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**Döring, Leif; Gonon, Lukas; Prömel, David J. and Reichmann, Oleg (2019): On Skorokhod embeddings and Poisson equations. In: Annals of Applied Probability, Vol. 29, No. 4**

**Full text not available from 'Open Access LMU'.**

## Abstract

The classical Skorokhod embedding problem for a Brownian motion W asks to find a stopping time τ so that Wτ is distributed according to a prescribed probability distribution μ. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let X be a Markov process with initial marginal distribution μ0 and let μ1 be a probability measure. The task is to find a stopping time τ such that Xτ is distributed according to μ1. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given μ0,μ1 and the task of giving a solution which is as explicit as possible. If μ0 and μ1 have positive densities h0 and h1 and the generator A of X has a formal adjoint operator A∗, then we propose necessary and sufficient conditions for the existence of an embed- ding in terms of the Poisson equation A∗H = h1 − h0 and give a fairly explicit construction of the stopping time using the solution of the Poisson equation. For the class of L ́evy processes we carry out the procedure and extend a result of Bertoin and Le Jan to L ́evy processes without local times.

Item Type: | Journal article |
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Faculties: | Mathematics, Computer Science and Statistics > Mathematics > Workgroup Financial Mathematics |

Subjects: | 500 Science > 510 Mathematics |

ISSN: | 1050-5164 |

Language: | English |

Item ID: | 110071 |

Date Deposited: | 26. Mar 2024, 12:34 |

Last Modified: | 08. Sep 2024, 18:18 |