Identifying the Predictable Drift of a Semimartingale from Marginal Laws
Abstract
A special semimartingale admits a unique decomposition X=X0+M+A into a local martingale M and a predictable finite-variation part A. We consider the identification of A when X is observed only through repeated cross-sections. The estimand is then the projection of the sampled predictable compensator onto the observable feature filtration, namely the current state together with whatever randomness is shared across the population, so that at a fixed diffusion coefficient the marginal flow identifies the drift only up to a Markovian projection. If the drift is an affine functional of an observed lag window, the joint problem is a convex quadratic programme whose solution is the pseudo-panel regression of econometrics. Our principal concern is the case, which we believe not to have been treated before, in which the drift is the output of a hidden linear dynamical system whose dynamics are themselves to be identified from the marginals. The joint problem is then a bilinear quadratically constrained programme, which we solve to certified global optimality by spatial branch and bound; with unpenalised state disturbances and a drift basis growing with the grid it is NP-hard already in latent dimension one, by reduction from ℓ1 rank-one matrix approximation, whereas the complexity of the deterministic system at fixed latent dimension remains open. A block-coordinate decomposition offers a cheaper alternative. For the estimator itself, we obtain rates at a fixed mesh, separated into Monte-Carlo, estimation and grid contributions.
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BibTeX
@misc{marecek2026identifying,
title = {{Identifying the Predictable Drift of a Semimartingale from Marginal Laws}},
author = {Jakub Marecek and Enrico Biffis and Abigail Langbridge and Robert Shorten},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33372},
eprint = {2609.33372},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33372},
}
RIS
TY - UNPB
TI - Identifying the Predictable Drift of a Semimartingale from Marginal Laws
AU - Jakub Marecek
AU - Enrico Biffis
AU - Abigail Langbridge
AU - Robert Shorten
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33372
UR - https://arxiv.org/abs/2609.33372
AB - A special semimartingale admits a unique decomposition X=X_0+M+A into a local martingale M and a predictable finite-variation part A. We consider the identification of A when X is observed only through repeated cross-sections. The estimand is then the projection of the sampled predictable compensator onto the observable feature filtration, namely the current state together with whatever randomness is shared across the population, so that at a fixed diffusion coefficient the marginal flow identifies the drift only up to a Markovian projection. If the drift is an affine functional of an observed lag window, the joint problem is a convex quadratic programme whose solution is the pseudo-panel regression of econometrics. Our principal concern is the case, which we believe not to have been treated before, in which the drift is the output of a hidden linear dynamical system whose dynamics are themselves to be identified from the marginals. The joint problem is then a bilinear quadratically constrained programme, which we solve to certified global optimality by spatial branch and bound; with unpenalised state disturbances and a drift basis growing with the grid it is NP-hard already in latent dimension one, by reduction from ℓ^1 rank-one matrix approximation, whereas the complexity of the deterministic system at fixed latent dimension remains open. A block-coordinate decomposition offers a cheaper alternative. For the estimator itself, we obtain rates at a fixed mesh, separated into Monte-Carlo, estimation and grid contributions.
ER -