Simulation of stochastic systems using antithetic multilevel Monte Carlo on GPUs

Abstract

Numerous natural processes ranging from chemical reactions to wind speed can be modeled using systems of stochastic differential equations (SDEs) [1]. Like ordinary differential equations (ODEs), stochastic differential equations model the rate of change or a variable in relation to one or more variables. However, SDEs differ from ODEs in that they contain a term that introduces an element of randomness to a normally deterministic process. Because of this uncertainty, only the simplest SDEs have analytical solutions, and one must turn to computational simulations in order to approximate their solutions. [2]

Monte Carlo methods are a common and useful means of solving systems of SDEs. The SDE is simulated by taking small steps through time or space until the terminating condition is met - either the final time or distance. This process is repeated numerous times after which the mean and variance of the solutions obtained are calculated in order to estimate the confidence interval containing the true value. [3]

When there are multiple processes involved (e.g., multiple assets in a financial portfolio or multiple ports on a server), there can be correlation among the individual processes. Unfortunately simulating numerous processes is computationally expensive unless advanced computational and numeric methods are employed. The goal of this dissertation is to propose a high performance computational framework for solving systems of stochastic differential equations with correlation in finance that utilizes antithetic multilevel Monte Carlo on a cluster of GPU-enabled servers.

The dissertation includes the following sections. In chapter one, an introduction to stochastic differential equations is provided along with relevant background. Next, various computing architectures are reviewed. The third chapter introduces the concept of Monte Carlo simulations and some of its variants including antithetic multilevel Monte Carlo. The previously discussed topics are applied to the area of financial options in chapter four. Forecasting, specifically exponential smoothing in a high performance environment, is covered in chapter five. The final chapter summarizes the results and provides an overview of future research.