A Note on Fuzzy Set--Valued Brownian Motion

TL;DR

This paper demonstrates that fuzzy set--valued Brownian motion can be represented as the indicator function of a standard Wiener process, simplifying its analysis by linking it to classical stochastic processes.

Contribution

It establishes that fuzzy set--valued Brownian motion is equivalent to an indicator function of a Wiener process, providing a new perspective and simplifying its mathematical handling.

Findings

Fuzzy set--valued Brownian motion can be represented as an indicator of a Wiener process.

This representation simplifies the analysis of fuzzy stochastic processes.

The paper bridges fuzzy set theory and classical stochastic calculus.

Abstract

In this paper, we prove that a fuzzy set--valued Brownian motion $B_t$, as defined in [1], can be handle by an $R^d$--valued Wiener process $b_t$, in the sense that $B_t =\indicator{b_t}$; i.e. it is actually the indicator function of a Wiener process.