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By definition, your $S$ is a continuous process and $$d\langle S\rangle_t=\sigma_S^2\,S_t^2\,dt\,.$$ The applicable Ito formula is the traditional one $$dV(t,S_t)=\partial_tV(t,S_t)\,dt+\partial_SV(t,S_t)\,dS_t+\frac{1}{2}\partial^2_SV(t,S_t)\,d\langle S\rangle_t\,.$$ Now plug in your expressionfor $dS_t\,.$