Monte Carlo Approximate Tensor Moment Simulations (Numerical Linear Algebra with Applications - Vol
Abstract
An algorithm to generate samples with approximate first-order, second-order, third-order, and fourth-ordermoments is presented by extending the Cholesky matrix decomposition to a Cholesky tensor decompositionof an arbitrary order. The tensor decomposition of the first-order, second-order, third-order, and fourth-orderobjective moments generates a non-linear system of equations. The algorithm solves these equations by numerical methods. The results show that the optimization algorithm delivers samples with an approximate residual error of less than 10^-16 between the components of the objective and the sample moments. The algorithm is extended for a n-th-order approximate tensor moment version, and simulations of non-normal samples replicated from distributions with asymmetries and heavy tails are presented. An application for sensitivity analysis of portfolio risk assessment with Value-at-Risk (VaR) is provided. A comparison with previous methods available in the literature suggests that the methodology proposed reduces the error of the objective moments in the generated samples.