  We use Young integration (resp, bounded $p,q$-variation theory introduced in
\cite{Feng-Zhao}) to establish integration of determinate functions with
respect to local time of symmetric $\alpha$-stable L\'evy process, for $\alpha
\in ]1,2]$, in one parameter case (resp, in two parameter case). We then apply
these integrals to write the corresponding generalized It\^{o} formula.
Furthermore, some approximations schemes of the area integral w.r.t local time
are given.
