# On the martingale property of stochastic exponentials

@article{Wong2004OnTM,
title={On the martingale property of stochastic exponentials},
author={Bernard Wong and Chris C. Heyde},
journal={Journal of Applied Probability},
year={2004},
volume={41},
pages={654 - 664}
}
• Published 1 September 2004
• Mathematics
• Journal of Applied Probability
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…
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In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some
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## References

SHOWING 1-10 OF 22 REFERENCES

The existence of a unique equivalent measure up to an explosion time is proved by means of results that give a handle on situations where an equivalent martingale measure cannot exist.
• Mathematics
• 2002
Using the Ray-Knight theorem we give conditions for anonnegative diffusion without drift to reach zero or not. These results also givenecessary and sufficient conditions for such a diffusion process
• Mathematics
• 1977
1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process,
• Mathematics
• 1979
This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first,
1 Stochastic Processes: Basic Concepts and Definitions.- 2 Martingales and the Wiener Process.- 3 Stochastic Integrals.- 4 The Ito Formula.- 5 Stochastic Differential Equations.- 6 Functionals of a
• C. Sin
• Mathematics, Economics