Numerical methods for stochastic differential equations (SDEs)

<p>Written exam in on a scale from 18 to 30.</p><p>Oral exam in on a scale from 18 to 30.</p>

<p>Intermediate tests or individual projects.</p>

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.

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

- 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