Purely pathwise probability-free Ito integral

V. Vovk

doi:10.15330/ms.46.1.96-110

This paper gives a simple construction of the pathwise Ito integral $\int_0^t\phi d\omega$ for an integrand $\phi$ and an integrator $\omega$ satisfying various topological and analytical conditions. The definition is purely pathwise in that neither $\phi$ nor $\omega$ are assumed to be paths of processes, and the Ito integral exists almost surely in a non-probabilistic finance-theoretic sense. For example, one of the results shows the existence of $\int_0^t\phi d\omega$ for a cadlag integrand $\phi$ and a cadlag integrator $\omega$ with jumps bounded in a predictable manner.

1. A. Ananova, R. Cont, Pathwise integration with respect to paths of finite quadratic variation, arXiv:1603.03305, 2016.

2. M. Beiglbock, A.M.G. Cox, M. Huesmann, N. Perkowski, D.J. Promel, Pathwise super-replication via Vovks outer measure, arXiv:1504.03644, 2016.

3. K. Bichteler, Stochastic integration and Lp-theory of semimartingales, Annals of Probability, 9 (1981), 49-89.

4. R. Cont, D.-A. Fournie, Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, 259 (2010), 1043-1072.

5. Ph. Courrege, P. Priouret, Temps darret dune fonction aleatoire: relations dequivalence associees et proprietes de decomposition, Publications de lInstitut de statistique de Universite de Paris, 14 (1965) 245-274.

6. M. Davis, J. Ob loj, P. Siorpaes, Pathwise stochastic calculus with local times, arXiv:1508.05984, 2015.

7. C. Dellacherie Quelques exemples familiers, en probabilites, densembles analytiques non boreliens, Seminaire de probabilites de Strasbourg, 12 (1978) 746-756.

8. C. Dellacherie, P.-A. Meyer, Probabilities and Potential, North-Holland, Amsterdam, 1978, Chapters I-IV. French original: 1975; reprinted in 2008.

9. H. Follmer, Calcul Ito sans probabilites, Seminaire de probabilites de Strasbourg, 15 (1981) 143-150.

10. R.L. Karandikar On pathwise stochastic integration, Stochastic Processes and their Applications, 57 (1995), 11-18.

11. A.N. Kolmogorov, Sur la loi des grands nombres, Atti della Reale Accademia Nazionale dei Lincei. Classe di scienze fisiche, matematiche, e naturali. Rendiconti Serie VI, 185 (1929) 917-919.

12. R.M. Lochowski, Asymptotics of the truncated variation of model-free price paths and semimartingales with jumps, arXiv:1508.01269, 2016.

13. R.M. Lochowski, Integration with respect to model-free price paths with jumps arXiv:1511.08194, 2015.

14. R.M. Lochowski, N. Perkowski, D.J. Promel, A superhedging approach to stochastic integration, arXiv:1609.02349, 2016.

15. M. Nutz, Pathwise construction of stochastic integrals, Electronic Communications in Probability, 17 (2012), no.24, 1-7.

16. N. Perkowski, D.J. Promel, Local times for typical price paths and pathwise Tanaka formulas, Electronic Journal of Probability, 20 (2015), no.46, 1-15.

17. N. Perkowski, D.J. Promel, Pathwise stochastic integrals for model free finance, Bernoulli, 22 (2016), 2486-2520.

18. N. Perkowski, D.J. Prmel, Private communication, 2016.

19. C. Riga, A pathwise approach to continuous-time trading, arXiv:1602.04946, 2016.

20. T. Sasai, K. Miyabe, A. Takemura, Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales: a game-theoretic approach, arXiv:1504.06398, 2015.

21. D. Sondermann, Introduction to stochastic calculus for finance: a new didactic approach, Lecture Notes in Economics and Mathematical Systems, V.579, Springer, Berlin, 2006.

22. V. Vovk, Continuous-time trading and the emergence of probability, Finance and Stochastics, 16 (2012), 561-609; arXiv:0904.4364, 2015.

23. V. Vovk, Ito calculus without probability in idealized financial markets, Lithuanian Mathematical Journal, 55 (2015), 270-290.

24. V. Vovk, Getting rich quick with the Axiom of Choice, arXiv:1604.00596, 2016; Journal version (subject to a minor revision): Finance and Stochastics, under the title “The role of measurability in game-theoretic probability”.

	Purely pathwise probability-free Ito integral
Author	V. Vovk v.vovk@rhul.ac.uk Department of Computer Science, Royal Holloway, University of London
Abstract	This paper gives a simple construction of the pathwise Ito integral $\int_0^t\phi d\omega$ for an integrand $\phi$ and an integrator $\omega$ satisfying various topological and analytical conditions. The definition is purely pathwise in that neither $\phi$ nor $\omega$ are assumed to be paths of processes, and the Ito integral exists almost surely in a non-probabilistic finance-theoretic sense. For example, one of the results shows the existence of $\int_0^t\phi d\omega$ for a cadlag integrand $\phi$ and a cadlag integrator $\omega$ with jumps bounded in a predictable manner.
Keywords	box-counting dimension; Ito integral; Ito’s formula; pathwise integration; quadratic variation
DOI	doi:10.15330/ms.46.1.96-110
Reference	1. A. Ananova, R. Cont, Pathwise integration with respect to paths of finite quadratic variation, arXiv:1603.03305, 2016. 2. M. Beiglbock, A.M.G. Cox, M. Huesmann, N. Perkowski, D.J. Promel, Pathwise super-replication via Vovks outer measure, arXiv:1504.03644, 2016. 3. K. Bichteler, Stochastic integration and Lp-theory of semimartingales, Annals of Probability, 9 (1981), 49-89. 4. R. Cont, D.-A. Fournie, Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, 259 (2010), 1043-1072. 5. Ph. Courrege, P. Priouret, Temps darret dune fonction aleatoire: relations dequivalence associees et proprietes de decomposition, Publications de lInstitut de statistique de Universite de Paris, 14 (1965) 245-274. 6. M. Davis, J. Ob loj, P. Siorpaes, Pathwise stochastic calculus with local times, arXiv:1508.05984, 2015. 7. C. Dellacherie Quelques exemples familiers, en probabilites, densembles analytiques non boreliens, Seminaire de probabilites de Strasbourg, 12 (1978) 746-756. 8. C. Dellacherie, P.-A. Meyer, Probabilities and Potential, North-Holland, Amsterdam, 1978, Chapters I-IV. French original: 1975; reprinted in 2008. 9. H. Follmer, Calcul Ito sans probabilites, Seminaire de probabilites de Strasbourg, 15 (1981) 143-150. 10. R.L. Karandikar On pathwise stochastic integration, Stochastic Processes and their Applications, 57 (1995), 11-18. 11. A.N. Kolmogorov, Sur la loi des grands nombres, Atti della Reale Accademia Nazionale dei Lincei. Classe di scienze fisiche, matematiche, e naturali. Rendiconti Serie VI, 185 (1929) 917-919. 12. R.M. Lochowski, Asymptotics of the truncated variation of model-free price paths and semimartingales with jumps, arXiv:1508.01269, 2016. 13. R.M. Lochowski, Integration with respect to model-free price paths with jumps arXiv:1511.08194, 2015. 14. R.M. Lochowski, N. Perkowski, D.J. Promel, A superhedging approach to stochastic integration, arXiv:1609.02349, 2016. 15. M. Nutz, Pathwise construction of stochastic integrals, Electronic Communications in Probability, 17 (2012), no.24, 1-7. 16. N. Perkowski, D.J. Promel, Local times for typical price paths and pathwise Tanaka formulas, Electronic Journal of Probability, 20 (2015), no.46, 1-15. 17. N. Perkowski, D.J. Promel, Pathwise stochastic integrals for model free finance, Bernoulli, 22 (2016), 2486-2520. 18. N. Perkowski, D.J. Prmel, Private communication, 2016. 19. C. Riga, A pathwise approach to continuous-time trading, arXiv:1602.04946, 2016. 20. T. Sasai, K. Miyabe, A. Takemura, Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales: a game-theoretic approach, arXiv:1504.06398, 2015. 21. D. Sondermann, Introduction to stochastic calculus for finance: a new didactic approach, Lecture Notes in Economics and Mathematical Systems, V.579, Springer, Berlin, 2006. 22. V. Vovk, Continuous-time trading and the emergence of probability, Finance and Stochastics, 16 (2012), 561-609; arXiv:0904.4364, 2015. 23. V. Vovk, Ito calculus without probability in idealized financial markets, Lithuanian Mathematical Journal, 55 (2015), 270-290. 24. V. Vovk, Getting rich quick with the Axiom of Choice, arXiv:1604.00596, 2016; Journal version (subject to a minor revision): Finance and Stochastics, under the title “The role of measurability in game-theoretic probability”.
Pages	96-110
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html