Issues in the Software Implementation of Stochastic Numerical Runge–Kutta

Abstract

This paper discusses stochastic numerical methods of Runge-Kutta type with weak and strong convergences for systems of stochastic differential equations in Itô form. At the beginning we give a brief overview of the stochastic numerical methods and information from the theory of stochastic differential equations. Then we motivate the approach to the implementation of these methods using source code generation. We discuss the implementation details and the used programming languages and libraries

An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox

“…There exist several software packages to simulate SDEs available in various programming languages. A short search brought up the following 20 toolboxes: for the C++ programming language [4,25], for the Julia programming language [2,48,49,50], for Mathematica [54], for Matlab [17,22,44,52], for the Python programming language [1,3,15,16,36] and for the R programming language [6,18,23,24]. However, only four of these toolboxes seem to contain an implementation of an approximation for the iterated stochastic integrals using Wiktorsson's algorithm and none of them provide an implementation of the recently proposed Mrongowius-Rößler algorithm.…”

“…In order to get some better approximation, one may take into account either some parts or even the whole tail sum as well. Adding the exact simulation of R (p) 1 (h) in (16), which belongs to the Fourier coefficient a i 0 , results in the second algorithm presented in Section 3.2 which is due to Milstein [40]. In contrast to Milstein, in the seminal paper by Wiktorsson [53] the whole…”

“…For some prescribed p ∈ N, the rest term approximation employed by Milstein [40] is concerned with the exact simulation of the term R (p) 1 (h) given by (16). Utilizing the independence of coefficients a i r and their distributional properties (10), the random vector ∞ r=p+1 a r possesses a multivariate Gaussian distribution with zero mean and covariance matrix h…”

An analysis of approximation algorithms for iterated stochastic integrals and a Julia and Matlab simulation toolbox

“…In order to get some better approximation, one may take into account either some parts or even the whole tail sum as well. Adding the exact simulation of R (p) 1 (h) in (16), which belongs to the Fourier coefficient a i 0 , results in the second algorithm presented in Section 3.2 which is due to Milstein [40]. In contrast to Milstein, in the seminal paper by Wiktorsson [53] the whole tail sum R (p)…”

“…For some prescribed p ∈ N, the rest term approximation employed by Milstein [40] is concerned with the exact simulation of the term R (p) 1 (h) given by (16). Utilizing the independence of the coefficients a i r and their distributional properties (10), the random vector ∞ r=p+1 a r possesses a multivariate Gaussian distribution with zero mean and covariance matrix h…”