It\^o integral for a two-sided L\'evy process

In this article, we construct an It\^o integral with respect to a two-sided finite-variance L\'evy process $\{L(x)\}_{x\in \mathbb{R}}$, without a Gaussian component. Using Rosenthal inequality for discrete-time martingales, we give an estimate for the $p$-th moment of this integral, for any even integer $p\geq 2$. Then, using Poisson-Malliavin calculus, we show that the It\^o integral is an extension of the Hitsuda-Skorohod integral with respect to the compensated Poisson random measure associated to the L\'evy process.