An ito formula for processes with values in an abstract Wiener space

Lecture Notes in Mathematics - Stochastic Analysis and Related Topics ◽

10.1007/bfb0081933 ◽

2006 ◽

pp. 234-246

Author(s):

H. Korezlioglu ◽

A. S. Ustunel

Keyword(s):

Wiener Space ◽

Itô Formula ◽

Abstract Wiener Space ◽

Ito Formula

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The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Mathematics ◽

10.3390/math9040389 ◽

2021 ◽

Vol 9 (4) ◽

pp. 389

Author(s):

Jeong-Gyoo Kim

Keyword(s):

Hilbert Space ◽

Fourier Series ◽

Fourier Coefficients ◽

Double Sequence ◽

Intermediate Stage ◽

Wiener Space ◽

Abstract Wiener Space ◽

Abstract Space ◽

Double Sequences ◽

Two Parameter

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

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A generalization of the Itô formula

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202102018 ◽

2002 ◽

Vol 31 (8) ◽

pp. 477-496

Author(s):

Said Ngobi

Keyword(s):

Sobolev Space ◽

White Noise ◽

Itô Formula ◽

Ito Formula ◽

Integrable Functions ◽

Square Integrable Functions ◽

White Noise Space

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

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