Abstract
In this paper, we formulate a version of Itô’s formula for the backwards Itô–Henstock integral of an operator-valued stochastic process. Itô’s formula is the stochastic analogue of the change of variable for deterministic integrals.
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References
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, New York (2004)
Arcede, J., Cabral, E.: An equivalent definition for the backwards Itô integral. Thai J. Math. 9, 619–630 (2011)
Arcede, J., Cabral, E.: On integration-by-parts and Itô formula for backwards Itô integral. MINDANAWAN J. Math. 3, 112–132 (2012)
Aubin, J.P.: Applied Functional Analysis, 2nd edn. Wiley, New York (2000)
Chew, T.S., Toh, T.L., Tay, J.Y.: The non-uniform Riemann approach to Itô’s integral. Real Anal. Exch. 27, 495–514 (2002-2003)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Dieudonné, J.: Foundations of Modern Analysis. Academic Press, London (1969)
Gawarecki, L., Mandrekar, V.: Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations. Springer, Berlin (2011)
Graves, L.M.: Riemann integration and Taylor’s theorem in general analysis. Trans. Am. Math. Soc. 29, 163–177 (1927)
Labendia, M., De Lara-Tuprio, E., Teng, T.R.: Itô-Henstock integral and Itô’s formula for the operator-valued stochastic process. Math. Bohemica 143, 135–160 (2018)
McShane, E.J.: Stochastic integrals and stochastic functional equations. SIAM J. Appl. Math. 17, 287–306 (1969)
Nashed, M.Z.: Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis. In: Rall, L. (ed.) Nonlinear Functional Analysis and Applications, pp. 103–309. Academic Press, New York (1971)
Pop-Stojanovic, Z.R.: On McShane’s belated stochastic integral. SIAM J. Appl. Math. 22, 87–92 (1972)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, London (1980)
Rulete, R., Labendia, M.: Backwards Itô-Henstock integral for the Hilbert–Schmidt–Valued stochastic process. Eur. J. Pure Appl. Math. 12, 58–78 (2019)
Toh, T.L., Chew, T.S.: Henstock’s version of Itô’s formula. Real Anal. Exch. 35, 375–390 (2009-2010)
Toh, T.L., Chew, T.S.: On the Henstock–Fubini theorem for multiple stochastic integrals. Real Anal. Exch. 30, 295–310 (2004-2005)
Toh, T.L., Chew, T.S.: The Riemann approach to stochastic integration using non-uniform meshes. J. Math. Anal. Appl. 280, 133–147 (2003)
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The authors would like to thank the anonymous referees for their valuable comments for the improvement of this paper.
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Communicated by Feng Dai.
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Rulete, R.F., Labendia, M.A. Backwards Itô–Henstock’s version of Itô’s formula. Ann. Funct. Anal. 11, 208–225 (2020). https://doi.org/10.1007/s43034-019-00014-3
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DOI: https://doi.org/10.1007/s43034-019-00014-3