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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.

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