Previous |  Up |  Next


belated differentiation; Henstock-Kurzweil-Itô integral; integrable processes
The Henstock-Kurzweil approach, also known as the generalized Riemann approach, has been successful in giving an alternative definition to the classical Itô integral. The Riemann approach is well-known for its directness in defining integrals. In this note we will prove the Fundamental Theorem for the Henstock-Kurzweil-Itô integral, thereby providing a characterization of Henstock-Kurzweil-Itô integrable stochastic processes in terms of their primitive processes.
[1] Chew T. S., Tay, J. Y., Toh, T. L.: The non-uniform Riemann approach to Itô’s integral. Real Anal. Exch. 27 (2001/2002), 495–514. MR 1922665
[2] Henstock, R.: The efficiency of convergence factors for functions of a continuous real variable. J. London Math. Soc. 30 (1955), 273–286. DOI 10.1112/jlms/s1-30.3.273 | MR 0072968 | Zbl 0066.09204
[3] Henstock, R.: Lectures on the Theory of Integration. World Scientific, Singapore, 1988. MR 0963249 | Zbl 0668.28001
[4] Lee, P. Y.: Lanzhou Lectures on Henstock Integration. World Scientific, Singapore, 1989. MR 1050957 | Zbl 0699.26004
[5] McShane, E. J.: Stochastic Calculus and Stochastic Models. Academic Press, New York, 1974. MR 0443084 | Zbl 0292.60090
[6] Oksendal, B.: Stochastic Differential Equation: An Introduction with Applications. 4th edition. Springer, 1996.
[7] Pop-Stojanovic, Z. R.: On McShane’s belated stochastic integral. SIAM J. Appl. Math. 22 (1972), 89–92. DOI 10.1137/0122010 | MR 0322954 | Zbl 0243.60035
[8] Toh, T. L., Chew, T. S.: A variational approach to Itô’s integral. Proceedings of SAP’s 98, Taiwan, World Scientific, Singapore, 1999, pp. 291–299. MR 1819215
[9] Toh, T. L., Chew, T. S.: The Riemann approach to stochastic integration using non-uniform meshes. J. Math. Anal. Appl. 280 (2003), 133–147. DOI 10.1016/S0022-247X(03)00059-3 | MR 1972197
[10] Xu, J. G., Lee, P. Y.: Stochastic integrals of Itô and Henstock. Real Anal. Exch. 18 (1992/3), 352–366. MR 1228401
Partner of
EuDML logo