Polynomial Propagation of Moments in Stochastic Differential Equations Articles uri icon

publication date

  • June 2023

start page

  • 1153

end page

  • 1180

issue

  • 2

volume

  • 22

International Standard Serial Number (ISSN)

  • 1536-0040

abstract

  • We address the problem of approximating the moments of the solution,
    , of an Itô stochastic differential equation (SDE) with drift and diffusion terms over a time grid
    . In particular, we assume an explicit numerical scheme for the generation of sample paths
    and then obtain recursive equations that yield any desired noncentral moment of
    as a function of the initial condition
    . The core of the methodology is the decomposition of the numerical solution
    into a 'central part” and an 'effective noise” term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe the proposed methodology based on an Euler–Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of
    based on a Gram–Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e.,
    for some deterministic and known
    ) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes in order to approximate the target moments. The methodology has been inspired by the PCE and differential algebra methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

subjects

  • Telecommunications

keywords

  • uncertainty propagation; moment approximation; density estimation; polynomial chaos expansion; gram–charlier expansion; sde.