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
- Jump Processes
- SDE in Mathematical Finance
- Monte Carlo and Multilevel Monte Carlo Methods
- Geometric Numerical Integration of Stochastic Hamiltonian Systems
- Numerical Methods for SDE Systems
- Notes on the Numerical Integration of Stochastic PDEs
- Fractional Brownian Motion and Simulation Methods for Fractional Brownian Motion
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