Numerical methods for stochastic differential equations (SDEs)
Prerequisiti
The course is designed for students who have had a first introduction to the concepts and techniques of numerical calculation of ODEs, ODE analysis, and probability calculus.
Year of studies recommended: IV-V year and PhD students.
Programma
Part I, Fundamentals of Stochastic Calculus:
- Random Variables and Stochastic Processes
- Quadratic Variation
- Martingale
- Markov Processes
- Ito and Stratonovich Integrals
- SDEs
- Kolmogorov Equations
Part II, Numerical Methods for SDEs:
- Discretization of Brownian Motion
- Numerical Analysis of Ito and Stratonovich Integrals
- Stochastic One-Step Methods (Euler-Maruyama, Milstein, Runge-Kutta)
- Analysis of One-Step Methods
- Linear Stability Analysis
- Principles of Stochastic Geometric Numerical Integration
Obiettivi formativi
The course aims to provide students with the basic tools for the numerical analysis of SDEs. Topics will be covered from both a theoretical and algorithmic perspective.
Riferimenti bibliografici
- Numerical Solution of Stochastic Differential Equations (Stochastic Modelling and Applied Probability (23)), Springer, Stochastic Modelling and Applied Probability, Corrected, 1995, Peter E. Kloeden, Eckhard Platen
- Numerical Approximation of Ordinary Differential Problems. From Deterministic to Stochastic Numerical Methods, Springer, 2023, Raffaele D'Ambrosio
- Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)
- Higham, D.J., Kloeden, P.E.: An Introduction to the Numerical Simulation of Stochastic Differential Equations. SIAM, Philadelphia (2021)
-Additional references will be provided in the course of the lectures