Simulating first-passage times and the maximum of stochastic differential equations: An error analysis

This dissertation investigates the error inherent in using the Euler scheme to simulate stochastic (Ito) differential equations (SDE's) and certain functionals thereof. This work mainly belongs to the class of research which investigates the asymptotic distribution of the error as the step length used in the Euler scheme becomes small.;I have investigated the error both on the grid of approximation times and off the grid using appropriate interpolations. I have proved that when the step length is small the on-grid error evolves in time approximately as an SDE driven by two independent Brownian motions. The convergence results I have found for the off-grid errors allow me to rank the different interpolation schemes in terms of efficiency.;In addition, I have found the asymptotic distribution of the error when simulating first passage times, particularly level crossing times. In fact, the asymptotic error distribution arises as a special case of a more general theorem.;Moreover, I have extended previous work on simulating the maximum of Brownian motion with drift and found the limiting error distribution for the maximum of general SDE's. As an intermediate result I have found, consistent with previous work, that close to the maximum the amount by which the SDE is smaller than the maximum has approximately the same distribution as a scaled ;The second type of endeavor in this thesis pertains to investigating the bias when using the Euler scheme to estimate expectations. I consider the value of the SDE at k times and have obtained an expansion of the bias in powers of the step length. This extends a similar and well-known expansion for the value of the SDE at only one time.;The major contribution is the work on first passage times and the maximum.