Part I: Martingales - Quasimartingales - Semimartingales -- Chapter 1: Basic notions on stochastic processes -- 1. Stochastic basic - Stochastic processes -- 2. Examples and construction of stochastic processes -- 3. Well-measurable (or optional) and predictable processes -- 4. Stopping times -- 5. The σ-algebras FT and FT -- 6. Admissible measures -- 7. Decomposition theorems for stopping times -- Chapter 2: Martingales and quasimartingales - Basic inequalities and convergence theorem - Application to stochastic algorithms -- 8. Martingales, submartingales, supermartingales, quasimartingales: elementary properties -- 9. Doob's inequalities for real quasimartingales and the almost sure convergence theorem -- 10. Uniform integrability - Convergence in LP - Regularity properties of trajectories -- 11. Convergence theorems for vector-valued quasimartingales -- 12. A typical application of quasimartingale convergence theorems: convergence of stochastic algorithms -- Chapter 3: Quasimartingales form class [L. D] - Predictable and dual predictable projection of processes -- 13. Doleans measure of an [L. D] - quasimartingale -- 14. Predictable projection of a process and dual predictable projection of an admissible measure -- 15. The predictable F.V. process of an admissible measure on A and the Doob-Meyer decomposition of a quasimartingale -- Chapter 4: Square integrable Martingales and semimartingales -- 16. Spaces of real L2-martingales -- 17. The first increasing process and orthogonality of L2-martingales -- 18. TheL2-stochastic integral and the quadratic variation of an L2-martingale -- 19. Stopped martingales. Inequalities -- 20. Spaces of Hibert valued martingales -- 21. The process ≪M≫ of a square integrable Hilbert-valued martingale -- 22. The isometric stochastic integral with respect to Hilbert-valued martingales.

23. Localisation of processes and semimartingales -- Part II: Stochastic Calculus -- Chapter 5: Stochastic integral with respect to semimartingales and the transformation formula -- 24. Stochastic integral in the real case -- 25. Quadratic variation and the transformation theorem -- 26. Stochastic integral with respect to multidimensional semimartingales and tensor quadratic variation -- 27. The transformation formula in the multidimensional case -- Chapter 6: First applications of the transformation theorem -- 28. Characterizations of Brownian and Poisson processes -- 29. Exponential formulas and linear stochastic differential equations -- 30. Absolutely continuous changes of probablity -- Chapter 7: Random measures and local characteristics of a semimartingale -- 31. Stochastic integral with respect to a white random measures -- 32. Local characteristics of a semimartingale-Diffusions-Martingale problems -- Chapter 8: Stochastic differential equations -- 33. Examples of stochastic equations - Definitions -- 34. Strong solutions under Lipschitz hypotheses -- 35. Conditions for non-explosion -- 36. Pathwise regularity of solutions of equations depending on a parameter -- 37. Weak solutions of some stochastic differential equations -- Bibliography -- Index of notation -- Index.