Stratonovich-Henstock integral for the operator-valued stochastic process
DOI:
https://doi.org/10.22199/issn.0717-6279-5018Keywords:
Stratonovich-Henstock integral, Q-Wiener process, Itô’s formulaAbstract
In this paper, we introduce the Stratonovich-Henstock integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also formulate a version of Ito's formula for this integral.
Downloads
References
J.-P. Aubin, Applied functional analysis: Aubin/applied, 2nd ed. Nashville, TN: John Wiley & Sons, 2011. https://doi.org/10.1002/9781118032725
T. S. Chew, T. L. Toh, and J. Y. Tay, “The non-uniform Riemann approach to Itô’s integral,” Real Analysis Exchange, vol. 27, no. 2, pp. 495-514, 2002. https://doi.org/10.14321/realanalexch.27.2.0495
G. Da Prato and J. Zabczyk, Encyclopedia of mathematics and its applications volume 44: Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press, 1992. https://doi.org/10.1017/CBO9780511666223
J. Dieudonné, Foundations of modern analysis. San Diego, CA: Academic Press, 1969. https://doi.org/10.1126/science.132.3441.1759-a
L. Gawarecki and V. Mandrekar, Stochastic differential equations in infinite dimensions: With applications to stochastic partial differential equations. Berlin: Springer, 2013. https://doi.org/10.1007/978-3-642-16194-0
R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron and Henstock. Providence, RI: American Mathematical Society, 1994. https://doi.org/10.1090/gsm/004
R. Henstock, Lectures on the theory of integration. Singapore: World Scientific Publishing, 1988. https://doi.org/10.1142/0510
J. Kurzweil, Henstock-Kurzweil integration: Its relation to topological vector spaces. Singapore, Singapore: World Scientific Publishing, 2000. https://doi.org/10.1142/0510
J. Kurzweil and S. Schwabik, “McShane equi-integrability and Vitali’s convergence theorem”, Mathematica Bohemica, vol. 129, no. 2, pp. 141-157, 2004. https://doi.org/10.21136/MB.2004.133903
M. Labendia and J. Arcede, “A descriptive definition of the Itô-Henstock integral for the operator-valued stochastic process”, Advances in Operator Theory, vol. 4, no. 2, pp. 406-418, 2019. https://doi.org/10.15352/aot.1808-1406
M. Labendia, T. R. Teng, and E. De Lara-Tuprio, “Itô-Henstock integral and Itô’s formula for the operator-valued stochastic process”, Mathematica Bohemica, vol. 143, no. 2, pp. 135-160, 2018. https://doi.org/10.21136/MB.2017.0084-16
P. Y. Lee, Lanzhou lectures on Henstock integration. Singapore: World Scientific Publishing, 1989. https://doi.org/10.1142/0845
P. Y. Lee and R. Výborný, The Integral: An Easy Approach after Kurzwiel and Henstock. Australian mathematical society lecture series 14. Cambridge: Cambridge University Press, 2000.
T. Y. Lee, Henstock-Kurzweil integration on euclidean spaces. Singapore: World Scientific Publishing, 2011. https://doi.org/10.1142/7933
J. T. Lu and P. Y. Lee, “The primitives of Henstock integrable functions in Euclidean space”, Bulletin of the London Mathematical Society, vol. 31, no. 2, pp. 173-180, 1999. https://doi.org/10.1112/S0024609398005347
Z. M. Ma, Z. Grande, and P. Y. Lee, “Absolute integration using Vitali covers”, Real Analysis Exchange, vol. 18, no. 2, pp. 409-419, 1992. https://doi.org/10.2307/44152284
E. J. McShane, “Stochastic integrals and stochastic functional equations”, SIAM Journal on Applied Mathematics, vol. 17, no. 2, pp. 287-306, 1969. https://doi.org/10.1137/0117029
Z. R. Pop-Stojanovic, “On McShane’s belated stochastic integral”, SIAM Journal on Applied Mathematics, vol. 22, no. 1, pp. 87-92, 1972. https://doi.org/10.1137/0122010
C. Prévót and M. Röckner, A concise course on stochastic partial differential equations. Berlin: Springer, 2007. https://doi.org/10.1007/978-3-540-70781-3
M. Reed and B. Simon, Methods of modern mathematical physics: Functional analysis, v. 1. San Diego, CA: Academic Press, 1972. https://doi.org/10.1016/B978-0-12-585001-8.X5001-6
R. Rulete and M. Labendia, “Backwards Itô-Henstock integral for the Hilbert-Schmidt-valued stochastic process”, European Journal of Pure and Applied Mathematics, vol. 12, no. 1, pp. 58-78, 2019. https://doi.org/10.29020/nybg.ejpam.v12i1.3342
R. Rulete and M. Labendia, “Backwards Itô-Henstock’s version of Itô’s formula”, Annals of Functional Analysis, vol. 11, no. 1, pp. 208-225, 2020. https://doi.org/10.1007/s43034-019-00014-3
C. W. Swartz, Introduction to gauge integrals. Singapore: World Scientific Publishing, 2001. https://doi.org/10.1142/4361
T. L. Toh and T. S. Chew, “On belated differentiation and a characterization of Henstock-Kurzweil-Itô integrable processes”, Mathematica Bohemica, vol. 130, no. 1, pp. 63-72, 2005. https://doi.org/10.21136/MB.2005.134223
T. L. Toh and T. S. Chew, “On Itô-Kurzweil-Henstock integral and integration-by-part formula”, Czechoslovak Mathematical Journal, vol. 55, no. 3, pp. 653-663, 2005. https://doi.org/10.1007/s10587-005-0052-7
T. L. Toh and T. S. Chew, “On the Henstock-Fubini theorem for multiple stochastic integrals”, Real Analysis Exchange, vol. 30, no. 1, pp. 295-310, 2005. https://doi.org/10.14321/realanalexch.30.1.0295
T. L. Toh and T. S. Chew, “The Riemann approach to stochas tic integration using non-uniform meshes”, Journal of Mathematical Analysis and Applications, vol. 280, no. 1, pp. 133-147, 2003. https://doi.org/10.1016/S0022-247X(03)00059-3
H. Yang and T. L. Toh, “On Henstock-Kurzweil method to Stratonovich integral”, Mathematica Bohemica, vol. 141, no. 2, pp. 129-142. https://doi.org/10.21136/MB.2016.11
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Recson Canton, Mhelmar Labendia, Tin Lam Toh
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.