Dimensional Asymptotics of Euler-Based Simulated Likelihood for Multidimensional Diffusions

Abstract

A widely used simulation method for likelihood inference in discretely observed diffusions approximates each transition density by simulating an Euler scheme to one step before the observed endpoint and averaging the final Gaussian density. We show that this endpoint estimator is a shrinking-bandwidth kernel estimator whose effective simulation size is S/MK/2, not S, where K is the state dimension, M the number of Euler substeps and S the number of simulations. We derive exact Brownian moments, the general local moment law, the corrected density-level normalisation, and the pointwise mean squared error, minimised at M ≈ S2/(K+4). We then show that the simulated density converges in probability to zero, at every pair of points and although the true density is strictly positive, along every sequence with S/MK/2 → 0. The threshold is sharp: the quantity that must diverge for the central limit theorem to hold is the one that must vanish for the estimator to collapse, with no gap between the two conditions. The diagonal M = S, which satisfies the rate condition of Brandt and Santa-Clara (2002) and their own regularity assumptions, is then the corollary that lands inside their hypotheses for every K > 2. This refutes the consistency asserted in their Lemma 2 and, since √S times the centred density then diverges, the central-limit statement of their Lemma 3. The proofs of their parameter-estimator theorems consequently fail, and in four dimensions their rate condition is incompatible with mean-square consistency of the simulated density. We do not prove that the resulting argmax estimator is inconsistent. In the opposite direction we prove that in a Gaussian location model the simulated maximiser is consistent along the very sequences on which the density collapses, in every dimension K > 2 and so in the application's four, at the cost of a polylogarithmic requirement on the number of observations. The counterexample therefore does not transfer from the density to the estimator there, and the conclusion of the published consistency theorem survives the failure of its proof. The rate is a separate matter and remains open. This record contains two documents. The theory paper (main.pdf) is the above. A companion note (companion.pdf), Mathematical Status of the Exchange-Rate Application in Brandt and Santa-Clara (2002), reports the separate question of what can and cannot be established about that paper's four-dimensional exchange-rate specification: the square-root boundaries, the reality and invertibility of the incompleteness state, the validity of the Brownian correlation matrix, and the minimum-incompleteness identification rule. Several of its findings are negative, establishing that no problem arises where one might have been suspected. Nothing in the theory paper depends on it; the dependence runs one way. The record also contains the complete LaTeX and Python sources and a reproducibility bundle. All 53 generated figures and tables regenerate bit-for-bit from the included code.