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We consider the probabilistic approach to the problems treated in (7). We focus on the difiusion models generated by L~;a u(x) := µ2 x 2a u 00 + ( µ2 a x 2ai1 + µ1 x a ) u 0 ; ~ = (µ1; µ2) T 2 R £ (0; +1), when a = 1 or a = 1 and face the problem of flnding optimal (in the asymptotic sense) estimators of the unknown parameter vector ~. Lµ;a u(x) := x 2a u 00 (x) + (ax 2ai1 +µx a )u 0 (x); where µ 2 R, a 2 R, acting on suitable spaces of real valued continuous functions. For 0 • a • 1 we obtained explicit representations of the semigroups generated by Lµ;a via perturbations of squares of suitable generators of groups. In this paper we focus on the difiusion models generated by L~;a u(x) := µ2x 2a u 00 (x) + (µ2ax 2ai1 +µ1x a )u 0 (x); where ~ = (µ1;µ2) T 2 R £ (0; +1), and either a = 1 or a = 1. Such a choice of a is motivated by the applications to genetics and flnancial mathematics. The problem of flnding optimal (in the asymptotic sense) estimators of the unknown parameter vector ~ is considered for the case a = 1 (the case a = 1 is studied in (10)).

Markov semigroups and estimating functions, with applications to some financial models