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On the martingale property of stochastic exponentials

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Bernard Wong  and
C. C. Heyde
Bernard Wong*
Affiliation:
University of New South Wales and Australian National University
C. C. Heyde*
Affiliation:
Australian National University and Columbia University
*
∗ Postal address: School of Actuarial Studies, University of New South Wales, Sydney, NSW 2052, Australia. Email address: bernard.wong@unsw.edu.au
∗∗ Postal address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
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Abstract

We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property are related to whether a related process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a ‘candidate measure’ the integrand process is square integrable over time. Applications of our theorem to problems arising in mathematical finance are also given.

Keywords

Explosionmartingale propertyequivalent local martingale measure

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 654 - 664
Copyright
Copyright © Applied Probability Trust 2004 

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