The Sum and Difference of Two Lognormal Random Variables

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.