Consider
$$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$
such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and is a stable process, and $\Lambda:t\rightarrow\mathbb{R}^+$ is deterministic. Numerically, the discrete form is
$$S_{t+\Delta t}=\Delta t\left(\sigma_S S_t\xi_t+(\zeta_t\Delta t+\omega_t)(\Lambda-S_t)\right)+S_t$$
where $\zeta\sim\mathcal{S}(\alpha,0,c,0)$ is Lévy alpha-stable distributed. Note that the characteristic function of $\zeta_t$ is
$$\varphi=e^{-\left|ct\right|^{\alpha}}.$$
So, when $\alpha=1$, we have the Cauchy distribution. Does a suitably modified version of Ito's lemma exist for the value $dV(t,S_t)$ of an option $V(t,S_t$)?