Efficient Computation of the Stochastic Behavior of Partial Sum Processes

TL;DR

This paper presents a recursive integration method combined with Fast Fourier Transforms to efficiently compute probabilities of dynamical partial sum processes, enabling exact calculations in high-dimensional stochastic applications.

Contribution

It introduces a novel recursive integration approach with FFT to perform exact probability calculations for complex dynamical partial sum expressions.

Findings

Recursive integration reduces high-dimensional problems to two-dimensional calculations.

FFT accelerates the computation of probability distributions.

Method applicable to reliability, quality assessment, and stochastic control.

Abstract

In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in the fields of reliability, product quality assessment, and stochastic control. While these probability calculations are ostensibly of a high dimension, and consequently intractable in general, it is shown how a recursive integration methodology can be implemented to obtain exact calculations as a series of two-dimensional calculations. The computational aspects of the implementaion of this methodology, with the adoption of Fast Fourier Transforms, are discussed.