The Sum and Difference of Two Lognormal Random Variables

Journal of Applied Mathematics, Volume 2012, Article ID 838397

13 Pages Posted: 23 May 2012 Last revised: 14 May 2013

See all articles by Chi-Fai Lo

Chi-Fai Lo

The Chinese University of Hong Kong

Date Written: May 10, 2013

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.

Keywords: Lognormal random variables, probability distribution functions, backward Kolmogorov equation, Lie-Trotter splitting approximation

Suggested Citation

Lo, Chi-Fai, The Sum and Difference of Two Lognormal Random Variables (May 10, 2013). Journal of Applied Mathematics, Volume 2012, Article ID 838397. Available at SSRN: https://ssrn.com/abstract=2064829 or http://dx.doi.org/10.2139/ssrn.2064829

Chi-Fai Lo (Contact Author)

The Chinese University of Hong Kong ( email )

Department of Physics
Shatin, N.T., Hong Kong
China

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