Continuity of imprecise stochastic processes with respect to the pointwise convergence of monotone sequences

TL;DR

This paper proves the continuity of the joint lower expectation of finite-state imprecise stochastic processes with respect to pointwise convergence of monotone sequences, using Ville-Vovk-Shafer and Williams extensions.

Contribution

It establishes new continuity results for lower expectations in imprecise stochastic processes under monotone convergence, extending prior theoretical understanding.

Findings

Continuity holds for non-decreasing sequences of functions.

Similar continuity results are shown for non-increasing sequences converging to bounded functions.

Results apply to both Ville-Vovk-Shafer and Williams natural extensions.

Abstract

We consider the joint lower expectation of a finite-state imprecise stochastic process, defined using either the Ville-Vovk-Shafer natural extension or the Williams natural extension. In both cases, we show that it is continuous with respect to the pointwise convergence of non-decreasing sequences of real-valued functions $f_n$, $n\in\mathbb{N}_0$, where each $f_n$ is $n$-measurable. For the Ville-Vovk-Shafer natural extension, a similar result is shown to hold for non-increasing sequences, provided that they converge to a bounded function.