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Type: Text × Subject: Laplace transform ×

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Now showing 1 - 3 of 3
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    Generalized Post-Widder inversion formula with application to statistics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Belomestny, Denis; Mai, Hilmar; Schoenmakers, John
    In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models.We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.
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    Memory equations as reduced Markov processes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Stephan, Artur; Stephan, Holger
    A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.
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    Statistical Skorohod embedding problem and its generalizations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Belomestny, Denis; Schoenmakers, John G.M.
    Given a Lévy process L, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time T based on i.i.d. sample from LT. Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose consistent estimators for the density of T; derive their conver-gence rates and prove their optimality. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of T. We also consider the application of our results to the problem of statistical inference for variance-mean mixture models and for time-changed Lévy processes.