Polynomial propagation of moments in stochastic differential equations

We address the problem of approximating the moments of the solution, $\boldsymbol{X}(t)$, of an It\^o stochastic differential equation (SDE) with drift and a diffusion terms over a time-grid $t_0, t_1, \ldots, t_n$. In particular, we assume an explicit numerical scheme for the generation of sample paths $\hat{\boldsymbol{X}}(t_0), \ldots, \hat{\boldsymbol{X}}(t_n), \ldots$ and then obtain recursive equations that yield any desired non-central moment of $\hat{\boldsymbol{X}}(t_n)$ as a function of the initial condition $\boldsymbol{X}_0$. 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 algorithms 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 $\hat{\boldsymbol{X}}(t_n)$ 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., $\boldsymbol{X}_0 = \boldsymbol{x}_0$ for some known $\boldsymbol{x_0}$) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes to approximate the target moments. The methodology has been inspired by the PCE and differential algebra (DA) 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.