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
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
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
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
Harrison, J. M. & Kreps, D. M. (1979), `Martingales and arbitrage in
multiperiod securities markets', Journal of Economic Theory
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,
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
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.