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Martingale property of generalized stochastic exponentials
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Mijatović, Aleksandar, Novak, Nika and Urusov, Mikhail (2012) Martingale property of generalized stochastic exponentials. In: Séminaire de Probabilités XLIV. Lecture Notes in Mathematics, Volume 2046 . Berlin Heidelberg: Springer, pp. 41-59. ISBN 9783642274602
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Official URL: http://dx.doi.org/10.1007/978-3-642-27461-9_2
Abstract
For a real Borel measurable function b, which satisfies certain integrability conditions, it is possible to define a stochastic integral of the process b(Y ) with respect to a Brownian motion W, where Y is a diffusion driven by W. It is well-known that the stochastic exponential of this stochastic integral is a local martingale. In this paper we consider the case of an arbitrary Borel measurable function b where it may not be possible to define the stochastic integral of b(Y ) directly. However the notion of the stochastic exponential can be generalized. We define a non-negative process Z, called generalized stochastic exponential, which is not necessarily a local martingale. Our main result gives deterministic necessary and sufficient conditions for Z to be a local, true or uniformly integrable martingale.
Item Type: | Book Item | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Series Name: | Lecture Notes in Mathematics | ||||
Publisher: | Springer | ||||
Place of Publication: | Berlin Heidelberg | ||||
ISBN: | 9783642274602 | ||||
ISSN: | 0075-8434 | ||||
Book Title: | Séminaire de Probabilités XLIV | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Volume 2046 | ||||
Page Range: | pp. 41-59 | ||||
DOI: | 10.1007/978-3-642-27461-9_2 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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