Delivered at:: Laboratory of Stochastic Analysis and its Applications

Course type:: Elective course

When:: 1 year, 3 module

Instructors

Jabir, Jean-Francois M.

Morozova, Ekaterina

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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.
Review the most fundamental simulation techniques (Euler-Maruyama and Milstein schemes) and statistical methods (MLE,QMLE, GMM) related to SDEs from a theoritical and practical point of view.
Present some recents applications of numerical probability and SDEs in modern sciences (notably Economy, climate and artificial intelligence) and advanced methods related to the theoretical and numerical resolution of SDEs

Course Contents

Introduction to Numerical Probability.
Basic Elements of Stochastic Processes.
Stochastic calculus.
Fundamentals on Stochastic Differential Equations (SDEs) and their applications.
Simulation and estimation methods for SDEs.
Advanced topics.

Assessment Elements

Seminar participation
Homeworks
Quizzes
Evaluation will be counted through the student participation to seminars and presentation of solutions to seminar tasks to the class audience
Exam
Exam evaluating the understanding and work of the students at the end of the course

Interim Assessment

2024/2025 3rd module
Final grade = 50% * N1+50%*N2 for N2=Exam grade and N1=max(50% * Homeworks+50% * Quizzes, 40% Homeworks+40% * Quizzes+20% * Seminar Participation)

Bibliography

Recommended Core Bibliography

Brownian motion and stochastic calculus, Karatzas, I., 1998
Damien Lamberton, & Bernard Lapeyre. (2011). Introduction to Stochastic Calculus Applied to Finance: Vol. 2nd ed. Chapman and Hall/CRC.
Gilles Pagès. (2018). Numerical Probability : An Introduction with Applications to Finance (Vol. 1st ed. 2018). Springer.
Ikeda, N., & Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North Holland.
Interest rate models- theory and practice : with smile, inflation and credit, Brigo, D., 2006
Levy processes and stochastic calculus, Applebaum, D., 2009
Partial differential equations for probabilists, Stroock, D. W., 2012
Simulation and inference for stochastic differential equations : with R examples, Iacus, S. M., 2010
Stochastic differential equations : an introduction with applications, Oksendal, B., 1998
Stochastic integration and differential equations, Protter, P. E., 2005
Understanding game theory : introduction to the analysis of many agent systems with competition and cooperation, Kolokoltsov, V. N., 2010

Authors

ZHABIR Zhan-Fransua Mekhdi

Master’s Programme 'Stochastic Modeling in Economics and Finance'

Introduction to Stochastic Differential Equations and Numerical Probability