Analysis of a discrete-time Markov process with a bounded continuous state space by the Fredholm integral equation of the second kind

Abstract

A discrete-time Markov process with a bounded continuous state space is considered. We show that the equilibrium equations on steady-state probability and densities form Fredholm integral equations of the second kind. Then, under a sufficient condition that the transition densities from one state to another state inside the boundaries of the state space can be expressed in the same separate forms, the steady-state probability and density functions can be obtained explicitly. We use it to demonstrate an economic production quantity model with stochastic production time, derive the expressions of the steady-state probabilities and densities, and find the optimal maximum stock level. A sensitivity analysis of the optimal stock level is performed using production time and cost parameters. The optimal stock level decreases with respect to the holding cost and the production cost, whereas it increases with respect to the lost sale cost and the arrival rate.