AMCS 336 Numerical Methods for Stochastic Differential Equations with connections to Machine Learning

stochastic differential equations Monte Carlo Multilevel Monte Carlo Variance Reduction Stochastic Optimal Control

The goal of this course is to give basic knowledge of stochastic differential equations and their numerical solution, useful for scientific and engineering modeling, guided by some problems in applications in financial mathematics, material science, geophysical flow problems, turbulent diffusion, control theory, and Monte Carlo methods.

We will discuss basic questions for numerical approximation of stochastic differential equations, for example:
To determine the price of an option is it more efficient to solve the deterministic Black and Scholes partial differential equation or use a Monte Carlo method based on a stochastic representation?

The course treats basic theory of stochastic differential equations including weak and strong approximation, efficient numerical methods and error estimates, the relation between stochastic differential equations and partial differential equations, variance reduction, etc. It also addresses Optimal control for ODEs and SDEs and connections to the Hamilton-Jacobi-Bellman nonlinear PDE.

Prerequisite: knowledge of basic probability, numerical analysis, and programming.

Upon completion of the course the student should be able to

formulate basic properties of the Wiener process
define the Ito integral
apply Ito's formula for stochastic differentials
discretize a given SDE and check resulting approximation properties
implement the Monte Carlo and Multilevel Monte Carlo methods for discretizations of SDEs
apply variance reduction techniques, including the multilevel Monte Carlo method
state Kolmogorov's backward and forward equations for a given SDE
control the discretization errors arising in a Monte Carlo method for the weak approximation of SDEs
formulate and discretize Optimal control problems for ODEs and SDEs