Abstract
In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type \[X_{t}=e^{-\xi_{t}}\left(X_{0}+\int_{0}^{t}e^{\xi_{u-}}du \right), \] where $\xi_{s}$ is a Lévy process. Our primal goal is to estimate the characteristics of the Lévy process $\xi$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the Lévy triplet of $\xi$, which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.
Highlights
Lett≥0 be a compound Poisson process (CP Pt)t≥0 with drift μ ∈ R, that is, ξt = μt + CP Pt, NtCP Pt := Yk, k=1 where Nt is a Poisson process with intensity λ, and Y1, Y2, . . . are i.i.d. r.v’s independent of Nt
We present a novel approach towards estimating the Levy triplet of ξ, which is based on the Mellin transform technique
A comprehensive study of the generalized Ornstein-Uhlenbeck (GOU) processes and an extended list of references can be found in the thesis of Behme [2], where, in particular, it is shown that Xt satisfies the following SDE: dXt = Xt−dUt + dt, where Ut := −ξt +
Summary
One important result from the theory of GOU processes is that, under some conditions, the process (1) is stationary with invariant stationary distribution given by the distribution of the following exponential functional of ξ :. To the best of our knowledge, the statistical inference for GOU processes of the form (1) from their low-frequency observations has not been yet studied in the literature. Let us mention that the problem of statistical inference for Levy processes (or some of their generalizations) observed at low frequency was the subject of many studies, see, e.g. Neumann and Reiß [28], Reiß [29], Kappus [17], Trabs [32] and Jongbloed et al [16].
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