Generators of Feller semigroups with coefficients depending on parameters and optimal estimators. (English) Zbl 1152.47032

Goldstein, Jerome A.; Mininni, Rosa Maria; Romanelli, Silvia

Discrete Contin. Dyn. Syst., Ser. B 8, No. 2, 511-527 (2007).

This paper follows one Feller’s classical boundary theory on the realization of semigroups of one-dimensional elliptic operators to the special operators with diffusion coefficient of power function \(x^{2 \alpha}\) and drift \(a x^{2 \alpha -1}+ \theta x^ \alpha \) on the interval \((0, + \infty)\) under Wentzell’s boundary conditions. The most interesting part is for the GMM estimation (Generalized Method of Moments by L. P. Hansen) of parameters \(\alpha\) and \(\theta\), knowing \(n_T\) values of the diffusion processes at \(n_T\) fixed times, in the special case of \( \alpha = 1/2\), which is connected to financial mathematics. A simulation study of the estimators is performed by discrete approximation of the SDEs and the simulation results obtained are discussed in the paper.

Reviewer: Qian Minping (Beijing)

MSC:

47D07	Markov semigroups and applications to diffusion processes
60J60	Diffusion processes
62M05	Markov processes: estimation; hidden Markov models
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
91B70	Stochastic models in economics

Keywords:

Feller semigroups; generator; elliptic operators; optimal estimator; generalized method of moments