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.
AMS Classification: 60G44, 60G48, 60H10, 60J60
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Acknowledgements
We would like to thank the anonymous referee for a thorough reading of the paper and many useful suggestions, which significantly improved the paper.
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Mijatović, A., Novak, N., Urusov, M. (2012). Martingale Property of Generalized Stochastic Exponentials. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIV. Lecture Notes in Mathematics(), vol 2046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27461-9_2
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DOI: https://doi.org/10.1007/978-3-642-27461-9_2
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