High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

Algorithm 1

gPC-based density estimation for multivariate Gaussian variables [3].

(i) As integral transformation (Equation (6)).

(ii) For and , solve (Equation (4)) with to obtain .

(iii) Calculated based on Equation (10).

(iv) Approximate .

(v) Approximate on a sample of points which are according to .

(vi) Histogram method calculated PDF for :

If moment estimation is needed for : where .

If density estimation needed for :

(vii)

(viii) histogram method calculated for , the same as .