On the martingale property of certain local martingales
By Aleksandar Mijatovic, Mikhail Urusov. 2009
146 citations recorded by the source corpus.
The stochastic exponential $${Z_t= {\rm exp}\{M_t-M_0-(1/2)\langle M,M\rangle_t\}}$$ of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where $${M_t=\int_0^t b(Y_u)\,dW_u}$$ and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.
Subfields: Finance, Mathematical Physics, Modeling and Simulation
View Paper: https://www.semanticscholar.org/paper/78a3c3f7b5a86ed8abb34baff5df0cb98e644852
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In Finance
- Ioannis Karatzas
- J. Ruf
- Soumik Pal
- Mykhaylo Shkolnikov
- Constantinos Kardaras
- Tomoyuki Ichiba
- Gordan Zitkovic
- Marcel Nutz
In Mathematical Physics
Citations in the corpus
- Convergence of integral functionals of one-dimensional diffusions (2011, unknown): “In the case D = ∅ the local martingale Z is studied in [17], where deterministic necessary and sufficient conditions for Z to be a martingale and to be a uniformly integrable martingale are established.”
- Distribution of the time to explosion for one-dimensional diffusions (2013, background): “By [41], the local martingale M(·) is a true martingale (use Theorem 2.”