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

We consider the realization of the operator L(theta,a)u(x) :=x(2a)u ''(x) + (ax(2a-1) + theta x(a))u'(x), acting on C[0, + infinity], for theta is an element of R, a G is an element of R. We show that L-theta,L-a, with the so called Wentzell boundary conditions, generates a Feller semi-group for any theta is an element of R, a is an element of 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, + infinity] is the space of all real valued continuous functions on [0, + infinity) which admit finite limit at + infinity.