Real Analysis Exchange

Henstock's Version of Itô's Formula

Tuan Seng Chew and Tin Lam Toh

Full-text: Open access


Itô's Formula is the stochastic analogue of the change of variable formula for deterministic integrals. It is a useful tool in dealing with stochastic integration. In this paper, using Henstock's approach, we derive two versions of Itô's Formula. Henstock's or generalized Riemann approach has been successful in giving an alternative definition of stochastic integral, which is more explicit, intuitive and less measure theoretic. Henstock's approach provides a simpler and more direct proof of Itô's Formula, although we do not claim that it is a generalization of the classical results.

Article information

Real Anal. Exchange, Volume 35, Number 2 (2009), 375-390.

First available in Project Euclid: 22 September 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

It\^o's formula Henstock's stochastic integral generalized Riemann approach


Toh, Tin Lam; Chew, Tuan Seng. Henstock's Version of Itô's Formula. Real Anal. Exchange 35 (2009), no. 2, 375--390.

Export citation


  • T. S. Chew, P. Y. Lee, Nonabsolute integration using Vitali covers, New Zealand J. Math., 23 (1994), 25–36.
  • T. S. Chew, T. L. Toh, J. Y. Tay, The non-uniform Riemann approach to Itô's integral, Real Anal. Exchange, 27(2) (2002-03), 495–514.
  • K. L. Chung, R. J. Williams, Introduction to Stochastic Integration, Second edition, Probability and its Applications, Birkhäuser Boston, Boston, MA, 1990.
  • R. Henstock, The efficiency of convergence factors for functions of a continuous real variable, J. London Math. Soc., 30 (1955), 273–286.
  • R. Henstock, Lectures on the theory of integration, Series in Real Analysis, 1, World Scientific Publishing, Singapore, 1988.
  • R. Henstock, The general theory of integration, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.
  • R. Henstock, Stochastic and other functional integrals, Real Anal. Exchange, 16 (1990-01), 460–470.
  • M. Hitsuda, Formula for Brownian partial derivatives, Publ. Fac. Integrated Arts and Sciences, Hiroshima University, Series III, 4 (1978), 1 - 15.
  • P. Y. Lee, R. Vyborny, Integral: an easy approach after Kurzweil and Henstock, English summary, Australian Mathematical Society Lecture Series, 14, Cambridge University Press, Cambridge, 2000.
  • T. W. Lee, On the generalised Riemann integral and stochastic integral, J. Aust. Math. Soc., Series A, 21 (1976), 64–71.
  • E. J. McShane, Stochastic Calculus and Stochastic Models, Probability and Mathematical Statistics, 25, Academic Press, New York-London, 1974.
  • P. Muldowney, A general theory of integration in function spaces, including Wiener and Feynman integration, Pitman Research Notes in Mathematics Series, 153, Longman Scientific & Technical, Harlow, John Wiley & Sons, New York, 1987.
  • D. Nualart, The Malliavin calculus and related topics, English summary, Probability and its Applications (New York), Springer-Verlag, New York, 1995.
  • D. Nualart, E. Pardoux, Stochastic calculus with anticipating integrands, Probab. Th. Rel. Fields, 78 (1988), 535–581.
  • E. Pardoux, P. Protter, A two-sided stochastic integral and its calculus, Probab. Th. Rel. Fields, 76 (1987), 15–49.
  • Z. R. Pop-Stojanovic, On McShane's belated stochastic integral, SIAM J. Appl. Math., 22 (1972), 87–92.
  • P. Protter, A comparison of stochastic integrals, Ann. Probab., 7 (1979), 276–289.
  • P. Protter, Stochastic integration and differential equations. A new approach, Applications of Mathematics (New York), 21, Springer-Verlag, Berlin, 1990.
  • D. Revuz, M. Yor, Continuous martingales and Brownian motion, Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1994.
  • A. V. Skorohod, On a generalisation of a stochastic integral, Theory Probab. Appl., 20 (1975), 219–233.
  • R. L. Stratonovich, A new representation for stochastic integrals and equations, SIAM J. Control Optim., 4 (1966), 362–371.
  • T. L. Toh, T. S. Chew, A variational approach to Itô's integral, Trends in probability and related analysis (Taipei, 1998), 291–299, World Sci. Publ., River Edge, NJ, 1999.
  • T. L. Toh, T. S. Chew, The Riemann Approach to Stochastic Integration using Non-uniform Mesh, J. Math. Anal. Appl., 280 (2003), 133–147.
  • T. L. Toh, T. S. Chew, The Riemann Approach to Multiple Wiener Integral, Real Anal. Exchange, 29 (2004), 275–289.
  • T. L. Toh, T. S. Chew, On Henstock-Fubini's Theorem for Multiple Stochastic Integral, Real Anal. Exchange, 30 (2005), 295–310.
  • T. L. Toh, T. S. Chew, On Henstock's Multiple Wiener Integral and Henstock's Version of Hu-Meyer Theorem, Math. Comput. Modelling, 42 (2005), 139–149.
  • T. L. Toh, T. S. Chew, On Itô-Kurzweil-Henstock Integral and Integration-by-part Formula, Czechoslovak Math. J., 55 (2005), 653–663.
  • T. L. Toh, T. S. Chew, On Belated Differentiation and a Characterization of Henstock-Kurzweil-Itô Integrable Processes, Math. Bohem., 130 (2005), 63–73.
  • H. Weizsäcker, G. Winkler, Stochastic integrals. An introduction, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1990.
  • E. Wong, M. Zakai, An extension of stochastic integrals in the plane, Ann. Proabab., 5 (1977), 770–778.
  • J. G. Xu, P. Y. Lee, Stochastic integrals of Itô and Henstock, Real Anal. Exchange, 18 (1992-93), 352–366.
  • M. Zähle, Integration with respect to fractal functions and stochastic Calculus I, Probab. Th. Rel. Fields, 111 (1998), 337–374.