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5 - Consistency of Multivariate Special Semimartingales

from Part One - Consistencies

Published online by Cambridge University Press:  18 September 2020

Tomasz R. Bielecki
Affiliation:
Illinois Institute of Technology
Jacek Jakubowski
Affiliation:
Uniwersytet Warszawski, Poland
Mariusz Niewȩgłowski
Affiliation:
Politechnika Warszawska, Poland
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Summary

In this chapter we introduce and discuss various concepts of consistency for multivariate special semimartingales. The results here are mainly based on Theorem 5.1, which generalizes to the case of semimartingales that are not special. Thus, these results themselves generalize in a straightforward manner to the case of semimartingales that are not special. We chose to work with special semimartingales in order to ease somewhat the presentation. Throughout this chapter the semimartingale truncation functions will be considered to be standard truncation functions of appropriate dimensions. In what follows, the semimartingale characteristics will be always computed with respect to the relevant standard truncation functions. Thus, the semimartingale characteristics for all the semimartingales showing in the rest of this chapter are considered to be unique (as functions of the trajectories on the canonical space) once the filtration is chosen with respect to which the characteristics are computed. The theory is illustrated by various examples.

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Publisher: Cambridge University Press
Print publication year: 2020

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