An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model

TL;DR

This paper introduces an approximate maximum likelihood estimator for drift parameters in multidimensional diffusion models, demonstrating its convergence to the continuous-time estimator as observation intervals shrink, under uniform ellipticity conditions.

Contribution

The paper develops and analyzes an approximate maximum likelihood estimator for multidimensional diffusion models with nonlinear drift dependence, proving its stable convergence to the true estimator.

Findings

Estimator converges to the continuous-time MLE as sampling frequency increases.

Convergence is stable in law, accounting for randomness in the diffusion path.

Uniform ellipticity is essential for the theoretical results.

Abstract

For a fixed $T$ and $k \geq 2$, a $k$-dimensional vector stochastic differential equation $dX_t=\mu(X_t, \theta)dt+\nu(X_t)dW_t,$ is studied over a time interval $[0,T]$. Vector of drift parameters $\theta$ is unknown. The dependence in $\theta$ is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter $\overline{\theta}_n\equiv \overline{\theta}_{n,T}$ obtained from discrete observations $(X_{i\Delta_n}, 0 \leq i \leq n)$ and maximum likelihood estimator $\hat{\theta}\equiv \hat{\theta}_T$ obtained from continuous observations $(X_t, 0\leq t\leq T)$, when $\Delta_n=T/n$ tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on $\hat{\theta}$ and on path $(X_t, 0 \leq t\leq T)$. The uniform ellipticity of diffusion matrix $S(x)=\nu(x)\nu(x)^T$ emerges as the main assumption on the diffusion coefficient function.