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Least squares estimation for \(\alpha\)-fractional bridge with discrete observations. (English) Zbl 1474.62302

Summary: We consider a fractional bridge defined as \(d X_t = - \alpha (X_t /(T - t)) d t + d B_t^H\), \(0 \leq t < T\), where \(B^H\) is a fractional Brownian motion of Hurst parameter \(H > 1 / 2\) and parameter \(\alpha > 0\) is unknown. We are interested in the problem of estimating the unknown parameter \(\alpha > 0\). Assume that the process is observed at discrete time \(t_i = i \Delta_n, i = 0, \ldots, n\), and \(T_n = n \Delta_n\) denotes the length of the “observation window.” We construct a least squares estimator \(\widehat{\alpha}_n\) of \(\alpha\) which is consistent; namely, \(\widehat{\alpha}_n\) converges to \(\alpha\) in probability as \(n \rightarrow \infty\).

MSC:

62M09 Non-Markovian processes: estimation
60G22 Fractional processes, including fractional Brownian motion
60H07 Stochastic calculus of variations and the Malliavin calculus
62M05 Markov processes: estimation; hidden Markov models
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