Abstract
We consider systems of interacting diffusion processes which generalize the volatility-stabilized market models introduced in Fernholz and Karatzas (Ann Finance 1(2):149–177, 2005). We show how to construct a weak solution of the underlying system of stochastic differential equations. In particular, we express the solution in terms of time changed squared-Bessel processes, and discuss sufficient conditions under which one can show that this solution is unique in distribution (respectively, does not explode). Sufficient conditions for the existence of a strong solution are also provided. Moreover, we discuss the significance of these processes in the context of arbitrage relative to the market portfolio within the framework of Stochastic Portfolio Theory.
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Acknowledgments
The author is indebted to Professor Ioannis Karatzas for his suggestion to study this topic, his careful reading of preliminary versions of the manuscript and invaluable advice. The author is grateful to Robert Fernholz, Mike Hogan, Tomoyuki Ichiba, Soumik Pal, Johannes Ruf, Mykhaylo Shkolnikov, and Phillip Whitman for discussions on the subject matter of this paper. The author would also like to thank an anonymous referee for his/her helpful comments and suggestions. Research partially supported by the National Science Foundation grant DMS-09-05754.
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Picková, R. Generalized volatility-stabilized processes. Ann Finance 10, 101–125 (2014). https://doi.org/10.1007/s10436-013-0230-9
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DOI: https://doi.org/10.1007/s10436-013-0230-9