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Ito calculus without probability in idealized financial markets

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  • Vladimir Vovk

Abstract

We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks only one monetary unit and brings infinite capital if quadratic variation does not exist. This result allows one to apply numerous known results in pathwise Ito calculus to typical price paths; we give a brief overview of such results.

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  • Vladimir Vovk, 2011. "Ito calculus without probability in idealized financial markets," Papers 1108.0799, arXiv.org, revised Aug 2014.
    • Handle: RePEc:arx:papers:1108.0799
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    References listed on IDEAS

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    1. Ryuichi Nakajima & Masayuki Kumon & Akimichi Takemura & Kei Takeuchi, 2010. "Approximations and asymptotics of upper hedging prices in multinomial models," Papers 1007.4372, arXiv.org, revised Jun 2011.
    2. Kumon, Masayuki & Takemura, Akimichi & Takeuchi, Kei, 2011. "Sequential optimizing strategy in multi-dimensional bounded forecasting games," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 155-183, January.
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