# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Main Results--Fuzzy Stochastic Differential Equation When H \(>\) %  
We will now focus on the special case of H \(>\) % and apply the Liouville form fBm [16; 23]. This is a type of fractional Gaussian process that is defined by a covariance function that ex hibits power-law scaling. Specifically, the covariance function is given by \(C(t,\ s)=\left|t-s\right|^{(2H-2)}\), where \(0<H<1\) is the Hurst exponent. Now, in this case, \(H>1\%\) for the process to have finite moments of order greater than or equal to two. If \(H<1\%\), the process has infinitely many moments and the covariance function becomes singular. This means that the covariance function does not satisfy the conditions of positive definiteness that are typically expected for covariance functions. When \(H>1\%\), the Liouville fBM does not have mean reversion. When \(H<1\%\), the fBM process does not possess the self-similarity and stationarity properties required for the Liouville form. A fractional Brownian motion with \(H<1\%\) is often referred to as \"rough\" or \"pathologically irregular.\" It exhibits strong long-range dependence, which means that distant points in the process are highly correlated. In the Liouville form, fBm is expressed as a stochastic integral involving a deterministic function and a standard Brownian motion. It has the following form [24]:  
\[X(t)=X_{0}+\int_{0}^{t}f(s,\ X(s))ds+\left\langle\int_{0}^{t}g(s,\ X(s))dB_{H}( s)\right\rangle,\ X0=X(0) \tag{10}\]  
where the FSDE is given by Equation (10), X(t) is the solution of the equation at time t, X\({}_{0}\) is the initial condition of X at time 0, f(s, X(s)) is a function that describes the drift of X, g(s, X(s)) is a function that describes the diffusion of X, and B\({}_{\rm H}\) is the Liouville fBm that drives the equation. The integral term in the equation represents the stochastic integral of g(s, X(s)) with respect to B\({}_{\rm H}\)(s), which is a type of integral that is defined in terms of the Ito integral and the covariance structure of the fBm. The angle brackets around the integral indicate that it is taken in the sense of the Stratonovich integral. The corresponding approximation equation to (10) is given by X\({}^{\xi}\)(t), where X\({}^{\xi}\)(t) is an approximation to X(t) that is obtained by discretising the integral term using a numerical scheme. The integral term in the approximation equation is approximated using a discrete version of the stochastic integral, which is defined in terms of the Riemann sum and the covariance structure of the discretised fBm, denoted by BH\({}^{\xi}\)(s) [16]. \[X^{\xi}(t)=X_{0}+\int_{0}^{t}f(s,\ X^{\xi}(s))ds+\left\langle\int_{0}^{t}g(s, \ X^{\xi}(s))dB^{\xi}{}_{H}(s)\right\rangle \tag{11}\]  
This section discusses a class of stochastic differential equations that are driven by a Liouville fBm, which is a type of fractional Gaussian process with a power-law covariance structure. Building on [16], we will now make a couple of assumptions to prove that (11) has a strong unique solution. The first assumption (**A1**) is that, for every \(u\), \(v\in F\)(R) and every \(t\in I\), there exists L \(>0\) such that max{d\({}_{\infty}\)(f(t, w, u)\), f(t, w, v)), \(\mid\) g(t, w, u) \(-\) g(t, w, v) \(\mid\) \(\mid\) \(\leq\) Ld\({}_{\infty}\)(u,v). This assumption helps to estimate the upper bound using constant L. The second assumption (**A2**) is the Cauchy-Schwarz inequality and supposes that there is C \(>0\) such that max{d\({}_{\infty}\)(f(t, w,u(0)), \(\mid\) g(t, w, u) \(\mid\) \(\leq\) C(1 + d\({}_{\infty}\)(u,\(0)). The last assumption (**A3**) is that the mappings f: \(1\times\Omega\times\) F(R) \(\sim\) F(R) and g: \(1\times\Omega\times\) F(R) \(\sim\) F(R) are measurable with respect to standard Brownian motion, formally, N \(\otimes\) B\({}_{\rm ds}\) \(\mid\) B\({}_{\rm ds}\)-measurable and N \(\otimes\) B\({}_{\rm ds}\) \(\mid\) B(R)-measurable, respectively. The product sigma-algebra is a way to combine two sigma-algebras (sets of events) from different sources.