Electronic Journal of Probability

Stochastic integration and differential equations for typical paths

Daniel Bartl, Michael Kupper, and Ariel Neufeld

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Abstract

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega $ but also in $\int \omega \,d\omega :=\omega ^{2} -\langle \omega \rangle $ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega ^{2} -\langle \omega \rangle $ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 97, 21 pp.

Dates
Received: 24 May 2018
Accepted: 17 July 2019
First available in Project Euclid: 18 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1568793790

Digital Object Identifier
doi:10.1214/19-EJP343

Zentralblatt MATH identifier
07107404

Subjects
Primary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60] 91G20: Derivative securities

Keywords
Föllmer integration pathwise stochastic integral pathwise SDE infinite dimensional stochastic calculus Vovk’s outer measure

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bartl, Daniel; Kupper, Michael; Neufeld, Ariel. Stochastic integration and differential equations for typical paths. Electron. J. Probab. 24 (2019), paper no. 97, 21 pp. doi:10.1214/19-EJP343. https://projecteuclid.org/euclid.ejp/1568793790


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References

  • [1] B. Acciaio, M. Beiglböck, F. Penkner, and W. Schachermayer. A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance, 26(2):233–251, 2016.
  • [2] B. Acciaio, M. Beiglböck, F. Penkner, W. Schachermayer, and J. Temme. A trajectorial interpretation of Doob’s martingale inequalities. The Annals of Applied Probability, 23(4):1494–1505, 2013.
  • [3] A. Ananova and R. Cont. Pathwise integration with respect to paths of finite quadratic variation. Journal de Mathématiques Pures et Appliquées, 107(6):737–757, 2017.
  • [4] D. Bartl, M. Kupper, and A. Neufeld. Pathwise superhedging on prediction sets. To appear in Finance and Stochastics, arXiv:1711.02764, 2017.
  • [5] D. Bartl, M. Kupper, D. J. Prömel, and L. Tangpi. Duality for pathwise superhedging in continuous time. Finance and Stochastics, 23(3):697–728, 2019.
  • [6] M. Beiglböck, A. M. Cox, M. Huesmann, N. Perkowski, and D. J. Prömel. Pathwise superreplication via Vovk’s outer measure. Finance and Stochastics, 21(4):1141–1166, 2017.
  • [7] M. Beiglböck and M. Nutz. Martingale inequalities and deterministic counterparts. Electronic Journal of Probability, 19, 2014.
  • [8] M. Beiglböck and P. Siorpaes. Pathwise versions of the Burkholder–Davis–Gundy inequality. Bernoulli, 21(1):360–373, 2015.
  • [9] K. Bichteler. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab., 9:49–89, 1981.
  • [10] M. Burzoni, M. Frittelli, and M. Maggis. Universal arbitrage aggregator in discrete time under uncertainty. Finance and Stochastics, 20(1):1–50, 2016.
  • [11] R. Cont and D.-A. Fournié. Change of variable formulas for non-anticipative functionals on path space. Journal of Functional Analysis, 259:1043–1072, 2010.
  • [12] R. Cont and N. Perkowski. Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity. Transactions of the American Mathematical Society, Series B, Vol. 6., 2019.
  • [13] M. Davis, J. Obloj, and V. Raval. Arbitrage bounds for prices of weighted variance swaps. Math. Finance, 24:821–854, 2014.
  • [14] Y. Dolinsky and A. Neufeld. Super-replication in fully incomplete markets. Math. Finance, 28(2):483–515, 2018.
  • [15] Y. Dolinsky and H. M. Soner. Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields, 160(1–2):391–427, 2014.
  • [16] B. Dupire. Functional Itô calculus. Portfolio Research Paper 2009-04, Bloomberg, 2009.
  • [17] H. Föllmer. Calcul d’Itô sans probabilités. Séminaire de probabilités de Strasbourg, 15:143–150, 1981.
  • [18] P. Friz and M. Hairer. A Course on Rough Paths. Springer, 2014.
  • [19] L. Ch. Galane, R. M. Lochowski, and F. J. Mhlanga. On SDEs with Lipschitz coefficients, driven by continuous, model-free price paths Preprint, arXiv:1807.05692, 2018.
  • [20] Y. Hirai. Remarks on Föllmer’s pathwise Itô calculus. Osaka Journal of Mathematics, 56(3):631–660, 2019.
  • [21] Z. Hou and J. Obłój. Robust pricing–hedging dualities in continuous time. Finance and Stochastics, 22(3):511–567, 2018.
  • [22] R. L. Karandikar. On pathwise stochastic integration. Stochastic Processes and their Applications, 57:11–18, 1995.
  • [23] R. M. Lochowski, N. Perkowski, and D. J. Prömel. A superhedging approach to stochastic integration. Finance and Stochastics, 21(4):1141–1166, 2017.
  • [24] T. J. Lyons. Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance, 2:117–133, 1995.
  • [25] M. Métivier. Semimartingales: A Course on Stochastic Processes, volume 2. Walter de Gruyter, 1982.
  • [26] P. Mykland. Financial options and statistical prediction intervals. Ann. Statist., 31(5):1413–1438, 2003.
  • [27] A. Neufeld. Buy-and-hold property for fully incomplete markets when super-replicating markovian claims. International Journal of Theoretical and Applied Finance, 21(7):1850051, 1–12, 2018.
  • [28] A. Neufeld and M. Nutz. Superreplication under volatility uncertainty for measurable claims. Electronic Journal of Probability, 18(48):1–14, 2013.
  • [29] M. Nutz. Pathwise construction of stochastic integrals. Electronic Communcations in Probability, 17(24):1–7, 2012.
  • [30] M. Nutz and H. M. Soner. Superhedging and dynamic risk measures under volatility uncertainty. SIAM Journal on Control and Optimization, 50(4):2065–2089, 2012.
  • [31] S. Peng. G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In Stochastic analysis and applications, pages 541–567. Springer, 2007.
  • [32] S. Peng. Nonlinear expectations and stochastic calculus under uncertainty. Preprint, arXiv:1002.4546, 2010.
  • [33] N. Perkowski and D. J. Prömel. Pathwise stochastic integrals for model free finance. Bernoulli, 22(4):2486–2520, 2016.
  • [34] P. Protter. Stochastic integration and differential equations. Math. Appl., Second edition, Springer-Verlag, 2004.
  • [35] C. Riga. A pathwise approach to continuous-time trading. Preprint, arXiv:1602.04946, 2016.
  • [36] H. M. Soner, N. Touzi, and J. Zhang. Dual formulation of second order target problems. The Annals of Applied Probability, 23(1):308–347, 2013.
  • [37] V. Vovk. Continuous-time trading and the emergence of probability. Finance and Stochastics, 16(4):561–609, 2012.
  • [38] V. Vovk. Purely pathwise probability-free Itô integral. Preprint, arXiv:1512.01698, 2015.
  • [39] V. Vovk and G. Shafer. Towards a probability-free theory of continuous martingales. Preprint, arXiv:1703.08715, 2017.