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 reparametrizing 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.
Highlights
For any word τ = (τ1, . . . , τk) constituted of the letter τi ∈ {1, . . . , n}, i = 1, . . . , k. Such expansions play an important role in the theory of rough paths, allowing one to define such a differential equation for a large class of drivers X ([3])
It immediately follows from equation (8) that any power of an iterated integral is a sum of iterated integrals and that a polynomial of iterated integrals is a linear combination of single iterated integrals
The solution of the stochastic differential equation can be approximated by a series of iterated integral of the drivers, whose coefficients are a function of the parameters
Summary
We introduce the mathematical background and motivation for manipulating expansions and iterated integrals. The subsection introduces the Picard procedure, a simple iterative way to derive local approximation of the solution of a differential equation. Iterated integrals are introduced and the two are combined to define the expansions. ∗Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Iterated integrals are introduced and the two are combined to define the expansions. ∗Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. †Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK
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