On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case

Paolo Tella ()
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Paolo Tella: Technische Universitat Dresden Internationales Hochschulinstitut Zittau

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2194-2216

Abstract: Abstract In this paper, we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an $${\mathbb {F}}$$ F -semimartingale X possessing the WRP with respect to $${\mathbb {F}}$$ F and an $${\mathbb {H}}$$ H -semimartingale Y possessing the WRP with respect to $${\mathbb {H}}$$ H . Assuming that $${\mathbb {F}}$$ F and $${\mathbb {H}}$$ H are independent, we show that the $${\mathbb {G}}$$ G -semimartingale $$Z=(X,Y)$$ Z = ( X , Y ) has the WRP with respect to $${\mathbb {G}}$$ G , where $${\mathbb {G}}:={\mathbb {F}}\vee {\mathbb {H}}$$ G : = F ∨ H . In our setting, X and Y may have simultaneous jump-times. Furthermore, their jumps may charge the same predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations.

Keywords: Weak representation property; Semimartingales; Progressive enlargement of filtrations; Independent semimartingales; Random measures; Stochastic integration; Jacod’s equivalence condition; 60G44; 60G57; 60H05; 60H30 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01145-x

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