Approximation of stochastic dynamics with application to molecular dynamics, kinetic Monte Carlo methods and financial mathematics

A differential equation model can be stochastic by two reasons: if estimated parameters implies this, as e.g. in financial mathematics, or if fundamental microscopic laws generate stochastics when coarse-grained, as in molecular dynamics for chemistry, material science and biology.

The course starts with basic background on stochastic differential equations theory and numerics, including the Basics below. The focus of the course is on some applications to:
-- determine stochastic differential equation parameters from data, by solving inverse problems using optimal control theory and Hamilton-Jacobi equations,
-- derive coarse-grained models and systematically choose computational accurate stochastic models in a hierarchy of models consisting of the Schrödinger equation, stochastic molecular dynamics, kinetic Monte Carlo methods, stochastic partial differential equations, deterministic partial differential equations.

Basics (two times 90 minutes):

• definition of Brownian motion,
• Ito integrals, Ito SDEs as forward Euler (and Stratonovich as trapezoidal),
• Kolmogorov's equations, weak and strong approximation,
• statistical and time-discretization errors.

• Derive the Black-Scholes equation,
• Optimal portfolio (and optimal control theory),
• Molecular dynamics (Smoluchowski and Langevin equations),
• Reaction rates and paths (with large deviations in an optimal control way).

Welcome
Anders Szepessy,
szepessy@kth.se, 790 7494