Euler scheme for SDEs with non-standard coefficients

Abstract

It is well known that the Euler-scheme method of approximations converges well for stochastic differential equations (SDEs) in the case where the coefficients obey a Lipschitz assumption. In this thesis we investigate the performance of the Euler scheme for three classes of non-Lipschitz coefficients: coefficients that are locally but not globally Lipschitz, coefficients that are discontinuous, and coefficients that are not locally integrable. For each class of non-standard coefficient considered above we provide an original contribution to the literature. Additionally, we provide an extensive literature review for the performance of the Euler scheme for Lipschitz coefficients, and for the three non-standard cases listed above. Finally, we provide a literature review of the performance of the Unadjusted Langevin Algorithm (ULA), given as the Euler scheme discretisation of a certain SDE on an unbounded interval.