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