Кто читает:: Международная лаборатория стохастического анализа и его приложений

Статус:: Курс по выбору

Когда читается:: 1-й курс, 2, 3 модуль

Преподаватели

Жабир Жан-Франсуа Мехди

Морено Франко Гарольд Андрес

Full Syllabus

Abstract

This course aims to provide a solid introduction on the conceptual, theoretical and practical aspects of probabilistic numerical methods and the eld of stochastic differential equations (SDEs). A SDE is typically a dynamical system endowing random components that models the evolution over time of particular phenomena subject to uncertainty (for instance the evolution of a nancial asset, risk assessment in insurance policy, . . . ). The course will present the importance of using SDEs to model random phenomenons, from their origin in Physics to their modern applications in Finance, Economy, Machine learning and other eld of Engineering, and surveys in depth the fundamental analytical tools which enables to investigate such models. Along this presentation, general methods to simulate random variables (discrete, real, multivariate), some essential randomized algorithms, and approximation techniques for simulating and investigating fundamental SDEs arising in Finance (e.g. Black and Scholes models, interest rates and bond model) will be reviewed. This course is primarily designed for students possessing a solid background in probability theory and some knowledge and understanding on mathematical modeling, mathematical analysis, differential equations, and computer programming. Although some knowledge on stochastic processes will be useful, part of the course will be dedicated to review/recall the fundamentals of the theory and applications on basic stochastic processes (martingales, Markov processes, Brownian motion) which will be used throughout the course.

Learning Objectives

This course aims to provide a solid introduction on the conceptual, theoretical and practical aspects of numerical methods based on probability and random systems, and the field of stochastic differential equations.

Expected Learning Outcomes

To present and study some elementary models of Stochastic Differential Equations which are used in Finance, Physics, Economy,..
To develop students' ability to apply the knowledge acquired during the course to study and use Stochastic Differential Equations for concrete modeling purposes, recognizing the appropriate frameworks and analytical tools related to these equations.
To introduce the fundamental methods of numerical probability for the approximation of integration calculus, the simulation of given distribution and discrete time stochastic processes.
To provide students with the knowledge of fundamental techniques to analyze the solutions of general SDEs, grounding their explanations on intuitive and analytical approaches.
To provide students with the knowledge of the theoretical, modeling and numerical aspects related to stochastic differential equations.

Course Contents

Introduction to Numerical Probability.
Markov decision Processes.
Stochastic Integration.
Basic Elements of Stochastic Processes.
Introduction to Stochastic Dierential Equations (SDEs) and related properties.
Advanced material.
Discretization and estimation methods for SDEs.

Assessment Elements

Homework, quizzes and work at seminars
Exam

Interim Assessment

2021/2022 3rd module
0.55 * Exam + 0.45 * Homework, quizzes and work at seminars

Bibliography

Recommended Core Bibliography

Introduction to stochastic calculus applied to finance, Lamberton, D. M., 2008

Recommended Additional Bibliography

Stochastic simulation and applications in finance with MATLAB programs, Huynh, H. T., 2008

Магистерская программа «Стохастическое моделирование в экономике и финансах»

Introduction to Stochastic Differential Equations and Numerical Probability