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 9, 9:45-11:45.

Detailed planning

18.09 - stochastic processes, Brownian motion, martingales
21.09 - quadratic variation, local martingales
25.09 - isometric extension, Wiener's integral
28.09 - progressive processes, Ito's integral, generalized Ito integral
02.10 - Itô processes, quadratic variation, integration-by-parts formula
05.10 - Itô's formula, examples
09.10 - exponential martingales, Novikov's criterion, Girsanov's theorem
12.10 - introduction to stochastic differential equations
16.10 - examples of stochastic differential equations, with explicit resolution
19.10 - 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 2023 and solution
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 - 2023)

Course description

Practical information

Detailed planning

Documents

Bibliography