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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 838397, 13 pages
http://dx.doi.org/10.1155/2012/838397
Research Article

The Sum and Difference of Two Lognormal Random Variables

Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Hong Kong

Received 15 May 2012; Accepted 19 July 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 C. F. Lo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of N lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.