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Martingale estimating functions for Feller diffusion processes generated by degenerate elliptic operators. (English) Zbl 1082.47036
The authors study a nonergodic diffusion process \(X=(X_t)_{t\geq 0}\) taking values in \([0,+\infty)\) which is a weak solution of the SDE \[ dX_t=\theta X_t\,dt+\sqrt{2}X_t\,dW_t, \quad X_0=x_0, \] with parameter \(\theta\in\mathbb R\). The associated (analytic) Feller semigroup on \(C[0,+\infty]\) has generator \(A_{\theta}\), \(D(A_{\theta})=\{u\in C[0,+\infty] \cap C^2(0,+\infty);\, A_{\theta}u:= x^2 u''+\theta x u'\to 0\) as \(x\to 0\) or \(x\to\infty\}\), a case not considered in [S. Karlin and H. M. Taylor, “A second course in stochastic processes” (Academic Press, New York etc.) (1981; Zbl 0469.60001)]. To construct optimal estimators for \(\theta\) based on discrete observations of \(X\), the methods proposed by M. Sørensen and coauthors in [Bernoulli 1, 17–39 (1995; Zbl 0830.62075)] and [ibid. 5, 299–314 (1999; Zbl 0980.62074)] are applied and compared. The estimators obtained are asymptotically normal and consistent, their behaviour is illustrated by simulation.

47D07 Markov semigroups and applications to diffusion processes
60J60 Diffusion processes
60G44 Martingales with continuous parameter
62M05 Markov processes: estimation; hidden Markov models