# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Preliminaries
#### 3.1.2 Fuzzy Random Variable, Fuzzy Stochastic Process, and Integral Overview
This implies that the fuzzy integral exhibits continuity with respect to the d1 metric, capturing the dependence and smoothness of the integral over time. We will now define an inequality that provides a bound on the maximum difference between fuzzy sets L1(u) and L1, y(u) over the interval [0, t] in terms of the cumulative difference between fuzzy sets X(s) and Y(s) over the sample interval. The inequality captures the relationship between the supremum and integral of the d1 metric almost everywhere (a.e.). This inequality states that the supremum of the d1 metric between \(L_{t,x}(u)\) and \(L_{t,y}(u)\) for \(u\) in in the interval [0, t] is bounded by the scaled integral of the d\({}_{\infty}\) metric between X(s) and Y(s) for \(s\) in the interval [0, t], with the scaling factor depending on the value of p.  
\[\sup_{u\in[0,t]}d_{\infty}^{p}\big{(}L_{t,\ x}(u),\ L_{t,y}(u)\big{)}\leq t^{p-1 }\int_{0}^{t}d_{\infty}^{p}(X(s),\ Y(s))ds,\ a.e. \tag{9}\]  
The L.H.S. represents the supremum (maximum) of the d\({}_{\infty}\) metric between \(L_{t,x}(u)\) and \(L_{t,y}(u)\), where \(t\) is a fixed value and \(u\) ranges from 0 to t. Here, d\({}^{p}\infty\)(a, b) denotes the d\({}_{\infty}\) metric between two fuzzy sets or elements \(a\) and \(b\). d\({}^{p}\infty\)(L\({}_{t,x}(u)\), L\({}_{t,y}(u)\)) is the d\({}_{\infty}\) metric between the fuzzy sets L\({}_{t,x}(u)\) and L\({}_{t,y}(u)\) at each \(u\). It measures the maximum difference in membership degrees between corresponding elements of the fuzzy sets at a given \(u\). The R.H.S. represents an integral involving the d\({}_{\infty}\) metric between the fuzzy sets X(s) and Y(s), where s ranges from 0 to t. The integral is scaled by t\({}^{p-1}\), where \(p\) is a positive real number. It is the integral of the d\({}_{\infty}\) metric between X(s) and Y(s) at each s. It measures the cumulative difference in membership degrees between corresponding elements of the fuzzy sets over the interval [0, t]. t\({}^{p-1}\) scales the integral by a factor that depends on the value of \(p\). The value of \(p\) determines how the integral contributes to the overall inequality. The inequality holds almost everywhere (a.e), meaning that it holds for all values of \(t\) except possibly for a set of measure zero. We will now define the fuzzy stochastic Ito integral by using fuzzy random variable \(\int_{0}^{T}X(s)dW(s)\), where W is a Wiener process. Let X \(\in\) L2(I \(\times\)\(\Omega\), N; R); then, \(\big{(}\big{\langle}\int_{0}^{T}X(s)dW(s)\big{\rangle}\big{)}\), where t \(\in\) I. This represents a fuzzy stochastic process defined over the interval I. For each fixed t, \(\big{\langle}\int_{0}^{T}X(s)dW(s)\big{\rangle}\) is a fuzzy set. The process captures the evolution of these fuzzy sets as t varies. Here, dW(s) represents the differential of a Wiener process or Brownian motion. The next part indicates that the fuzzy stochastic process \(\big{\langle}\int_{0}^{T}X(s)dW(s)\big{\rangle}\) belongs to the L\({}^{2}\) space over the product space I \(\times\)\(\Omega\). The fuzzy sets generated by the integral have finite second moments or variances and take values in the space of fuzzy sets over the real numbers, denoted as F(R). The last proposition states that if X belongs to a family of square integrables, then the next integration will become a fuzzy stochastic differential equation and then that integration also belongs to the family of square integrables. Formally, let X \(\in\) L\({}^{2}\)(I \(\times\)\(\Omega\), N; R); then, X belongs to the L\({}^{2}\) space, which consists of processes that have finite second moments or variances. The process X takes real-valued values and \(\big{\langle}\int_{0}^{T}X(s)dW(s)\big{\rangle},t\in I\) is d\({}_{\infty}\)-continuous. In other words, as t varies within the interval I, the fuzzy sets \(\big{\langle}\int_{0}^{T}X(s)dW(s)\big{\rangle}\) exhibit continuity with respect to the d\({}_{\infty}\) metric.