Electronic Communications in Probability

Pathwise construction of stochastic integrals

Marcel Nutz

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Abstract

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.

Article information

Source
Electron. Commun. Probab. Volume 17 (2012), paper no. 24, 7 pp.

Dates
Accepted: 19 June 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263157

Digital Object Identifier
doi:10.1214/ECP.v17-2099

Mathematical Reviews number (MathSciNet)
MR2950190

Zentralblatt MATH identifier
1245.60054

Subjects
Primary: 60H05: Stochastic integrals

Keywords
Pathwise stochastic integral aggregation non-dominated model second order BSDE G-expectation medial limit

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Nutz, Marcel. Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17 (2012), paper no. 24, 7 pp. doi:10.1214/ECP.v17-2099. https://projecteuclid.org/euclid.ecp/1465263157.


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References

  • Bichteler, Klaus. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49–89.
  • Denis, Laurent; Martini, Claude. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 (2006), no. 2, 827–852.
  • H. Föllmer. Calcul d'Itô sans probabilités. In Séminaire de Probabilités XV, volume 850 of Lecture Notes in Math., pages 143–150, Springer, Berlin, 1981.
  • D. H. Fremlin. Measure Theory, volume 5. Torres Fremlin, Colchester, 2008.
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3
  • Karandikar, Rajeeva L. On pathwise stochastic integration. Stochastic Process. Appl. 57 (1995), no. 1, 11–18.
  • Larson, Paul B. The filter dichotomy and medial limits. J. Math. Log. 9 (2009), no. 2, 159–165.
  • Li, Xinpeng; Peng, Shige. Stopping times and related Itô's calculus with $G$-Brownian motion. Stochastic Process. Appl. 121 (2011), no. 7, 1492–1508.
  • Meyer, P. A. Limites médiales, d'après Mokobodzki. Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971–1972), pp. 198–204. Lecture Notes in Math., Vol. 321, Springer, Berlin, 1973.
  • M. Nutz and H. M. Soner. Superhedging and dynamic risk measures under volatility uncertainty. To appear in SIAM J. Control Optim.
  • S. Peng. Nonlinear expectations and stochastic calculus under uncertainty. Preprint arXiv:1002.4546v1, 2010.
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 (2011), no. 67, 1844–1879.
  • H. M. Soner, N. Touzi, and J. Zhang. Wellposedness of second order backward SDEs. To appear in Probab. Theory Related Fields.
  • H. M. Soner, N. Touzi, and J. Zhang. Dual formulation of second order target problems. To appear in Ann. Appl. Probab.
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 (2011), no. 2, 265–287.
  • Willinger, Walter; Taqqu, Murad S. Pathwise approximations of processes based on the fine structure of their filtrations. Séminaire de Probabilités, XXII, 542–599, Lecture Notes in Math., 1321, Springer, Berlin, 1988.
  • Willinger, Walter; Taqqu, Murad S. Pathwise stochastic integration and applications to the theory of continuous trading. Stochastic Process. Appl. 32 (1989), no. 2, 253–280.