Simulation and Integration of Stochastic Differential Equations

Methods of integrating Stochastic Differential Equations (EDEs). Ito-Taylor expansion and simulation of stochastic multiple integrals. Strong approach of trajectories and weak approach of functional of the solution. Numerical methods: Euler-Maruyama, Ito-Taylor, Runge-Kutta stochastic. Extrapolation methods, Exponential schemes. Convergence speed. Numerical stability: A-stability, MS-stability. Simulation of functional solutions using simplified weak schemes. Monte Carlo methods for EDEs. Simulation of stochastic Hamiltonian systems. Efficient computational implementation of numerical schemes for PDEs. Applications to the simulation of real models, probabilistic methods for PDEs.

#### Basic Information

Workload

45 hours

Requirements

Stochastic Differential Equations

**Mandatory: **

- Kloeden P, Platen. (1999) Numerical solution of stochastic differential equations. Springer.
- Milstein G, Tretyakov (2004) Stochastic Numerics for Mathematical Physics. Springer.
- Kloeden P, Platen E, Schurz H (2012) Numerical Solution of SDE Through Computer Experiments. Universitext.

**Complementary: **

- Han, X, Kloeden P (2017) Random Ordinary Differential Equations and Their Numerical Solution. Springer.
- Schurz H (2001) Numerical Analysis of Stochastic Differential Equations without Tears. Handbook of Stochastic Analysis and its Applications. Marcel Dekker.
- Sauer T (2013) Computational solution of stochastic differential equations, WIREs Comput Stat 2013. doi: 10.1002/wics.1272, 2013.
- Higham D. J (2001) An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, (43), 3, 525-546.
- Jentzen A, Kloeden P (2010) Taylor Approximations for stochastic Patrtial Diffrential Equations. SIAM.