How can I solve $$dX_t=|X_t|^\alpha dW_t\ \ ?$$
What I tried is to set $X_t=u_tdt+v_tdW_t$. Then, by Itô formula $$df(X_t)=f'(X_t)u_tdt+f'(X_t)v_tdW_t+\frac{1}{2}f''(X_t)v_t^2dt.$$
So, we should have $$f'(X_t)u_t=\frac{1}{2}f''(X_t)v_t^2$$ and $$f'(X_t)v_t=|u_t+v_tdW_t|^\alpha .$$
but how can I continue ?