Any suggestions on a rigorous stochastic differential equations book? I have been looking through some books and they are not very rigorous. Any suggestions would be great.
 A: Karatzas and Shreve 
Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics) (Volume 113)
http://www.amazon.com/gp/aw/d/0387976558
A: There are a lot of good books on the market, maybe you should describe more carefully what you want...
I think the following books don't lack rigor


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*C. Dellacherie and P.A. Meyer, Probabilités et Potentiel. Ch. I à IV. Hermann, Paris 1975, 290 pages

*C. Dellacherie and P.A. Meyer, Probabilités et Potentiel. Ch. V à VIII, Hermann, Paris 1980, 476 pages

*C. Dellacherie and P.A. Meyer, Probabilités et Potentiel. Ch IX à XI. Théorie discrète du potentiel. Hermann, Paris 1983, 229 pages

*C. Dellacherie and P.A. Meyer, Probabilités et Potentiel. Ch. XII à XVI. Théorie du potentiel associée à une résolvante, théorie des processus de Markov Hermann, Paris 1987, 377 pages

*C. Dellacherie, B. Maisonneuve and P.A. Meyer, Probabilités et Potentiel. Ch XVII à XXIV. Processus de Markov (fin). Compléments de calcul stochastique. Hermann, Paris 1992, 429 pages
see http://lmrs.univ-rouen.fr/Ouvrages/potentiel.html for details.
Or, if you want something more "modern", there are for example


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*D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004

*P. Protter. Stochastic integration and differential equations, volume 21 of
Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2005

*N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, 1989

*G. Di Nunno, B. Øksendal, and F. Proske. Malliavin Calculus for Levy Processes with Applications to Finance. Universitext. Springer-Verlag, Berlin, 2009.
