Mathematica Bohemica, first online, pp. 26, 2017


Itô-Henstock integral and Itô's formula for the operator-valued stochastic process

Mhelmar A. Labendia, Timothy Robin Y. Teng, Elvira P. de Lara-Tuprio

Received September 28, 2016.  First published June 1, 2017.

Abstract:  In this paper, we introduce the Itô-Henstock integral of an operator-valued stochastic process and formulate a version of Itô's formula.
Keywords:  Itô-Henstock integrable function; Itô's formula; $Q$-Wiener process
Classification MSC:  60H30, 60H05
DOI:  10.21136/MB.2017.0084-16

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References:
[1] J.-P. Aubin: Applied Functional Analysis. Exercises by Bernard Cornet and Jean-Michel Lasry. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, New York (2000). DOI 10.1002/9781118032725 | MR 1782330 | Zbl 0946.46001
[2] 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
[3] G. Da Prato: Introduction to Stochastic Analysis and Malliavin Calculus. Appunti. Scuola Normale Superiore di Pisa (Nuova Series) 13. Edizioni della Normale, Pisa (2014). DOI 10.1007/978-88-7642-499-1 | MR 3186829 | Zbl 06276062
[4] G. Da Prato, J. Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 152. Cambridge University Press, Cambridge (2014). DOI 10.1017/CBO9781107295513 | MR 3236753 | Zbl 1317.60077
[5] J. Dieudonné: Grundzüge der modernen Analysis. Band 9. Logik und Grundlagen der Mathematik, Bd. 25. Friedr. Vieweg & Sohn, Braunschweig (1987). DOI 10.1007/978-3-322-90009-8 | MR 0936838 | Zbl 0646.58001
[6] L. Gawarecki, V. Mandrekar: Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations. Probability and Its Applications. Springer, Heidelberg (2011). DOI 10.1007/978-3-642-16194-0 | MR 2560625 | Zbl 1228.60002
[7] S. Ghahramani: Fundamentals of Probability with Stochastic Processes. Chapman and Hall/CRC Press, Boca Raton (2016). Zbl 1336.60001
[8] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4. AMS, Providence (1994). DOI 10.1090/gsm/004 | MR 1288751 | Zbl 0807.26004
[9] L. M. Graves: Riemann integration and Taylor's theorem in general analysis. Transactions AMS 29 (1927), 163-177. DOI 10.2307/1989284 | MR 1501382 | Zbl 53.0234.03
[10] R. Henstock: Lectures on the Theory of Integration. Series in Real Analysis 1. World Scientific Publishing, Singapore (1988). DOI 10.1142/0510 | MR 0963249 | Zbl 0668.28001
[11] J. Kurzweil: Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces. Series in Real Analysis 7. World Scientific Publishing, River Edge (2000). DOI 10.1142/4333 | MR 1763305 | Zbl 0954.28001
[12] P. Y. Lee: Lanzhou Lectures on Henstock Integration. Series in Real Analysis 2. World Scientific Publishing, London (1989). DOI 10.1142/0845 | MR 1050957 | Zbl 0699.26004
[13] 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
[14] T. Y. Lee: Henstock-Kurzweil Integration on Euclidean Spaces. Series in Real Analysis 12. World Scientific Publishing, Hackensack (2011). DOI 10.1142/7933 | MR 2789724 | Zbl 1246.26002
[15] E. J. McShane: Stochastic integrals and stochastic functional equations. SIAM J. Appl. Math. 17 (1969), 287-306. DOI 10.1137/0117029 | MR 0246387 | Zbl 0182.51001
[16] T. Mikosch: Elementary Stochastic Calculus - With Finance in View. Advanced Series on Statistical Science & Applied Probability 6. World Scientific Publishing, Singapore (1998). DOI 10.1142/3856 | MR 1728093 | Zbl 0934.60002
[17] M. Z. Nashed: Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis. Nonlinear Functional Analysis Appl., Proc. Adv. Sem. Math. Res. Center, Univ. Wisconsin 1970. Publ. 26 Math. Res. Center Univ. Wisconsin (1971), 103-309. MR 0276840 | Zbl 0224.00015
[18] Z. R. Pop-Stojanovic: On McShane's belated stochastic integral. SIAM J. Appl. Math. 22 (1972), 87-92. DOI 10.1137/0122010 | MR 0322954 | Zbl 0243.60035
[19] C. Prévôt, M. Röckner: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1905. Springer, Berlin (2007). DOI 10.1007/978-3-540-70781-3 | MR 2329435 | Zbl 1123.60001
[20] M. Reed, B. Simon: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, New York-London (1972). MR 0493419 | Zbl 0242.46001
[21] M. Riedle: Cylindrical Wiener processes. Séminaire de Probabilités XLIII (C. Donati-Martin et al.). Lecture Notes in Mathematics 2006. Springer, Berlin (2011), 191-214. DOI 10.1007/978-3-642-15217-7_7 | MR 2790373 | Zbl 1228.60049
[22] 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
[23] T.-L. Toh, T.-S. Chew: On the Henstock-Fubini theorem for multiple stochastic integrals. Real Anal. Exch. 30 (2005), 295-310. MR 2127534 | Zbl 1068.60076
[24] T.-L. Toh, T.-S. Chew: Henstock's version of Itô's formula. Real Anal. Exch. 35 (2010), 375-390. MR 2683604 | Zbl 1221.26015
[25] S. R. S. Varadhan: Probability Theory. Courant Lecture Notes in Mathematics 7. American Mathematical Society, Providence; New York Univ., Courant Institute of Mathematical Sciences, New York (2001). DOI 10.1090/cln/007 | MR 1852999 | Zbl 0980.60002

Affiliations:   Mhelmar A. Labendia, Department of Mathematics & Statistic, Mindanao State University-Iligan Institute of Technology, Andres Bonifacio Avenue, Tibanga, 9200 Iligan City, Philippines, e-mail: mhelmar.labendia@g.msuiit.edu.ph; Timothy Robin Y. Teng, Elvira P. de Lara-Tuprio, Department of Mathematics, School of Science and Engineering, Katipunan Ave, Ateneo de Manila University, 1108 Quezon City, Philippines, e-mail: tteng@ateneo.edu, edelara-tuprio@ateneo.edu


 
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