Genon-Catalot, V.; Laredo, C. Limit theorems for the first hitting times process of a diffusion and statistical applications. (English) Zbl 0655.62088 Scand. J. Stat., Theory Appl. 14, 143-160 (1987). The authors study the problem of estimation of an unknown parameter \(\theta\) in the drift function of a diffusion process \(\{X_ t,\;t\geq 0\}\), \(X_ 0=x\) when only the first hitting times \(T_ a\) of levels a: \(T_ a=\inf \{t\geq 0;\;X_ t\geq a\}\) are observed. For diffusions with a positive drift, they obtain limit theorems for the hitting times process \(\{T_ a,\;x\leq a\leq A\}\), as the variance of the diffusion goes to zero. Applying these results, they define a contrast function based on the process \(\{T_ a,\;x\leq a\leq A\}\) and derive a minimum contrast estimator for \(\theta\). It is shown to be consistent, asymptotically normal and asymptotically equivalent to the maximum likelihood estimator based on complete observation of the process \(\{X_ t\}\) up to the first hitting time of \(A\). The main technical tool is a stochastic Taylor’s formula due to R. Azencott [Séminaire de probabilités XVI, Univ. Strasbourg 1980/81, Suppl.: Géométrie différentielle stochastique, Lect. Notes Math. 921, 237–285 (1982; Zbl 0484.60064)]. Reviewer: B. L. S. Prakasa Rao (Hyderabad) Cited in 4 Documents MSC: 62M05 Markov processes: estimation; hidden Markov models 60F05 Central limit and other weak theorems 62F12 Asymptotic properties of parametric estimators Keywords:drift parameter estimation; Skorokhod topology; diffusion process; hitting times process; contrast function; minimum contrast estimator; consistent; asymptotically normal; maximum likelihood estimator; stochastic Taylor’s formula Citations:Zbl 0484.60064 PDFBibTeX XMLCite \textit{V. Genon-Catalot} and \textit{C. Laredo}, Scand. J. Stat. 14, 143--160 (1987; Zbl 0655.62088)