2025/2026

Introduction to Stochastic Differential Equations and Numerical Probability

Type: Mago-Lego

When: 3 module

Open to: students of all HSE University campuses

Instructors: Jean-Francois Mehdi Jabir

Language: English

ECTS credits: 3

Contact hours: 54

Course 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 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 provide students with the knowledge of fundamental techniques to analyze the solutions of general SDEs, grounding their explanations on intuitive and analytical approaches.
• 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.

Course Contents

• Fundamental of Numerical Probability
• Basic Elements of Stochastic Processes.
• Stochastic Calculus
• Fundamentals on Stochastic Differential Equations (SDEs) and their applications.
• Simulation and estimation methods for SDEs.

Assessment Elements

• Seminar participation
  This control will evaluate students detailling solutions, on the board, of exercises to the seminar classes (quality of presentation and following discussions on the solution methods will be specifically considered).
• Quizzes
• Exam
  Final test of the course evaluating the student understanding of the lectures and seminars contents.

Interim Assessment

• 2025/2026 3rd module
  Final Grade = 50% * N1+50%*N2 for N2=Exam grade and N1=max(Average of Quizzes; 70% * Average of Quizzes+30% * 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
• 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

Authors

• Zhabir Zhan-Fransua Mekhdi