arXiv subjects

David Oechsler

Publications and source records attributed to David Oechsler.

4 recordsLinked to original sources

arXiv2023

L\'evy Langevin Monte Carlo

Analogue to the well-known Langevin Monte Carlo method, in this article we provide a method to sample from a target distribution $\pi$ by simulating a solution of a stochastic differential equation. Hereby, the stochastic differential equation is driven by a general L\'evy process which - other than in the case of Langevin Monte Carlo - allows for non-smooth targets. Our method will be fully explored in the particular setting of target distributions supported on the half-line $(0,\infty)$ and a compound Poisson driving noise. Several illustrative examples conclude the article.

math.PR↗

arXiv2022

Invariant measures of L\'evy-type operators and their associated Markov processes

A distributional equation as a criterion for invariant measures of Markov processes associated to L\'evy-type operators is established. This is obtained via a characterization of infinitesimally invariant measures of the associated generators. Particular focus is put on the one-dimensional case where the distributional equation becomes a Volterra-Fredholm integral equation, and on solutions to L\'evy-driven stochastic differential equations. The results are accompanied by various illustrative examples.

math.PR↗

arXiv2021

On $q$-scale functions of spectrally negative L\'evy processes

We obtain series expansions of the $q$-scale functions of arbitrary spectrally negative L\'evy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit $q$-scale functions. Moreover, we study smoothness properties of the $q$-scale functions of spectrally negative L\'evy processes with infinite jump activity. This complements previous results of Chan et al. [7] for spectrally negative L\'evy processes with Gaussian component or bounded variation.

math.PR↗

arXiv2020

On $q$-scale functions of spectrally negative compound Poisson processes

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes. For spectrally negative compound Poisson processes with positive drift, a new representation of the $q$-scale functions in terms of the characteristics of the process is derived. Moreover, similar representations of the derivatives and the primitives of the $q$-scale functions are presented. The obtained formulae for the derivatives allow for a complete exposure of the smoothness properties of the considered $q$-scale functions. Some explicit examples of $q$-scale functions are given for illustration.

math.PR↗