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 1/2 D II'. We show that for renewal shot noise with state-dependent gain, ẋ=−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∗ 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∗=−1/β if and only if ξβ>1 and collapses the whole line onto x∗ 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.
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BibTeX
@misc{bianucci2026itô,
title = {{Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus}},
author = {Marco Bianucci and Riccardo Mannella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39511},
eprint = {2610.39511},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39511},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus
AU - Marco Bianucci
AU - Riccardo Mannella
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39511
UR - https://arxiv.org/abs/2610.39511
AB - For Gaussian white noise the Itô and Stratonovich prescriptions differ by O(dt) per time step, which accumulates into the familiar spurious drift 1/2 D II'. We show that for renewal shot noise with state-dependent gain, ẋ=−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^∗ 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^∗=−1/β if and only if ξβ>1 and collapses the whole line onto x^∗ 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.
ER -