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June 2021 Semimartingales and shrinkage of filtration
Tomasz R. Bielecki, Jacek Jakubowski, Monique Jeanblanc, Mariusz Niewęgłowski
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Ann. Appl. Probab. 31(3): 1376-1402 (June 2021). DOI: 10.1214/20-AAP1621

Abstract

We consider a complete probability space (Ω,F,P), which is endowed with two filtrations, G and F, assumed to satisfy the usual conditions and such that FG. On this probability space we consider a real valued G-semimartingale X.

The purpose of this work is to study the following two problems:

A. If X is F-adapted, compute the F-semimartingale characteristics of X in terms of the G-semimartingale characteristics of X.

B. If X is a special G-semimartingale but not F-adapted, compute the F-semimartingale characteristics of the F-optional projection of X in terms of the G-canonical decomposition and the G-semimartingale characteristics of X.

In this paper problem B is solved under the assumption that the filtration F is immersed in G. Beyond the obvious mathematical interest, our study is motivated by important practical applications in areas such as finance and insurance (cf. Structured Dependence Between Stochastic Processes (2020) Cambridge Univ. Press).

Citation

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Tomasz R. Bielecki. Jacek Jakubowski. Monique Jeanblanc. Mariusz Niewęgłowski. "Semimartingales and shrinkage of filtration." Ann. Appl. Probab. 31 (3) 1376 - 1402, June 2021. https://doi.org/10.1214/20-AAP1621

Information

Received: 1 November 2019; Revised: 1 July 2020; Published: June 2021
First available in Project Euclid: 23 June 2021

Digital Object Identifier: 10.1214/20-AAP1621

Subjects:
Primary: 60G99 , 60H99

Keywords: Filtration shrinkage , optional projection , Semimartingale , semimartingale characteristics , special semimartingale

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 3 • June 2021
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