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Informatique et Mathématiques appliquées

# Mathematical bases for stochastic calculus - 5MMFMCS

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• #### Number of hours

• Lectures : 18.0
• Tutorials : -
• Laboratory works : -
• Projects : -
• Internship : -
• Written tests : -
ECTS : 2.0
• Officials : Pierre ETORE

### Goals

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

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

Prerequisites

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

Tests

N1=(2/3)E1+(1/3)CC
N2=E2
N=Max(N1,N2)

CC = Contrôle continu,
E1 = examen écrit,
E2 = examen écrit (session de rattrapage)

Calendar

The course exists in the following branches:

• Curriculum - Financial Engineering - Semester 9
see the course schedule for 2020-2021

Course ID : 5MMFMCS
Course language(s):

The course is attached to the following structures:

You can find this course among all other courses.

Bibliography

I. Karatzas, S.E. Shreve "Brownian motion and stochastic calculus", Springer

D. Revuz, M. Yor "Continuous martingales and Brownian Motion", Springer

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École nationale supérieure d'informatique et de mathématiques appliquées
681, rue de la passerelle - Domaine universitaire - BP 72
38402 SAINT MARTIN D'HERES