Computational Methods for Stochastic Differential Equations and Machine Learning

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In this course we will define what is meant by stochastic differential equations and their solutions. The definition uses numerical schemes, and in the process we also obtain convergence results for these schemes. Expected values as functions of starting positions for SDEs are given by solutions to partial differential equations. We will consider the two different possibilities of finding such expected values by Monte Carlo simulation and finite difference schemes.

Applications included are e.g. finance, where stock prices are modelled using SDEs, molecular dynamics, where SDEs are used to model systems with constant temperature, and machine learning  where the basic stochastic gradient descent algorithms is a numerical scheme for perturbed gradient flow. Optimal control will also be discussed in the course. It is used e.g. in optimal hedging,  finding reaction rates in molecular dynamics and analyzing machine learning convergence rates


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