Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus

For Gaussian white noise the Itô and Stratonovich prescriptions differ by $O(dt)$ per time step, which accumulates into the familiar spurious drift $\tfrac12 D\,II'$. We show that for renewal shot noise with state-dependent gain, $\dot x=-C(x)-I(x)ξ[t]$ with $ξ[t]$ a train of Dirac spikes of finite