Mathematical bases for stochastic calculus - 5MMFMCS3

Goal(s)

This course aims to provide the foundations of stochastic calculus. We insist on the mathematical aspects of this calculus, its applications to finance also being developed in other modules. The goal is to familiarize students with the language of stochastic analysis, used extensively in research articles in financial mathematics.

Content(s)

1. Overview of random processes: comparison of processes, C and D spaces, continuous time filtration, sigma-algebra of events prior to a stoping time.
2. Martingale: martingale in continuous time, Doob decomposition, local martingales.
3. Itô's integral: quadratic variation of random processes, integration by bounded martingales, Kunita-Watanabe's inequality, integration by local martingale
4. Itô's calculus: integration by semi-martingales, Itô's processes, Itô's formula, applications : SDE solution.
5. Girsanov's theorems : exponential martingale, martingale measure, applications to nondegenerate Brownian diffusions.
6. Stochastic Differential Equations : strong solutions, Itô's Theorem, examples and counter-examples, Applications: Black and Scholes model, weak solutions of SDE.
7. Diffusions and probabilistic interpretation of PDE: Markov property, gernerators, Feynman-Kac formulas

4MMPSAF Stochastic processes and application in finance (or another equivalent course)