An approximation method for the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable

TL;DR

This paper introduces an approximation approach for optimizing the $p$-th moment of $ ext{R}^n$-valued random variables, transforming the problem into solvable differential equations via variational and duality methods.

Contribution

It develops a novel approximation framework that converts the moment optimization into a sequence of nonlinear differential equations with proven existence and uniqueness.

Findings

The method successfully transforms the optimization into differential equations.

The canonical duality approach guarantees solution existence and uniqueness.

An approximation of the probability density function is constructed.

Abstract

This paper mainly addresses the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable. Through an ingenious approximation mechanism, one transforms the maximization problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational approach. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the dual transformation gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of the probability density function accordingly.