Technical Note - On Matrix Exponential Differentiation with Application to Weighted Sum Distributions

Abstract

On Modeling the Probability Distribution of Stochastic Sums In the “Technical Note—On Matrix Exponential Differentiation with Application to Weighted Sum Distributions,” Das, Tsai, Kyriakou, and Fusai propose an efficient methodology for approximating the unknown probability distribution of a weighted stochastic sum or time integral. Resulting from earlier contributions based on continuous-time Markov chain approximations of one-dimensional Markov processes is the Laplace transform of the unknown distribution available in exponential matrix form. In this paper, the authors develop a bona fide Pearson curve-fitting approach to this distribution based on the moments, which they recover from the derivatives of the Laplace transform. Motivated by the computational hurdles toward this, they derive computationally efficient closed-form expressions for the derivatives of the matrix exponential. They then apply to pricing average-based options.