Master
2020/2021
Introduction to Stochastic Differential Equations and Numerical Probability
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Elective course (Statistical Modelling and Actuarial Science)
Area of studies:
Economics
Delivered by:
Laboratory of Stochastic Analysis and its Applications
Where:
Faculty of Economic Sciences
When:
2 year, 1, 2 module
Mode of studies:
offline
Open to:
students of all HSE University campuses
Master’s programme:
Statistical Modelling and Actuarial Science
Language:
English
ECTS credits:
5
Contact hours:
60
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
- The main objective of the course is to introduce the fundamental methods of numerical probability for the approximation of integration calculus, the simulation of given distribution and continuous time stochastic processes.
Expected Learning Outcomes
- To introduce the fundamental methods of numerical probability for the approximation of integration calculus, the simulation of given distribution and continuous time stochastic processes.
- To provide students with the knowledge of the theoretical, modeling and numerical aspects related to Stochastic Differential 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
- 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
Course Contents
- Introduction to Numerical Probability- Pseudo-random number generators. - Simulation of random variables: The inverse transform method; the acceptance rejection algorithm; the Box-Muller algorithm. - Limit theorems in Probability and Markov Chain Monte Carlo (MCMC) methods: Markov Chains; state transition matrix; general MCMC methods; numerical simulation.
- Markov Decision Processes- Markov reward processes: return process; value function; Bellman equation. - Markov decision process: polices; value function on a policy; the action value function. - Optimal value function. - Monte Carlo Methods for solving Markov decision problems.
- Basic Elements of Stochastic Processes.- Time continuous Stochastic; Filtration; Stopping Time; Martingales. - The Poisson processes and their main properties. Numerical simulation of Poisson processes. - The Brownian motion and its main properties. Numerical simulation of BMs -Introduction to Lévy processe. - Martingales inequality.
- Stochastic Integration.- Ito's integral. - Ito's formula and its applications.
- Introduction to Stochastic Differential Equations (SDEs) and related properties.- Generic form of a SDE and basic properties. - Comparison between times series, ordinary differential equations and stochastic differential equations. - Construction methods and the principle of causality of a solution to a SDE. - Equations with affine coefficients and applications in Finance and Physics. - More application of SDEs in Finance: The Black and Scholes model and the theory of option pricing. - Modeling interest rates and bonds. - Transformation methods to handle SDEs with general coefficients.
- Discretization and estimation methods for SDEs.- Reminders on numerical methods for dynamical systems. - The Euler-Maruyama scheme and its error analysis. - Efficient estimators and application to the Black and Scholes model. - Stochastic gradient flows and a glimpse to learning processes.
- Advanced material.-Other applications of SDEs in engineering. - The notion of weak solution to a SDE and related properties.
Interim Assessment
- Interim assessment (2 module)0.45 * Grades for homework, quizzes and work at seminars + 0.55 * Score for the exam #1
- Interim assessment (3 module)
Bibliography
Recommended Core Bibliography
- Brownian motion and stochastic calculus, Karatzas, I., 1998
- Ikeda, N., & Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North Holland.
Recommended Additional Bibliography
- Gilles Pagès. (2018). Numerical Probability : An Introduction with Applications to Finance (Vol. 1st ed. 2018). Springer.