# Fuzzy Fractional Brownian Motion: Review and Extension
## 3 Proofs and Discussion
### Main Results--Fuzzy Stochastic Differential Equation When H \(>\) %
\begin{array}{ll}\operatorname*{E}[\sup_{u\in[0,t]}d_{\infty}^{2}\left(X(u),\right.&X^{\varepsilon}(u)\right)\left.\left.\right]\\ &\leq 2\operatorname*{E}\sup_{u\in[0,t]}\;\int_{0}^{u}d_{\infty}^{2} \left(f(s,\;X(s),\;f(s,X^{\varepsilon}(s)\;)ds\right.\right.\\ &+\left.8e^{2a}\text{Esup}\;|\int_{0}^{u}g(s,\;X^{\varepsilon}(s))(dB(s))|^{2} \right.\\ &+\left.8a^{2}\text{Esup}\;|\int_{0}^{u}(g(s,\;X^{\varepsilon}(s))(\int_{0}^{ s}[(s-r+\varepsilon)^{a-1}\right.\\ &\left.-\left(s-r\right)^{a-1}\big{]}dB(s)\big{]}ds|^{2}+8a^{2}\text{Esup}\;| \int_{0}^{u}(g(s,\;X^{\varepsilon}(s)\right.\\ &-\left.g(s,X^{\varepsilon}(s)\right)\int_{0}^{s}\Big{[}(s-r+\varepsilon)^{a- 1}dB(s)\Big{)}ds|^{2}\\ &+\left.8e^{2a}\text{Esup}\;|\int_{0}^{u}(g(s,\;X(s)-g(s,\;X^{ \varepsilon}(s))(dB(s)|^{2}\right.\end{array} \]  
We now have  
\[ \begin{array}{ll}\operatorname*{E}[\sup_{u\in[0,t]}d_{\infty}^{2}\left(X(u),&X^{\varepsilon}(u)\right)]\\ &\leq 2\text{Esup}\;\int_{0}^{u}d_{\infty}^{2}\left(f(s,\;X(s),\;f(s,X^{ \varepsilon}(s)\;)ds\right.\\ &+\left.8e^{2a}\text{Esup}\;|\int_{0}^{u}g(s,\;X^{\varepsilon}(s))(dB(s))|^{2} \right.\\ &\left.+\left.8a^{2}\text{Esup}\;|\int_{0}^{u}\int_{s}^{u}(g(r,\;X^{ \varepsilon}(r))\big{[}(r-s+\varepsilon)^{a-1}\right.\\ &\left.-\left(r-s\right)^{a-1}\big{]}drdB(s)\right)\big{]}^{2}\\ &+\left.8a^{2}\text{Esup}\;|\int_{0}^{u}\int_{s}^{u}(g(r,\;X^{ \varepsilon}(r)\right.\\ &\left.-\left(s,X^{\varepsilon}(s)\right)\Big{[}(r-s+\varepsilon)^{a-1}drdB(s) \Big{)}\right]^{2}\\ &+\left.8e^{2a}\text{Esup}\;|\int_{0}^{u}(g(s,\;X(s)-g(s,\;X^{ \varepsilon}(s))(dB(s)|^{2}\right.\end{array} \]  
We will now apply [PERSON]'s inequality for [PERSON] motion to the second, third, fourth, and fifth term, which states that \(\operatorname*{E}[\sup_{u\in[0,t]}d_{\infty}^{2}\;|B_{t}|]^{2}\leq 4E|B_{t}|^{2}\), [PERSON]'s inequality to the third and fourth term, which states that \(f{\subset}{\rm L}^{\rm P}\), \({\rm q}{\subset}{\rm L}^{\rm Q}\), \(\int fg\leq||f||_{P}||g||_{Q}\), and Ito isometry to the second, third, fourth, and last terms, which states that \({\rm E}\Big{(}\int_{0}^{t}\int f_{t}dW_{t})^{2}\Big{)}=\)\(E(\int_{0}^{t}(f_{t})^{2}dt)\). \[ \begin{array}{ll}{\rm E}[\sup_{u\in[0,t]}d_{\infty}^{2}\ (X(u),&X^{\varepsilon}(u))\ \ ]\\ &\leq\ 2{\rm E}\sup_{u\in[0,t]}\ \int_{0}^{t}d_{\infty}^{2}\ (f(s,\ X(s),\ f(s,X^{ \varepsilon}(s)\ )ds\\ &+\ 32\varepsilon^{2a}{\rm E}\int_{0}^{u}g^{2}(s,\ X^{\varepsilon}(s))ds\\ &+\ 32a^{2}{\rm E}\int_{0}^{t}\Bigl{[}\int_{s}^{t}g^{2}(r,\ X^{ \varepsilon}(r))\Bigl{[}(r-s+\varepsilon)^{a-1}\\ &-\ r-s)^{a-1}\Bigr{]}\,dr\int_{s}^{t}\Bigl{[}(r-s+\varepsilon)^{a-1}-\bigl{(} r-s)^{a-1}\bigr{]}dr\Bigr{)}ds\Bigr{)}\\ &+\ 32a^{2}{\rm E}\int_{0}^{t}\int_{s}^{t}\Bigl{(}g(r,\ X^{\varepsilon}(r)-g(s,X ^{\varepsilon}(s)\ )^{2}(r-s\\ &+\ \varepsilon)^{a-1}dr\int_{s}^{t}(r-s+\varepsilon)^{a-1}\,dr\Bigr{)}ds\\ &+\ 32\varepsilon^{2a}{\rm E}\int_{0}^{u}(g(s,\ X(s)-g(s,\ X^{ \varepsilon}(s))^{2}ds.\end{array} \]  
Using (17), (18), (23), and assumptions A1-A3,  
\[ \begin{array}{ll}{\rm E}[\sup_{u\in[0,t]}d_{\infty}^{2}\ (X(u),&X^{ \varepsilon}(u))]]\\ &\leq 2L^{2}E\ \int_{0}^{t}d_{\infty}^{2}\ (X(s),\ X^{\varepsilon}(s))ds\\ &+\ 64\varepsilon^{2a}C^{2}\int_{0}^{t}\Bigl{(}1+E\Bigl{|}\Bigl{|} \Bigl{[}X^{\varepsilon}(s)\Bigr{]}^{0}\Bigl{|}\Bigr{|}^{2}\Bigr{)}ds\\ &+\ 64C^{2}(T+\varepsilon)^{2a}\frac{a+1}{a}\varepsilon^{a}\int_{0}^{t} \bigl{(}1+E\Bigl{|}\bigl{|}\bigl{[}X^{\varepsilon}(s)\bigr{]}^{0}\bigl{|} \bigr{|}^{2}\bigr{)}ds\\ &+\ 32L^{2}(T+\varepsilon)^{2a}{\rm E}\int_{0}^{t}d_{\infty}^{2}\ (X(s),\ X^{ \varepsilon}(s))ds\\ &+\ 32L^{2}\varepsilon^{2a}{\rm E}\int_{0}^{t}d_{\infty}^{2}\ (X(s),\ X^{ \varepsilon}(s))ds\\ &\leq\ \Bigl{(}2L^{2}+32L^{2}(T+\varepsilon)^{2a}\\ &+\ 32L^{2}\varepsilon^{2a}\Bigr{)}\int_{0}^{t}E\sup_{u\in[0,s]}d_{ \infty}^{2}\ (X(u),\ X^{\varepsilon}(u))ds+(64\varepsilon^{2a}C^{2}\\ &+\ 64C^{2}(T+\varepsilon)^{2a}\frac{a+1}{a}\varepsilon^{a}\Bigr{)}\int_{0}^{t} \bigl{(}1+E\bigl{|}\bigl{[}X^{\varepsilon}(s)\bigr{]}^{0}\bigl{|}\bigr{|}^{2} \bigr{)}ds\end{array}