A Distributed Procedure for Computing Stochastic Expansions with Mathematica

TL;DR

This paper introduces a Mathematica-based distributed method for efficiently computing stochastic expansions of polynomial differential equations, reducing memory use and enabling parallel processing.

Contribution

It presents a novel re-parametrization and distribution approach for polynomial expansions, along with an iterative shuffle product and expectation calculation methods.

Findings

Reduces memory requirements for stochastic expansion calculations.

Enables parallelized computation of stochastic expansions.

Provides efficient algorithms for expectation of iterated integrals.

Abstract

The solution of a (stochastic) differential equation can be locally approximated by a (stochastic) expansion. If the vector field of the differential equation is a polynomial, the corresponding expansion is a linear combination of iterated integrals of the drivers and can be calculated using Picard Iterations. However, such expansions grow exponentially fast in their number of terms, due to their specific algebra, rendering their practical use limited. We present a Mathematica procedure that addresses this issue by re-parametrising the polynomials and distributing the load in as small as possible parts that can be processed and manipulated independently, thus alleviating large memory requirements and being perfectly suited for parallelized computation. We also present an iterative implementation of the shuffle product (as opposed to a recursive one, more usually implemented) as well as a fast way for calculating the expectation of iterated Stratonovich integrals for Brownian Motion.