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.
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
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