Stochastic Calculus - Long course - SFA (ENSTA)

Objectif

After the course, we expect that the student has the capabilities to work independently on a research project in stochastic analysis, in view of practical applications to finance.

Plan

The course includes the following chapters.

1. Motivations: stochastic modeling, probabilistic representations of linear PDEs, stochastic control, filtering, mathematical finance.
2. Stochastic processes in continuous time: Gaussian processes, Brownian motion, (local) martingales, semimartingales, Itˆo processes.
3. Stochastic integrals: forward and Itˆo integrals.

4. Itˆo and chain rule formulae, a first approach to stochastic differential equations.
5. Girsanov formulae. Novikov and Benˆes coondition. Predictable representation of Brownian martingales.
6. Stochastic differential equations with Lipschitz coefficients. Markov flows.
7. Stochastic differential equations without Lipschitz coefficients: strong existence, pathwise uniqueness, existence and uniqueness in law. Engelbert-Schmidt criterion. Non-explosion conditions
8. Bessel processes and Cox-Ingersoll-Ross model in mathematical finance.
9. Backward stochastic differential equations and connections with semilinear PDEs.

Références

Close related references to the course are the following monographs and articles: [2, 6, 3, 5]. For deeper considerations, we also refer to [7, 8, 1, 4].

[1] Jean Jacod. Calcul stochastique et probl`emes de martingales, volume 714 of Lecture Notes in Mathematics. Springer, Berlin, 1979.
[2] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.
[3] D. Lamberton and B. Lapeyre. Introduction au calcul stochastique appliqué à la finance. Ellipses ´Edition Marketing, Paris, second edition, 1997.
[4] David Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, New York, 1995.
[5] E. Pardoux. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Stochastic analysis and related topics, VI (Geilo, 1996), volume 42 of Progr. Probab., pages 79–127. Birkh¨auser Boston, Boston, MA, 1998.

[6] D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999.
[7] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itˆo calculus, Reprint of the second (1994) edition.
[8] Daniel W. Stroock and S. R. Srinivasa Varadhan. Multidimensional diffusion processes. Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1997 edition