Real Analysis Exchange

The Non-Uniform Riemann Approach to Itôʼs Integral

Tay Jing-Yi, Toh Tin-Lam, and Chew Tuan-Seng

Full-text: Open access

Abstract

In this paper, we shall consider two generalized Riemann approaches to the Itô integral; namely, the Itô-Henstock and the Itô-McShane approaches, and by establishing the equivalence of the Itô-Henstock integral with the classical Itô integral, prove the equivalence of all the three integrals.

Article information

Source
Real Anal. Exchange, Volume 27, Number 2 (2001), 495-514.

Dates
First available in Project Euclid: 2 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1212412852

Mathematical Reviews number (MathSciNet)
MR1922665

Zentralblatt MATH identifier
1067.60025

Subjects
Primary: 26A39: Denjoy and Perron integrals, other special integrals 60H05: Stochastic integrals

Keywords
Henstock McShane It\^o stochastic integrals non-uniform meshes

Citation

Tuan-Seng, Chew; Jing-Yi, Tay; Tin-Lam, Toh. The Non-Uniform Riemann Approach to Itôʼs Integral. Real Anal. Exchange 27 (2001), no. 2, 495--514. https://projecteuclid.org/euclid.rae/1212412852


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References

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  • Chew T. S., Lee P. Y., Nonabsolute Integration using Vitali Covers, New Zealand Journal of Mathematics 23, (1994) 25–36.
  • R. Henstock, Lectures on the theory of integration, World Scientific, Singapore, 1988.
  • R. Henstock, The general theory of integration, Oxford University Press, Oxford, 1991.
  • R. Henstock, Stochastic and other functional integrals, Real Analysis Exchange, 16 (1990-91), 460–470.
  • Lee P. Y., Lanzhou lectures on Henstock integration, World Scientific, Singapore, 1989.
  • Lee P. Y., R. Vyborny, The integral, an easy approach after Kurzweil and Henstock, Cambridge University Press, 2000.
  • Lee T. W.,On the generalised Riemann integral and stochastic integral, Journal Australian Math. Soc., 21(Series A) (1976), 64–71.
  • E. J. McShane, Stochastic Calculus and Stochastic Models, Academic Press, New York, 1974.
  • B. Oksendal, Stochastic differential equations: An introduction with applications, 4th edition, Springer, 1996.
  • P. Protter, A comparison of stochastic integrals, Annals of Probability, 7 (1979), 276–289.
  • Toh T. L., Chew T. S., A variational approach to Itô's integral, Proceedings of SAP's, 98 1999, Taiwan 291–299, World Scientific, Singapore.
  • Xu J. G., Lee P. Y., Stochastic integrals of Itô and Henstock, Real Analysis Exchange, 18 (1992-93), 352–366.
  • Yeh, J., Martingales and stochastic analysis, World Scientific. Singapore, 1995.
  • P. Muldowney, The Henstock integral and the Black-Scholes theory of derivative asset pricing, Real Analysis Exchange, 26 (2000-2001), 117–132.