Identifying the Predictable Drift of a Semimartingale from Marginal Laws

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 its marginal laws at a succession of times, so that the pairing between observations, and with it the likelihood of the path, is lost. 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. We estimate it jointly with couplings between adjacent empirical marginals, constrained so that their conditional first moment agrees with a parametric drift, the parameter being identified by a rank condition upon the mean features. 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. Its fixed point is available in closed form; it converges locally $Q$-linearly under either an exact global solution of the identification sub-problem or the acquiescence condition of Hardt, Ma and Recht; and it shows that the couplings inform the parameter through the mean displacements between marginals and through nothing else. For the estimator itself we obtain rates at a fixed mesh, separated into Monte-Carlo, estimation and grid contributions.