Approximating the Value Functions of Stochastic Knapsack Problems: A Homogeneous Monge-Amp\'ere Equation and Its Stochastic Counterparts

TL;DR

This paper develops approximation schemes for stochastic knapsack problems using nonlinear PDEs and SPDEs, providing a new mathematical framework for resource allocation models in telecommunications and revenue management.

Contribution

It introduces a novel approach linking stochastic knapsack problems to solvable PDEs and SPDEs, advancing the theoretical understanding of their value functions.

Findings

Derived a system of nonlinear PDEs as limits of the stochastic knapsack process.

Established stochastic PDEs as limits for the optimal solutions.

Provided approximation schemes for resource allocation problems.

Abstract

Stochastic knapsack problem originally was a versatile model for controls in telecommunication networks. Recently, it draws attentions of revenue management community by serving as a basic model for allocating resources over time. We develop approximation schemes for knapsack problems in this paper, a system of nonlinear but solvable partial differential equations and stochastic partial differential equation are shown to be the limit of the process that following the optimal solution of the stochastic knapsack problem.