Stochastic Calculus

Course description

This course is a practical introduction to the theory of stochastic calculus, with an emphasis on examples and applications rather than abstract subtleties. It consists of four parts:

Preliminaries
Stochastic integration
Stochastic differentiation
Stochastic differential equations

Practical information

The lectures take place on Mondays, 13:45-15:15 and Thursdays, 8:30-11:45.
The exercise sessions take place on Mondays, 15:30-18:45.
The final exam will take place on November 3, 8:45-11:45.

Detailed planning

19.09 - stochastic processes, Brownian motion, martingales
22.09 - quadratic variation, local martingales
26.09 - isometric extension, Wiener's integral
29.09 - progressive processes, Ito's integral, generalized Ito integral
03.10 - Itô processes, quadratic variation and quadratic covariation
06.10 - integration-by-parts formula, Itô's formula, examples
10.10 - exponential martingales, Novikov's criterion
13.10 - Girsanov's theorem, introduction to stochastic differential equations
17.10 - examples of stochastic differential equations, with explicit resolution
20.10 - Markov property, semi-group, generator, Kolmogorov equations, Fokker-Planck equation, Feynman-Kac's formula

Documents

Lecture notes (in construction, comments welcome!)
Exercises (by Julien Poisat)
Exam 2022 and solution
Exam 2021 and solution
Former lecture notes (by Djalil Chafaï)

Bibliography

Calcul stochastique et modèles de diffusions (F. Comets, T. Meyre)
Calcul stochastique et problèmes de martingales (J. Jacod)
Brownian motion and stochastic processes (I. Karatzas, S.E. Shreve)
An introduction to stochastic differential equations (L.C. Evans)
Stochastic calculus (R. Durrett)

Stochastic calculus (M2 - 2022)

Course description

Practical information

Detailed planning

Documents

Bibliography