Identifying the Predictable Drift of a Semimartingale from Marginal Laws

2026-09-27

AI Summary

This study addresses the identification of the predictable drift of a semimartingale solely from its marginal distributions, with particular emphasis on hidden linear dynamical systems. The proposed method estimates the drift from repeated cross-sectional data by formulating the task as a bilinear optimization problem. Establishing that this formulation is NP-hard, the authors develop a spatial branch-and-bound algorithm to compute globally optimal solutions, alongside a block coordinate decomposition alternative. This work makes the first contribution toward drift identification and computational complexity analysis for hidden linear dynamical systems. Furthermore, it derives convergence rates under a fixed grid discretization, precisely disentangling the respective contributions of Monte Carlo sampling error, estimation error, and grid approximation error to the overall accuracy.

Abstract

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 $\ell^1$ 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.

Problem

• semimartingale
• predictable drift
• marginal laws
• hidden linear dynamical system
• cross-sectional data

Innovation

• semimartingale
• hidden linear dynamical system
• bilinear quadratically constrained programme
• spatial branch and bound
• marginal laws