1
|
Kim P, Song R, Vondraček Z. Positive self-similar Markov processes obtained by resurrection. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.11.014] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
|
2
|
Abstract
AbstractWe study the existence, optimality, and construction of non-randomised stopping times that solve the Skorokhod embedding problem (SEP) for Markov processes which satisfy a duality assumption. These stopping times are hitting times of space-time subsets, so-called Root barriers. Our main result is, besides the existence and optimality, a potential-theoretic characterisation of this Root barrier as a free boundary. If the generator of the Markov process is sufficiently regular, this reduces to an obstacle PDE that has the Root barrier as free boundary and thereby generalises previous results from one-dimensional diffusions to Markov processes. However, our characterisation always applies and allows, at least in principle, to compute the Root barrier by dynamic programming, even when the well-posedness of the informally associated obstacle PDE is not clear. Finally, we demonstrate the flexibility of our method by replacing time by an additive functional in Root’s construction. Already for multi-dimensional Brownian motion this leads to new class of constructive solutions of (SEP).
Collapse
|
3
|
Double hypergeometric Lévy processes and self-similarity. J Appl Probab 2021. [DOI: 10.1017/jpr.2020.86] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
AbstractMotivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.
Collapse
|
4
|
Mucha J. Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin. ELECTRON J PROBAB 2021. [DOI: 10.1214/21-ejp594] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Jacek Mucha
- Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology ul. Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland
| |
Collapse
|
5
|
|
6
|
Döring L, Weissmann P. Stable processes conditioned to hit an interval continuously from the outside. BERNOULLI 2020. [DOI: 10.3150/19-bej1134] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
|
7
|
|
8
|
|
9
|
Forman N, Pal S, Rizzolo D, Winkel M. Uniform control of local times of spectrally positive stable processes. ANN APPL PROBAB 2018. [DOI: 10.1214/17-aap1370] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
|
10
|
Abstract
We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes.
Collapse
|
11
|
Abstract
We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes.
Collapse
|
12
|
|
13
|
Kuznetsov A, Kyprianou A, Pardo JC, Watson A. The hitting time of zero for a stable process. ELECTRON J PROBAB 2014. [DOI: 10.1214/ejp.v19-2647] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
|