engleski

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

An approximate maximum likelihood method of estimation of diffusion parameters (ϑ, σ) 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 dXt=μ(Xt, ϑ)dt+σ−−√b(Xt)dWt, with non-random initial condition. SDE is nonlinear in ϑ generally. Based on assumption that maximum likelihood estimator ϑˆT of the drift parameter based on continuous observation of a path over [0, T] exists we prove that measurable estimator (ϑˆn, T, σˆn, T) of the parameters obtained from discrete observations of X along [0, T] by maximization of the approximate log-likelihood function exists, σˆn, T being consistent and asymptotically normal, and ϑˆn, T−ϑˆT tends to zero with rate δ√n, T in probability when δn, T=max0≤i<n(ti+1−ti) tends to zero with T fixed. The same holds in case of an ergodic diffusion when T goes to infinity in a way that Tδn goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of ϑˆn, T, σˆn, T and asymptotic efficiency of ϑˆn, T in this case.

parameter estimation ; diffusion processes ; discrete observation

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