BDG inequalities and their applications for model-free continuous price paths with instant enforcement
Volume 10, Issue 4 (2023), pp. 425–457
Pub. online: 4 October 2023
Type: Research Article
Open Access
Open Access
Received
14 December 2021
14 December 2021
Revised
12 August 2023
12 August 2023
Accepted
19 August 2023
19 August 2023
Published
4 October 2023
4 October 2023
Abstract
Shafer and Vovk introduce in their book [8] the notion of instant enforcement and instantly blockable properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of “typical” price paths. In this paper an outer measure on the space $[0,+\infty )\times \Omega $ is introduced, which assigns zero value exactly to those sets (properties) of pairs of time t and an elementary event ω which are instantly blockable. Next, for a slightly modified measure, Itô’s isometry and BDG inequalities are proved, and then they are used to define an Itô-type integral. Additionally, few properties are proved for the quadratic variation of model-free continuous martingales, which hold with instant enforcement.
References
Bartl, D., Kupper, M., Neufeld, A.: Stochastic integration and differential equations for typical paths. Electron. J. Probab. 24(97), 1–21 (2019). MR4017115. https://doi.org/10.1214/19-ejp343
Beiglböck, M., Siorpaes, P.: Pathwise versions of the Burkholder-Davis-Gundy inequality. Bernoulli 21(1), 360–373 (2015). MR3322322. https://doi.org/10.3150/13-BEJ570
Hrbacek, K., Jech, T.: Introduction to Set Theory, 3rd and expanded edn. Marcel Dekker, Inc., New York, Basel (1999). MR1697766
Łochowski, R.M.: On a generalisation of the Banach Indicatrix Theorem. Colloq. Math. 148(2), 301–314 (2017). MR3660293. https://doi.org/10.4064/cm6583-3-2017
Łochowski, R.M., Obłój, J., Prömel, D.J., Siorpaes, P.: Local times and Tanaka–Meyer formulae for càdlàg paths. Electron. J. Probab. 26, 1–29 (2021). MR4269207. https://doi.org/10.1214/21-ejp638
Łochowski, R.M., Perkowski, N., Prömel, D.: A superhedging approach to stochastic integration. Stoch. Process. Appl. 128, 4078–4103 (2018). MR3906979. https://doi.org/10.1016/j.spa.2018.01.009
Perkowski, N., Prömel, D.J.: Pathwise stochastic integrals for model free finance. Bernoulli 22(4), 2486–2520 (2016). MR3498035. https://doi.org/10.3150/15-BEJ735
Shafer, G., Volk, V.: Game-Theoretic Foundations for Probability and Finance. Wiley Series in Probability and Statistics. Wiley (2019). MR3287403. https://doi.org/10.1002/9781118763117.ch6
Vovk, V.: Continuous-time trading and the emergence of volatility. Electron. Commun. Probab. 13, 319–324 (2008). MR2415140. https://doi.org/10.1214/ECP.v13-1383
Vovk, V.: Continuous-time trading and the emergence of randomness. Stochastics 81, 455–466 (2009). MR2569261. https://doi.org/10.1080/17442500802221712
Vovk, V.: Rough paths in idealized financial markets. Lith. Math. J. 51, 274–285 (2011). MR2805744. https://doi.org/10.1007/s10986-011-9125-5
Vovk, V.: Continuous-time trading and the emergence of probability. Finance Stoch. 16(4), 561–609 (2012). MR2972235. https://doi.org/10.1007/s00780-012-0180-5
Vovk, V.: Itô calculus without probability in idealized financial markets. Lith. Math. J. 55(2), 270–290 (2015). MR3347596. https://doi.org/10.1007/s10986-015-9280-1
Vovk, V., Schafer, G.: Towards a probability-free theory of continuous martingales. Working Paper no. 45 (2016). MR3761734. https://doi.org/10.1145/3130261.3130263
