Mean exit times and multi-level Monte Carlo with applications in finance

Rights statement
Awarding institution
  • University of Strathclyde
Date of award
  • 2013
Thesis identifier
  • T13439
Qualification Level
Qualification Name
Department, School or Faculty
  • This thesis is focused broadly on the stopped exit time problem in the stochastic differential equations setting. It currently appears that one of the most efficient and powerful approaches to solve this problem (especially in higher dimensions for which there are no explicit formulae) is the simulation of sample paths of time discrete approximations. We compute the mean exit time using a Monte Carlo technique, which has the advantage of being straightforward to implement. However, controlling the Monte Carlo sampling error and large biases in the numerical method make this method computationally expensive. In our work we employ a variance reduction technique called multi-level Monte Carlo which dramatically reduces the complexity of the method. Surprisingly, even though we need to compute the quantity in a weak sense (an expected value), the multi-level method also relies on a strong convergence property of the numerical scheme. In order to justify the multi-level method we then establish a rate of strong convergence for exit times. We also present an extension to the basic method which reduces the computational complexity even further. The extended version uses a Brownian bridge technique which is applied on the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one (the Milstein scheme.) Our results have been derived for multi-dimensional stochastic differential equations and can be applied to nonlinear stochastic differential equations models, including those arising in finance and chemical kinetics. Our theoretical work is complemented by computational tests for a number of practical problems.
Resource Type
Date Created
  • 2013
Former identifier
  • 995979