What are some easier books for studying martingale?

Question

What are some easier books for studying martingale?

They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales?

It should cover continuous-time martingale, stochastic integrals and most other basic topics for martingle

Answer 1

I think it's pretty hard to find a book which covers martingale theory; usually, books either give just an introduction or they focus on one particular aspect of martingale theory. I'll list some books which might be of interest and sketch (roughly) which parts they cover:

• David Williams: Probability with Martingales (Basic properties, optional stopping, convergence theorems, strong law of large numbers, uniform integrability, Radon-Nikodým & Kakutani theorem)
• René Schilling: Measures, Integrals and Martingales (Basic properties of discrete-time martingales; optional stopping, convergence theorems, uniform integrability, Radon-Nikodým; some martingale inequalities). As a side remark: There are full solutions to all exercises on the web.
• Daniel Revuz, Marc Yor: Continuous martingales and Brownian motion (Maximal inequalities, convergence theorems, optional stopping, quadratic variation, stochastic integrals, representation theorems)
• P.E. Kopp: Martingales and stochastic integrals (discrete-time and continuous time martingales, convergence theorems, decomposition theorems, optional stopping, Doob-Meyer decomppsition, stochastic integration)
• Robert Liptser, Albert Shiryaev: Statistics of Random Proceses I (discrete-time and continuous-time martingales, Doob-Meyer decomposition, stochastic integration, representation theorems)
• Stewart N. Ethier, Thomas G. Kurtz: Markov Processes (discrete-time and continuous-time martingales, local martingales, Doob-Meyer decomposition, quadratic variation, and some more advanced topics such as the martingale problem and the martingale central limit theorem)

And for the sake of completeness (... since I think that the book is comprehensive, but hardly an easy read):

• Robert Liptser, Albert Shiryaev: Theory of Martingales.

Answer 2

Samual Cohen, Robert Elliott: Stochastic Calculus and Applications

Nice book to start with. It begins with Measure Theoretic Probability, then considers Discrete and Continuous time Martingales (Optional Stopping, Inequalities, Convergence), proceeds to Stochastic Integration, and, finally, ends with Stochastic DE.