scholarly journals Basic properties of Brownian motion and a capacity on the Wiener space

1984 ◽  
Vol 36 (1) ◽  
pp. 161-176 ◽  
Author(s):  
Masatoshi FUKUSHIMA
Keyword(s):  
Brownian Motion ◽  
Wiener Space ◽  
Basic Properties

scholarly journals Another visit to two halflines

1989 ◽  
Vol 21 (4) ◽  
pp. 935-937
Author(s):  
G. Hooghiemstra
Keyword(s):  
Brownian Motion ◽  
Standard Brownian Motion ◽  
Basic Properties

We shall use three basic properties of Brownian motion to derive in an elegant and non-computational way the probability that standard Brownian motion, starting from 0, will ever cross the halflines t → αt + β or t → γt + δ where γ, δ < 0 < α, β.

scholarly journals Fractionally integrated Bessel process

2021 ◽  
Vol 477 (2250) ◽  
pp. 20200934
Author(s):  
Georgiy Shevchenko ◽  
Dmytro Zatula
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hölder Continuity ◽  
Bessel Process ◽  
Holder Continuity ◽  
Basic Properties

We consider a fractionally integrated Bessel process defined by Y s δ , H = ∫ 0 ∞ ( u H − ( 1 / 2 ) − ( u − s ) + H − ( 1 / 2 ) ) d X u δ , where X δ is the Bessel process of dimension δ  > 2. We discuss the relation of this process to the fractional Brownian motion at its maximum, study the basic properties of the process and prove its Hölder continuity.

On homogeneous chaos

1991 ◽  
Vol 110 (2) ◽  
pp. 353-363 ◽  
Author(s):  
Nigel Cutland ◽  
Siu-Ah Ng
Keyword(s):  
Brownian Motion ◽  
Wiener Space ◽  
Chaos Decomposition ◽  
Image Position

AbstractThis paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos decompositionFirst these are identified as Radon–Nikodým derivatives. Two elementary non-standard proofs of the Wiener–Itô chaos decomposition are given, based on Anderson's construction of Brownian motion and Itô integration.

scholarly journals Another visit to two halflines

1989 ◽  
Vol 21 (04) ◽  
pp. 935-937
Author(s):  
G. Hooghiemstra
Keyword(s):  
Brownian Motion ◽  
Standard Brownian Motion ◽  
Basic Properties

We shall use three basic properties of Brownian motion to derive in an elegant and non-computational way the probability that standard Brownian motion, starting from 0, will ever cross the halflines t → αt + β or t → γt + δ where γ, δ &lt; 0 &lt; α, β.

Scanning Brownian Processes

1997 ◽  
Vol 29 (02) ◽  
pp. 295-326 ◽  
Author(s):  
Robert J. Adler ◽  
Ron Pyke
Keyword(s):  
Brownian Motion ◽  
Random Field ◽  
Gaussian Random Field ◽  
Sample Path ◽  
Geometrical Properties ◽  
Basic Properties ◽  
Scanning Process ◽  
Sample Paths ◽  
Sample Path Properties ◽  
Path Properties

The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A 0 of some ‘scanning set' A 0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A 0. We ask if the set A 0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A 0 from realisations of the sample paths of the random field Z.

Scanning Brownian Processes

1997 ◽  
Vol 29 (2) ◽  
pp. 295-326 ◽  
Author(s):  
Robert J. Adler ◽  
Ron Pyke
Keyword(s):  
Brownian Motion ◽  
Random Field ◽  
Gaussian Random Field ◽  
Sample Path ◽  
Geometrical Properties ◽  
Basic Properties ◽  
Scanning Process ◽  
Sample Paths ◽  
Sample Path Properties ◽  
Path Properties

The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A0 of some ‘scanning set' A0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A0. We ask if the set A0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A0 from realisations of the sample paths of the random field Z.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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