Set-valued Brownian motion

TL;DR

This paper extends classical stochastic processes to set-valued functions in Banach spaces, exploring their properties and embedding results within the structure of compact convex subsets.

Contribution

It introduces set-valued Brownian motions and martingales in Banach spaces, utilizing embedding techniques and ordered structures of compact convex sets.

Findings

Development of set-valued Brownian motion and martingale theory

Embedding results for set-valued functions in Banach spaces

Application of ordered structure in the analysis of set-valued stochastic processes

Abstract

Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the paper of the second author in which an embedding result is obtained which considers also the ordered structure of $ck(X)$ and f-algebras.