]; Chapter 8 for Martingales in continuous time; and Appendix C for some fancy theory about analytic sets and measurability.
Pollard (2002), A User's Guide to Measure Theoretic Probability, Cambridge University Press.
The text for Stat 330/600. Look at: Chapter 7 for convergence in distribution in metric spaces; Chapter 9 for Brownian motion, including a first-principles account of option pricing; Appendix E for Martingales in continuous time (material that started out as part of a course on stochastic calculus that I taught in Fall1995).
Protter, P. (1990), Stochastic Integration and Differential Equations, Springer, New York.
Neat, often self-contained, proofs, but his treatment is unusual enough to make the book poorly suited to random access. Protter starts from the concept of semimartingales defined as processes for which simple stochastic integrals have desirable properties, then works towards the usual definition.
Revuz, D. & Yor, M. (1991), Continuous Martingales and Brownian Motion, Springer, New York.
Steele, J. M. (2001), Stochastic Calculus and Financial Applications, Springer.
Witty and elegant. Very readable. Covers Itô integrals wrt Brownian motion.
Stroock, D. W. & Varadhan, S. R. S. (1979), Multidimensional Diffusion Processes, Springer, New York.
Develops enough of the theory to cover diffusions. I learned a lot from this book. Solid classical probability.