Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the upper and lower covariances under multiple probabilities

Xinpeng Li, Jingxu Niu, Ke Zhou

Abstract

In this paper, we define the upper (resp. lower) covariance under multiple probabilities via a corresponding max-min-max (resp. min-max-min) optimization problem and the related properties of covariances are obtained. In particular, we propose a fast algorithm of calculation for upper and lower covariances under the finite number of probabilities. As an application, our algorithm can be used to solve a class of quadratic programming problem exactly, and we obtain a probabilistic representation of such quadratic programming problem.

On the upper and lower covariances under multiple probabilities

Abstract

In this paper, we define the upper (resp. lower) covariance under multiple probabilities via a corresponding max-min-max (resp. min-max-min) optimization problem and the related properties of covariances are obtained. In particular, we propose a fast algorithm of calculation for upper and lower covariances under the finite number of probabilities. As an application, our algorithm can be used to solve a class of quadratic programming problem exactly, and we obtain a probabilistic representation of such quadratic programming problem.
Paper Structure (5 sections, 18 theorems, 116 equations)

This paper contains 5 sections, 18 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper and Lower Covariances
  4. Calculation of upper and lower covariance
  5. Application: Quadratic programming problem

Key Result

Theorem 2.2

Let $X$ be a random variable on sublinear expectation space $(\Omega,\mathcal{F},{\mathbb{E}^{\mathcal{P}}})$ with ${\mathbb{E}^{\mathcal{P}}}[X^2]<\infty$. Then we have

Theorems & Definitions (35)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 25 more