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Tsallis value-at-risk: generalized entropic value-at-risk
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- Journal:
- Probability in the Engineering and Informational Sciences / Volume 38 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 29 November 2022, pp. 1-20
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Motivated by Ahmadi-Javid (Journal of Optimization Theory Applications, 155(3), 2012, 1105–1123) and Ahmadi-Javid and Pichler (Mathematics and Financial Economics, 11, 2017, 527–550), the concept of Tsallis Value-at-Risk (TsVaR) based on Tsallis entropy is introduced in this paper. TsVaR corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the Value-at-Risk. The main properties and analogous dual representation of TsVaR are investigated. These results partially generalize the Entropic Value-at-Risk by involving Tsallis entropies. Three spaces, called the primal, dual, and bidual Tsallis spaces, corresponding to TsVaR are fully studied. It is shown that these spaces equipped with the norm induced by TsVaR are Banach spaces. The Tsallis spaces are related to the
$L^p$ spaces, as well as specific Orlicz hearts and Orlicz spaces. Finally, we derive explicit formula for the dual TsVaR norm.
