# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
Here, [X]\({}^{\alpha}\) represents the equivalence class of X with respect to the parameter \(\alpha\), and K(R) denotes the set of fuzzy sets over the real numbers. A measurable multifunction [X]\({}^{\alpha}\): I \(\times\)\(\Omega\)\(\rightarrow\) K(R) maps a measurable set in the time domain I and the sample space \(\Omega\) to a set of measurable fuzzy sets in the value domain R. For example, let X(t) be a fuzzy stochastic process that represents the price of a stock at time t. The parameter \(\alpha\) can represent the level of confidence that we have in the price prediction. For example, \(\alpha\) = 0.95 could represent a 95% confidence level. We can define the equivalence class [X(t)]\(\alpha\) as follows: [X(t)]\({}^{\alpha}\) = {Y \(\in\) K(R) \(\mid\) Y(x) \(\geq\)\(\alpha\) for all x \(\in\) R}. The parameter \(\alpha\) represents the level of confidence in the price prediction. It quantifies the minimum membership value required for a fuzzy set to be considered as part of the equivalence class [X(t)]\({}^{\alpha}\). This equivalence class contains all fuzzy sets that are at least as precise as X(t) with respect to the confidence level \(\alpha\). The equivalence class [X(t)]\({}^{\alpha}\) is defined as the set of fuzzy sets Y in K(R) that have a membership value greater than or equal to \(\alpha\) for all x in the real numbers (R). In other words, [X(t)]\({}^{\alpha}\) contains all fuzzy sets that are at least as precise as X(t) with respect to the confidence level \(\alpha\).We can then show that the multifunction [X(t)]\({}^{\alpha}\): I \(\times\)\(\Omega\)\(\rightarrow\) K(R) is measurable. This means that the fuzzy stochastic process X(t) is measurable. By stating that the multifunction [X(t)]\({}^{\alpha}\): I \(\times\)\(\Omega\)\(\rightarrow\) K(R) is measurable, we are asserting that the mapping from the Cartesian product of the index set I and the sample space \(\Omega\) to the set of fuzzy sets K(R) is a measurable multifunction. This means that the pre-images of measurable sets in K(R) under [X(t)]\({}^{\alpha}\) are measurable sets in I \(\times\)\(\Omega\). The pre-images, in this context, refer to the subsets of the domain I \(\times\)\(\Omega\) that get mapped to a specific set in the target space K(R) under the mapping [X(t)]\({}^{\alpha}\): I \(\times\)\(\Omega\)\(\rightarrow\) K(R). To put it simply, if we have a set A in K(R), the pre-image of A under [X(t)]\({}^{\alpha}\) is the set of all points (i, \(\omega\)) in the domain I \(\times\)\(\Omega\) such that [X(t)]\({}^{\alpha}\)(i, \(\omega\)) belongs to A. In other words, the pre-image of a set A in K(R) consists of all the elements (i, \(\omega\)) in I \(\times\)\(\Omega\) that get mapped to A under the mapping [X(t)]\({}^{\alpha}\). By saying that the pre-images of measurable sets in K(R) under [X(t)]\({}^{\alpha}\) are measurable sets in I \(\times\)\(\Omega\), we mean that if we consider a measurable set A in K(R), the pre-image of A under [X(t)]\(\alpha\) is a measurable set in I \(\times\)\(\Omega\). This implies that the behaviour of the fuzzy stochastic process X(t) with respect to such pre-images is well behaved and can be characterised by measurable sets in the domain I \(\times\)\(\Omega\). We propose that the condition that [X]\({}^{\alpha}\) be measurable for all \(\alpha\) in the interval [0, 1] ensures that the fuzzy stochastic process X can be integrated with respect to time and the sample space. B(I) is the Borel \(\sigma\)-algebra, which consists of all subsets of the interval I. To determine measurability, [X]\({}^{\alpha}\): I \(\times\)\(\Omega\)\(\rightarrow\) K(R) should be B(I) \(\otimes\) A-measurable. Here, B(I) \(\otimes\) A represents the product \(\sigma\)-algebra, which is generated by the Borel \(\sigma\)-algebra B(I) and the \(\sigma\)-algebra A. This requirement ensures that the equivalence class [X]\({}^{\alpha}\) is measurable with respect to the joint \(\sigma\)-algebra generated by B(I) and A. In this context, K(R) denotes the set of fuzzy sets over the real numbers.