On parameter estimation of diffusion processes: sequential and fixed sample size estimation revisited

We derive new properties of the sequential parameter estimators for diffusion-type processes $\mathbf{X}=\{X_t,\, 0\leq t\leq \tau \}$, where $\tau $ is a stopping time (this includes the case of fixed sample size estimate). Some earlier theoretical results in this direction can be found in the book [R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes, Springer, 2001]. Under an essentially less restrictive setting, we derive formulas for the moments of the maximum likelihood estimator (MLE) $\widehat{\lambda}_{\tau }$ for the parameter $\lambda $ of the drift coefficient $f_{t}(\lambda )=a_{t}-\lambda b_{t}$ and prove the exponential boundedness of $\widehat{\lambda}_{\tau }$ under a mild condition. In the provided examples we consider the mean-reverting ergodic diffusion process $\mathbf{X}$, where $b_{t}=X_{t}$, and the diffusion coefficient $\sigma_{t}=\sigma X_{t}^{\gamma }$. In particular, we provide nonasymptotic analytical and numerical results for the bias and mean-square error of $\widehat{\lambda}_{\tau }$ for the Ornstein–Uhlenbeck (O–U) and Cox–Ingersoll–Ross (CIR) processes when $\tau =T$ is a fixed sample size, and $\tau =\tau_{H}$ is a specially chosen stopping time that guarantees a prescribed magnitude of $1/H$ for the variance of $\widehat{\lambda}_{\tau_{H}}$.