Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes

TL;DR

This paper investigates the asymptotic properties of the maximum likelihood estimator for the drift parameter in time-inhomogeneous diffusion processes, establishing conditions for convergence to Dickey-Fuller limit distributions.

Contribution

It provides new sufficient conditions for the MLE's asymptotic distribution in both standard and perturbed drift diffusion models, extending existing theoretical results.

Findings

MLE normalized by Fisher information converges to Dickey-Fuller distribution.

Conditions are identified for both standard and perturbed drift SDEs.

Results apply to processes with time-dependent coefficients and nonlinear drifts.

Abstract

We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=\alpha b(t)X_t dt + \sigma(t) dB_t$, $t\in[0,T)$, with a parameter $\alpha\in R$, where $T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process. We formulate sufficient conditions under which the MLE of $\alpha$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics. Next we study a SDE $dY_t=\alpha b(t)a(Y_t) dt + \sigma(t) dB_t$, $t\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^\gamma)$ with some $\gamma\in[0,1)$. We give again sufficient conditions under which the MLE of $\alpha$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics.