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September  2007, 8(2): 511-527. doi: 10.3934/dcdsb.2007.8.511

Generators of Feller semigroups with coefficients depending on parameters and optimal estimators

Jerome A. Goldstein 1, , Rosa Maria Mininni 2, and  Silvia Romanelli 2,

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States
2. 

Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona, 4, 70125 Bari, Italy, Italy

Received  September 2006 Revised  February 2007 Published  June 2007

We consider the realization of the operator $L_{\theta, a}u(x) $:$= x^{2 a}u''(x) \ + \ (a x^{2 a - 1} + \theta x^a)u'(x)$, acting on $C[0,+\infty]$, for $\theta\in\R$, $a\in\R$. We show that $L_{\theta, a}$, with the so called Wentzell boundary conditions, generates a Feller semigroup for any $\theta\in\R$, $a\in\R$. The problem of finding optimal estimators for the corresponding diffusion processes is also discussed, in connection with some models in financial mathematics. Here $C[0,+\infty]$ is the space of all real valued continuous functions on $[0,+\infty)$ which admit finite limit at $+\infty$.
Keywords: stochastic differential equations, optimal estimators., diffusion processes, Feller semigroups, operator semigroups.
Mathematics Subject Classification: Primary: 47D07, 60J60, 62M05; Secondary: 60H10, 91B7.
Citation: Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generators of Feller semigroups with coefficients depending on parameters and optimal estimators. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 511-527. doi: 10.3934/dcdsb.2007.8.511
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