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Continuous martingales and Brownian motion Daniel Revuz, Marc Yor
Publisher: Springer

Diffusions, Markov Processes, and Martingales: Volume 1. Watanabe : Stochastic differential equations and diffusion processes. Yor : Continuous martingales and Brownian motion. In this book, which is basically self-contained, the following topics are treated thoroughly: Brownian motion as a Gaussian process, Brownian motion as a Markov process, and Brownian motion as a Continuous Distributions - Probability Examples c-6 Related topics which are treated include Markov chains, renewal theory, the martingale problem, Itô calculus, cylindrical measures, and ergodic theory. Probability and its Applications Continuous martingales and brownian motion Continuous martingales and brownian motion,D. The martingale representation theorem states that any martingale adapted with respect to a Brownian motion can be expressed as a stochastic integral with respect to the same Brownian motion. Whence, the entire theory of stochastic calculus is built around brownian motion. North Holland (Second edition, 1988). Let N_t=e^{i\lambda M_t +\frac{1}{ . Of facts and formulae associated Brownian motion. Moreover, every continuous martingale is just brownian motion with a different clock. Then, to get a solid background in SDE's you can read Revuz, Yor "Continuous Martingales and Brownian Motion" which is more or a less the standard stoch calc book for pure mathematicians. Hm, it's covered in Yor's book "Continuous martingales and brownian motion" but only as an exercise, I also believe it's present in "Aspects of brownian motion" but I don't have access to this book as of now. [ReYo98] D.Revuz, M.Yor, Continuous Martingales and Brownian Motion, Grundlehren der mathematischen Wissenschaften, 3rd edition, Springer, 1998. Amazon.com: Handbook of Brownian Motion - Facts and Formulae. Be a continuous local martingale such that M_0=0 and such that for every t \ge 0 , \langle M \rangle_t =t . The process (M_t)_{t \ge 0} is a standard Brownian motion. Continuous martingales and Brownian motion, Revuz D., Yor M. Continuous Martingales and Brownian Motion (Grundlehren Der Mathematischen Wissenschaften, Vol 293).