Numerical approximations for stochastic differential equations

Thesis

In this thesis, we consider problems related to the numerical simulation of stochastic differential equations (SDEs). In particular, we are interested in methods that use both increments and areas of the Brownian path. For example, time integrals of Brownian motion carry useful information whilst being normally distributed and thus straightforward to generate. We will make precise the "path information" contained by such integrals and present a polynomial approximation theorem for Brownian motion. Since the Gaussian coefficients given by this expansion are independent, we can express Brownian motion as a polynomial with additional noise. We shall then use this simple observation to develop high order methods.

The majority of these methods approximate SDEs through the use of ODEs, which can be accurately discretized using state-of-the-art solvers. To numerically demonstrate these approaches, we will simulate four SDEs: Inhomogeneous Geometric Brownian Motion, Cox-Ingersoll-Ross model, underdamped Langevin equation and the Lévy area of Brownian motion. The code for these examples can be found at github.com/james-m-foster.

We also study problems related to variable step size methods for SDEs. Most notably, we investigate which numerical methods and variable step size strategies can ensure pathwise convergence to the true SDE solution. Our analysis is based on rough path theory and requires methods to be locally close to an ODE driven by a piecewise "Brownian polynomial". This result can therefore be applied to the Heun and midpoint methods. In order to prove convergence, we will require variable step size strategies to produce a nested sequence of partitions with mesh size tending to zero.

Authors: Foster, JM

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Orthogonal polynomials; Brownian motion; Stochastic differential equations; Numerical approximations; Lévy area; Variable step size methods; Langevin dynamics; Inhomogeneous geometric Brownian motion; Cox-Ingersoll-Ross model
• Subjects: Stochastic analysis; Rough path theory; Numerical analysis