2026/2027
Введение в стохастические дифференциальные уравнения и числовую вероятность
Язык:
русский
Кредиты:
3
Контактные часы:
54
Программа дисциплины
Аннотация
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.
Цель освоения дисциплины
- 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.
Планируемые результаты обучения
- 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.
- 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.
- To introduce the fundamentals of probabilistic numerical methods and illustrate the interest of such methods in integral calculus, the simulation of multivariate random variable and basic discrete time stochastic processes.
- To explain and apply the foundational concepts of stochastic analysis including: - describe and discuss the main theoretical aspects of primary stochastic processes (Brownian motion, jump processes, martingales) and their applications in Finance and other fields of sicence (physics and engineering) - examine and assess the fundamental of Itô's integration and Itô's calculus.
- Describe and analyse (briefly) some advanced concepts of SDEs (notably elemental links with partial differential equations, and the notion of weak solutions and related resolutions methods) and some recents applications of SDEs in the modeling of very-large dimensioned systems.
Содержание учебной дисциплины
- Fundamental of Numerical Probability
- Introduction to Stochastic Analysis
- Fundamentals on Stochastic Differential Equations (SDEs) and their applications.
- Simulation and estimation methods for SDEs.
- Advanced Topics in SDEs and Numerical Probability
Элементы контроля
- Seminar assessmentFor each seminar, students will be tasked to solve selected problems from the seminar worksheets and present their solutions to the class. Assessment will be based on the accuracy of the solutions, the clarity of the presentation, the quality of group discussion and of the student's knowledge of the course content.
- Quiz testsTwo quiz tests of a duration of 1h20 will be organized during the module. Each test will assessing student's understanding of a specific content of the course.
- Exam
Промежуточная аттестация
- 2026/2027 3rd moduleFinal Grade = 50% * N1+50%*N2 for N1=max(Average of the two Quizzes; 60% * Average of Quizzes+40% * Seminar assessment) and N2=Grade of Exam.
Список литературы
Рекомендуемая основная литература
- Bernt Øksendal. (2010). Stochastic Differential Equations : An Introduction with Applications (Vol. 6th ed. 2003). Springer.
- 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.
- Introduction to stochastic calculus applied to finance, Lamberton, D. M., 2008
- Monte Carlo methods in finance, Jackel, P., 2002
- Monte Carlo methods in financial engineering, Glasserman, P., 2004
- Numerical solution of stochastic differential equations, Kloeden, P.E., 1999
- Partial differential equations for probabilists, Stroock, D. W., 2012
- Stochastic differential equations : an introduction with applications, Oksendal, B., 1998
- Stochastic integration and differential equations, Protter, P. E., 2005