Title:
|
Itô-Henstock integral and Itô's formula for the operator-valued stochastic process (English) |
Author:
|
Labendia, Mhelmar A. |
Author:
|
Teng, Timothy Robin Y. |
Author:
|
de Lara-Tuprio, Elvira P. |
Language:
|
English |
Journal:
|
Mathematica Bohemica |
ISSN:
|
0862-7959 (print) |
ISSN:
|
2464-7136 (online) |
Volume:
|
143 |
Issue:
|
2 |
Year:
|
2018 |
Pages:
|
135-160 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
In this paper, we introduce the Itô-Henstock integral of an operator-valued stochastic process and formulate a version of Itô's formula. (English) |
Keyword:
|
Itô-Henstock integrable function |
Keyword:
|
Itô's formula |
Keyword:
|
$Q$-Wiener process |
MSC:
|
60H05 |
MSC:
|
60H30 |
idZBL:
|
Zbl 06890411 |
idMR:
|
MR3831483 |
DOI:
|
10.21136/MB.2017.0084-16 |
. |
Date available:
|
2018-06-11T11:00:31Z |
Last updated:
|
2020-07-01 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147241 |
. |
Reference:
|
[1] Aubin, J.-P.: 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). Zbl 0946.46001, MR 1782330, 10.1002/9781118032725 |
Reference:
|
[2] 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. Zbl 1067.60025, MR 1922665 |
Reference:
|
[3] Prato, G. Da: Introduction to Stochastic Analysis and Malliavin Calculus.Appunti. Scuola Normale Superiore di Pisa (Nuova Series) 13. Edizioni della Normale, Pisa (2014). Zbl 06276062, MR 3186829, 10.1007/978-88-7642-499-1 |
Reference:
|
[4] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions.Encyclopedia of Mathematics and Its Applications 152. Cambridge University Press, Cambridge (2014). Zbl 1317.60077, MR 3236753, 10.1017/CBO9781107295513 |
Reference:
|
[5] Dieudonné, J.: Grundzüge der modernen Analysis. Band 9.Logik und Grundlagen der Mathematik, Bd. 25. Friedr. Vieweg & Sohn, Braunschweig (1987). Zbl 0646.58001, MR 0936838, 10.1007/978-3-322-90009-8 |
Reference:
|
[6] Gawarecki, L., Mandrekar, V.: Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations.Probability and Its Applications. Springer, Heidelberg (2011). Zbl 1228.60002, MR 2560625, 10.1007/978-3-642-16194-0 |
Reference:
|
[7] Ghahramani, S.: Fundamentals of Probability with Stochastic Processes.Chapman and Hall/CRC Press, Boca Raton (2016). Zbl 1336.60001, MR 3887668 |
Reference:
|
[8] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics 4. AMS, Providence (1994). Zbl 0807.26004, MR 1288751, 10.1090/gsm/004 |
Reference:
|
[9] Graves, L. M.: Riemann integration and Taylor's theorem in general analysis.Transactions AMS 29 (1927), 163-177. Zbl 53.0234.03, MR 1501382, 10.2307/1989284 |
Reference:
|
[10] Henstock, R.: Lectures on the Theory of Integration.Series in Real Analysis 1. World Scientific Publishing, Singapore (1988). Zbl 0668.28001, MR 0963249, 10.1142/0510 |
Reference:
|
[11] Kurzweil, J.: Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces.Series in Real Analysis 7. World Scientific Publishing, River Edge (2000). Zbl 0954.28001, MR 1763305, 10.1142/4333 |
Reference:
|
[12] Lee, P. Y.: Lanzhou Lectures on Henstock Integration.Series in Real Analysis 2. World Scientific Publishing, London (1989). Zbl 0699.26004, MR 1050957, 10.1142/0845 |
Reference:
|
[13] Lee, P. Y., Výborný, R.: The Integral: An Easy Approach after Kurzweil and Henstock.Australian Mathematical Society Lecture Series 14. Cambridge University Press, Cambridge (2000). Zbl 0941.26003, MR 1756319 |
Reference:
|
[14] Lee, T. Y.: Henstock-Kurzweil Integration on Euclidean Spaces.Series in Real Analysis 12. World Scientific Publishing, Hackensack (2011). Zbl 1246.26002, MR 2789724, 10.1142/7933 |
Reference:
|
[15] McShane, E. J.: Stochastic integrals and stochastic functional equations.SIAM J. Appl. Math. 17 (1969), 287-306. Zbl 0182.51001, MR 0246387, 10.1137/0117029 |
Reference:
|
[16] Mikosch, T.: Elementary Stochastic Calculus---With Finance in View.Advanced Series on Statistical Science & Applied Probability 6. World Scientific Publishing, Singapore (1998). Zbl 0934.60002, MR 1728093, 10.1142/3856 |
Reference:
|
[17] Nashed, M. Z.: 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. Zbl 0224.00015, MR 0276840 |
Reference:
|
[18] Pop-Stojanovic, Z. R.: On McShane's belated stochastic integral.SIAM J. Appl. Math. 22 (1972), 87-92. Zbl 0243.60035, MR 0322954, 10.1137/0122010 |
Reference:
|
[19] Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations.Lecture Notes in Mathematics 1905. Springer, Berlin (2007). Zbl 1123.60001, MR 2329435, 10.1007/978-3-540-70781-3 |
Reference:
|
[20] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis.Academic Press, New York-London (1972). Zbl 0242.46001, MR 0493419 |
Reference:
|
[21] Riedle, M.: Cylindrical Wiener processes.C. Donati-Martin et al. Séminaire de Probabilités XLIII Lecture Notes in Mathematics 2006. Springer, Berlin (2011), 191-214. Zbl 1228.60049, MR 2790373, 10.1007/978-3-642-15217-7_7 |
Reference:
|
[22] Toh, T.-L., Chew, T.-S.: The Riemann approach to stochastic integration using non-uniform meshes.J. Math. Anal. Appl. 280 (2003), 133-147. Zbl 1022.60055, MR 1972197, 10.1016/S0022-247X(03)00059-3 |
Reference:
|
[23] Toh, T.-L., Chew, T.-S.: On the Henstock-Fubini theorem for multiple stochastic integrals.Real Anal. Exch. 30 (2005), 295-310. Zbl 1068.60076, MR 2127534, 10.14321/realanalexch.30.1.0295 |
Reference:
|
[24] Toh, T.-L., Chew, T.-S.: Henstock's version of Itô's formula.Real Anal. Exch. 35 (2010), 375-390. Zbl 1221.26015, MR 2683604, 10.14321/realanalexch.35.2.0375 |
Reference:
|
[25] Varadhan, S. R. S.: Probability Theory.Courant Lecture Notes in Mathematics 7. American Mathematical Society, Providence; New York Univ., Courant Institute of Mathematical Sciences, New York (2001). Zbl 0980.60002, MR 1852999, 10.1090/cln/007 |
. |