The Itô Integral

2020 ◽  
pp. 635-663
Author(s):  
Achim Klenke
Keyword(s):  
Itô Integral ◽  
Ito Integral

On the moments of a multiple wiener-ito integral and the space induced by the polynomials of the integral

Stochastics ◽  
1988 ◽  
Vol 25 (4) ◽  
pp. 233-340 ◽  
Author(s):  
D. Nualart ◽  
A. S. Ustunel ◽  
M. Zakai*
Keyword(s):  
Itô Integral ◽  
Ito Integral

Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

1996 ◽  
Vol 16 (5) ◽  
pp. 1087-1100
Author(s):  
Eric Slud ◽  
Daniel Chambers
Keyword(s):  
Large Class ◽  
Order Term ◽  
Sufficient Conditions ◽  
Polynomial Form ◽  
Itô Integral ◽  
Ito Integral ◽  
Weakly Mixing ◽  
Subordinated Processes ◽  
Infinite Sums ◽  
Necessary And Sufficient

abstractNecessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

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