Parameter Estimation in Multidimensional Diffusion Models with Low Regularity Coefficients

This paper studies parameter estimation for discretely observed multidimensional diffusion models with low-regularity coefficients. Since the transition density of such models is typically unavailable in closed form, likelihood-based inference becomes difficult, especially in multidimensional settings. To address this problem, we construct and computationally investigate a Hermite-based quasi-maximum likelihood estimator based on a parametrix-type decomposition of the transition density. The approach yields a continuously differentiable quasi-likelihood function and allows the construction of a quasi-maximum likelihood estimator for the unknown parameter vector. In addition, conditional least-squares estimators based on first- and second-order discretizations are considered, together with one-step and Rao-type corrections. The numerical study is carried out for two nonlinear multidimensional diffusion models. The results show that the main practical differences between the competing procedures arise in the estimation of diffusion parameters. In the presented examples, the Hermite-based quasi-maximum likelihood estimator provides the most accurate and best-centered recovery of the diffusion parameter, while the corrected conditional least-squares estimators improve the corresponding uncorrected procedures to varying degrees. The drift parameter is recovered with broadly comparable accuracy by several methods. The emphasis of the present study is on computational construction and finite-sample numerical performance rather than on establishing a new asymptotic theory for the resulting estimator. These results indicate that Hermite-based quasi-likelihood estimation is a viable and computationally implementable tool for inference in multidimensional diffusion models with reduced regularity. The proposed framework may be useful in broader problems of stochastic modelling and numerical identification for nonlinear systems.