Estimating a class of diffusions from discrete observations via approximate maximum likelihood method

ABSTRACT

An approximate maximum likelihood method of estimation of diffusion parameters $(\vartheta ,\sigma )$ based on discrete observations of a diffusion X along fixed time-interval $[0,T]$ and Euler approximation of integrals is analysed. We assume that X satisfies a stochastic differential equation (SDE) of form $\mathrm{d}{X}_t=\mu (X_t,\vartheta )\mathrm{d}t+\sqrt{\sigma }b\left(X_t\right)\mathrm{d}{W}_t$, with non-random initial condition. SDE is nonlinear in $\vartheta$ generally. Based on assumption that maximum likelihood estimator ${\hat{\vartheta }}_T$ of the drift parameter based on continuous observation of a path over $[0,T]$ exists we prove that measurable estimator $({\hat{\vartheta }}_{n,T},{\hat{\sigma }}_{n,T})$ of the parameters obtained from discrete observations of X along $[0,T]$ by maximization of the approximate log-likelihood function exists, ${\hat{\sigma }}_{n,T}$ being consistent and asymptotically normal, and ${\hat{\vartheta }}_{n,T}-{\hat{\vartheta }}_T$ tends to zero with rate ${\sqrt{\delta }}_{n,T}$ in probability when ${\delta }_{n,T}=\underset{0\le i<n}{max}(t_{i+1}-t_i)$ tends to zero with T fixed. The same holds in case of an ergodic diffusion when T goes to infinity in a way that $T{\delta }_n$ goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of ${\hat{\vartheta }}_{n,T}$, ${\hat{\sigma }}_{n,T}$ and asymptotic efficiency of ${\hat{\vartheta }}_{n,T}$ in this case.

KEYWORDS:

• Parameter estimation
• diffusion processes
• discrete observation

AMS subject classifications:

• 62M05
• 62F12
• 60J60

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work has been partially supported by Croatian Science Foundation under the project 3526, and by Ministry of Science, Education and Sports, Republic of Croatia [Grants 037-0372790-2800 and 037058].