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 give a direct counterexample to the joint transition-density statements of Brandt and Santa-Clara (2002): along M = S → ∞, a sequence satisfying their rate condition and their own regularity assumptions, the simulated density converges in probability to zero for every K > 2 although the true density is strictly positive. 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, and we set out what such a proof would require. Issues specific to the exchange-rate application are analysed separately in appendices. The deposit contains the manuscript, the complete LaTeX and Python sources, and a reproducibility bundle. All 31 figures and tables regenerate bit-for-bit from the included code.