preprints.org > computer science and mathematics > probability and statistics > doi: 10.20944/preprints202311.0563.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 7 November 2023 / Approved: 8 November 2023 / Online: 8 November 2023 (14:35:34 CET)

Financial time series and other human-driven, non-natural processes are known to exhibit fat-tailed outcome distributions. That is, such processes demonstrate a greater tendency for extreme outcomes than the normal distribution or other natural distributional processes would predict. We examine the mathematical expectation, or simply "expectation," traditionally the probability-weighted outcome, regarded since the seventeenth century as the mathematical definition of "expectation." However, when considering the "expectation" of an individual confronted with a finite sequence of outcomes, particularly existential outcomes (e.g., a trader with a limited time to perform or lose his position in a trading operation), we find this individual "expects" the median terminal outcome over those finite trials, with the classical seventeenth-century definition being the asymptotic limit as trials increase to this. Since such finite-sequence "expectations" often differ in values from the classic one, so do the optimal allocations (e.g., growth-optimal). We examine these for fat-tailed distributions. The focus is on implementation, and the techniques described can be applied to all distributional forms.

stopping times; expectation; finite sequences; non-asymptotic; optimal allocation; Kelly criterion; stable pareto; fat tails; generalized hyperbolic

Computer Science and Mathematics, Probability and Statistics

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