On Kurzweil-Henstock's double Itô-Wiener nonstochastic integral
Yong Khin TAY, Tin Lam TOH
Received June 30, 2025. Published online January 12, 2026.
Abstract: Motivated by the study of multiple stochastic integral using the Kurzweil-Henstock approach, in this paper we use this approach to study double Itô-Wiener nonstochastic integral on the square $[0,1]\times[0,1]$. We prove the properties of the integral, including the main convergence theorems. Finally, we prove that the double Itô-Wiener nonstochastic integral can be expressed as an iterated McShane integral.
References: [1] R. G. Bartle, D. R. Sherbert: Introduction to Real Analysis. John Wiley & Sons, New York (1992). MR 1135107 | Zbl 0810.26001
[2] T. S. Chew, Z. Huang, C. S. Wang: The non-uniform Riemann approach to anticipating stochastic integrals. Stoch. Anal. Appl. 22 (2004), 429-442. DOI 10.1081/SAP-120028598 | MR 2037380 | Zbl 1056.60062
[3] T.-S. Chew, P.-Y. Lee: Nonabsolute integration using Vitali covers. N. Z. J. Math. 23 (1994), 25-36. MR 1279123 | Zbl 0832.26005
[4] T.-S. Chew, J.-Y. Tay, T.-L. Toh: The non-uniform Riemann approach to Itô's integral. Real Anal. Exch. 27 (2001/2002), 495-514. MR 1922665 | Zbl 1067.60025
[5] P.-Y. Lee: Lanzhou Lectures on Henstock Integration. Series in Real Analysis 2. World Scientific, London (1989). DOI 10.1142/0845 | MR 1050957 | Zbl 0699.26004
[6] P. Y. Lee, R. Výborný: The Integral: An Easy Approach After Kurzweil and Henstock. Australian Mathematical Society Lecture Series 14. Cambridge University Press, Cambridge (2000). MR 1756319 | Zbl 0941.26003
[7] C. Y. Y. Lim, T. L. Toh: A note on Henstock-Itô's non-stochastic integral. Real Anal. Exch. 47 (2022), 443-460. DOI 10.14321/realanalexch.47.2.1637314733 | MR 4551045 | Zbl 1538.26022
[8] E. J. McShane: A Riemann-Type Integral That Includes Lebesgue-Stieltjes, Bochner and Stochastic Integrals. Memoirs of the American Mathematical Society 88. AMS, Providence (1969). DOI 10.1090/memo/0088 | MR 0265527 | Zbl 0188.35702
[9] Y. X. Ng, T. L. Toh: A note on Kurzweil-Henstock's anticipating non-stochastic integral. Math. Bohem. 150 (2025), 371-392. DOI 10.21136/MB.2024.0028-24 | MR 4964584 | Zbl 8127821
[10] T.-L. Toh, T.-S. Chew: A variational approach to Itô's integral. Trends in Probability and Related Analysis World Scientific, Singapore (1999), 291-299. MR 1819215 | Zbl 0981.60054
[11] T.-L. Toh, T.-S. Chew: 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 | Zbl 1022.60055
[12] T.-L. Toh, T.-S. Chew: On the Henstock-Fubini theorem for multiple stochastic integrals. Real Anal. Exch. 30 (2004/2005), 295-310. MR 2127534 | Zbl 1068.60076
[13] T.-L. Toh, T.-S. Chew: Henstock's multiple Wiener integral and Henstock's version of Hu-Meyer theorem. Math. Comput. Modelling 42 (2005), 139-149. DOI 10.1016/j.mcm.2004.03.008 | MR 2162393 | Zbl 1084.60523
[14] T.-L. Toh, T.-S. Chew: On belated differentiation and a characterization of Henstock-Kurzweil-Itô integrable processes. Math. Bohem. 130 (2005), 63-72. DOI 10.21136/MB.2005.134223 | MR 2128359 | Zbl 1112.26012
[15] T.-L. Toh, T.-S. Chew: On Itô-Kurzweil-Henstock integral and integration-by-parts formula. Czech. Math. J. 55 (2005), 653-663. DOI 10.1007/s10587-005-0052-7 | MR 2153089 | Zbl 1081.26005
[16] T. L. Toh, T. S. Chew: Henstock's version to Itô's formula. Real Anal. Exch. 35 (2009/2010), 375-390. MR 2683604 | Zbl 1221.26015
[17] T.-L. Toh, T.-S. Chew: The Kurzweil-Henstock theory of stochastic integration. Czech. Math. J. 62 (2012), 829-848. DOI 10.1007/s10587-012-0048-z | MR 2984637 | Zbl 1265.26020
[18] H. Yang, T. L. Toh: On Henstock method to Stratonovich integral with respect to continuous semimartingale. Int. J. Stoch. Anal. 2014 (2014), Article ID 534864, 7 pages. DOI 10.1155/2014/534864 | MR 3293832 | Zbl 1325.60082
[19] H. Yang, T. L. Toh: On Henstock-Kurzweil method to Stratonovic integral. Math. Bohem. 141 (2016), 129-142. DOI 10.21136/MB.2016.11 | MR 3499780 | Zbl 1389.26016
Affiliations: Yong Khin Tay, Tin Lam Toh (corresponding author), National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Singapore, e-mail: tinlam.toh@nie.edu.sg
