Itô Calculus without Probability in Idealized Financial Markets*

We consider idealized financial markets in which price paths of the traded securities are càdlàg 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 Itô calculus to typical price paths; we give a brief overview of such results.

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