Stochastic integration with respect to cylindrical Lévy processes

Adam Jakubowski ^[1] ; Markus Riedle ^[2]
1. [1] Nicolaus Copernicus University
  
  Nicolaus Copernicus University
  
  Toruń, Polonia
2. [2] King's College London
  
  King's College London
  
  Reino Unido
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 2, 2017, págs. 4273-4306
Idioma: inglés
DOI: 10.1214/16-AOP1164
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Resumen
- A cylindrical Lévy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Lévy processes in Hilbert spaces. The space of admissible integrands consists of càglàd, adapted stochastic processes with values in the space of Hilbert–Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semimartingale with càdlàg trajectories.