The SDE is $dX_t=4X_t^{\frac{3}{4}}dB_t+6X_t^{\frac{1}{2}}dt,\ X_0=0$.
The solution obviously is $X_t=B_t^4$.
However is it the unique (strong) solution for this SDE?
I mean if it is true that there are no $X_t'$ such that it satisfies the SDE and $P(X_t=X_t'\ \forall t\geq 0)=1$.
Surely the Lipschitz condidions doesn't hold here, so the uniquness and existence theorem can't be used...
The solution is not unique.
Define $\tau_1 = \inf\{t\ge 0: B_t=1\}$, $\tau_0 = \inf\{t\ge \tau_1: B_t = 0\}$. Then $X_t' = B^4_{t\wedge \tau_0}$ is another strong solution.