The Probability Space of Brownian Motion

Consider a fractional Brownian motion (fBM) [Formula: see text] with Hurst index [Formula: see text]. We construct a probability space supporting both BH and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one for any user-specified error bound [Formula: see text]. When [Formula: see text], we further enhance our error guarantee to the α-Hölder norm for any [Formula: see text]. This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations [Formula: see text]. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently.

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REPRESENTATION OF PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC BURGERS' EQUATIONS

Stochastics and Dynamics ◽

10.1142/s0219493709002798 ◽

2009 ◽

Vol 09 (04) ◽

pp. 613-634 ◽

Cited By ~ 13

Author(s):

YONG LIU ◽

HUAIZHONG ZHAO

Keyword(s):

Dynamical System ◽

Integral Equation ◽

Brownian Motion ◽

Linear Operator ◽

Probability Space ◽

Burgers Equation ◽

Stationary Solutions ◽

Random Dynamical System ◽

Ergodic Transformations ◽

Stochastic Burgers Equation

In this paper, we show that the stationary solution u(t, ω) of the differentiable random dynamical system U: ℝ+ × L2[0, 1] × Ω → L2[0, 1] generated by the stochastic Burgers' equation with large viscosity, denoted by ν, driven by a Brownian motion in L2[0, 1], is given by: u(t, ω) = U(t, Y(ω), ω) = Y(θ(t, ω)), where Y(ω) can be represented by the following integral equation: [Formula: see text] Here θ is the group of P-preserving ergodic transformations on the canonical probability space [Formula: see text] such that θ(t, ω)(s) = W(t + s) - W(t), where W is the L2[0, 1]-valued Brownian motion on the probability space [Formula: see text], Tν is the linear operator semigroup on L2[0, 1] generated by νΔ.

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Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

Jurnal Matematika Integratif ◽

10.24198/jmi.v15i2.23350 ◽

2019 ◽

Vol 15 (2) ◽

pp. 81 ◽

Cited By ~ 1

Author(s):

Herry Pribawanto Suryawan

Keyword(s):

Brownian Motion ◽

White Noise ◽

Fractional Brownian Motion ◽

Short Range ◽

Probability Space ◽

Noise Analysis ◽

Local Times ◽

White Noise Analysis ◽

Self Similarity ◽

Sample Paths

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

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FREE BROWNIAN MOTION AND EVOLUTION TOWARDS ⊞-INFINITE DIVISIBILITY FOR k-TUPLES

International Journal of Mathematics ◽

10.1142/s0129167x09005303 ◽

2009 ◽

Vol 20 (03) ◽

pp. 309-338 ◽

Cited By ~ 5

Author(s):

SERBAN T. BELINSCHI ◽

ALEXANDRU NICA

Keyword(s):

Brownian Motion ◽

Probability Space ◽

Infinite Divisibility ◽

The Other ◽

Transformation Formula ◽

Infinitely Divisible ◽

Other Hand ◽

Free Brownian Motion

Let [Formula: see text] be the space of non-commutative distributions of k-tuples of self-adjoint elements in a C*-probability space. For every t ≥ 0 we consider the transformation [Formula: see text] defined by [Formula: see text] where ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on [Formula: see text]. We prove that 𝔹s ◦ 𝔹t = 𝔹s + t, for all s, t ≥ 0. For t = 1, we prove that [Formula: see text] is precisely the set [Formula: see text] of distributions in [Formula: see text] which are infinitely divisible with respect to ⊞, and that the map [Formula: see text] coincides with the multi-variable Boolean Bercovici–Pata bijection put into evidence in our previous paper [1]. Thus for a fixed [Formula: see text], the process {𝔹t(μ)|t ≥ 0} can be viewed as some kind of "evolution towards ⊞-infinite divisibility". On the other hand, we put into evidence a relation between the transformations ⊞t and free Brownian motion. More precisely, we introduce a map [Formula: see text] which transforms the free Brownian motion started at an arbitrary [Formula: see text] into the process {𝔹t(μ)|t ≥ 0} for μ = Φ(ν).

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Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

Jurnal Matematika Integratif ◽

10.24198/jmi.v15.n2.23350.81 ◽

2020 ◽

Vol 15 (2) ◽

pp. 81

Author(s):

Herry Pribawanto Suryawan

Keyword(s):

Brownian Motion ◽

White Noise ◽

Fractional Brownian Motion ◽

Short Range ◽

Probability Space ◽

Noise Analysis ◽

Local Times ◽

White Noise Analysis ◽

Self Similarity ◽

Sample Paths

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

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Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200003041 ◽

2007 ◽

Vol 44 (02) ◽

pp. 393-408 ◽

Cited By ~ 1

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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Brownian motion with quadratic killing and some implications

Journal of Applied Probability ◽

10.1017/s0021900200116079 ◽

1986 ◽

Vol 23 (04) ◽

pp. 893-903 ◽

Cited By ~ 2

Author(s):

Michael L. Wenocur

Keyword(s):

Brownian Motion ◽

Weibull Distribution ◽

Special Function ◽

Function Theory ◽

Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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