Abstract
After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, 1974) and by Øksendal (Stochastic Differential Equations. Springer, New York, 1998). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.
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References
R. S. Liptzer and A. N. Shiryaev. Statistics of Stochastic Processes, Nauka, Moscow (1974).
B. Øksendal. Stochastic Differential Equations, Springer, New York (1998).
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© 2011 Springer-Verlag Berlin Heidelberg
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Gawarecki, L., Mandrekar, V. (2011). Stochastic Calculus. In: Stochastic Differential Equations in Infinite Dimensions. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16194-0_2
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DOI: https://doi.org/10.1007/978-3-642-16194-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16193-3
Online ISBN: 978-3-642-16194-0
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