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

Abstract

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 random amplitude, the two prescriptions differ by a finite amount at every event, and that where $I$ vanishes the difference can become qualitative. The Marcus kick, the jump-process counterpart of the Stratonovich prescription, is the flow map of $dx/ds=I(x)$: no kick can cross a zero $x^\ast$ of $I$, for any jump and waiting-time statistics. The Itô kick is only the tangent map of that flow. For $I(x)=1+βx$ it crosses $x^\ast=-1/β$ if and only if $ξβ>1$ and collapses the whole line onto $x^\ast$ at $ξβ=1$; below threshold it is itself a Marcus kick with a remapped jump law. Bounded jump laws thus have a threshold, while unbounded ones leak across the barrier for \emph{any} parameters. The discrepancy cannot be absorbed into a renormalised drift beyond lowest order, and only the Marcus kick is generated by a single operator, which makes the master equation of Bianucci, Bologna and Mannella (2026) intrinsically Marcus. Numerically, the standard Heun kick is a second-order truncation of the Marcus map that preserves the barrier only because its order is even; it shortens every jump, biasing second moments in proportion to the squared jump amplitude and the jump kurtosis, and depleting the region near the barrier. We give the exact-flow integrator that replaces it.