MACROSCOPIC QUANTITIES FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH A LEVY NOISE IN TWO DIMENSIONS Albert, Hannah integro-differential equation Lévy motion mean exit time stochastic differential equation Stochastic dynamical systems The mean exit time and transition probability density function are macroscopic quantities used to determine the behavior of stochastic di↵erential equations (SDEs). The integral-di↵erential equations determining these quantities for SDEs with non- Gaussian ↵-stable L´evy motions involve a nonlocal term consisting of a singular integral, which is a manifestation of the ’flights’ or ’jumps’ due to the non-Gaussian noise. A two-dimensional SDE with radially symmetric ↵-stable L´evy motion is considered, and an efficient second-order accurate numerical scheme is developed for calculating the mean exit time and transition probability density function. The scheme is numerically verified by testing the results of the deterministic integral-di↵erential equations with a known, smooth function u(x) in place of themean exit time, and by calculating an unknown mean exit time u(x). M.S. in Applied Mathematics, May 2016 Li, Xiaofan 2016 2016-05 Thesis application/pdf islandora:6472 http://hdl.handle.net/10560/3872 MATH / Applied Mathematics Illinois Institute of Technology en In Copyright http://rightsstatements.org/page/InC/1.0/ Restricted Access