scholarly journals Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values

2011 ◽  
Vol 81 (8) ◽  
pp. 1326-1335 ◽  
Author(s):  
Qidi Peng
Keyword(s):  
Gaussian Process ◽  
Hölder Exponent ◽  
Stationary Increments

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.

An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments

1985 ◽  
Vol 22 (02) ◽  
pp. 454-460 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Asymptotic Formula ◽  
Gaussian Process ◽  
Variance Function ◽  
Stationary Increments ◽  
Distribution Of The Maximum

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ 2(t) is convex and σ 2(t) = o(t) for t → 0, then P(max[o,t] X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

scholarly journals Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments

2017 ◽  
Vol 46 (3-4) ◽  
pp. 67-78 ◽  
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Sergiy Shklyar
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Gaussian Process ◽  
Unknown Parameter ◽  
Strong Consistency ◽  
Likelihood Function ◽  
Likelihood Estimator ◽  
Stationary Increments ◽  
Drift Estimation ◽  
Mixed Fractional Brownian Motion

The paper deals with the regression model X_t = \theta t + B_t , t\in[0, T ],where B=\{B_t, t\geq 0\} is a centered Gaussian process with stationary increments.We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equation.Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.

scholarly journals Gaussian Sample Functions: Uniform Dimension and Hölder Conditions Nowhere

1972 ◽  
Vol 46 ◽  
pp. 63-86 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Gaussian Process ◽  
Stationary Increments ◽  
Uniform Dimension ◽  
Gaussian Sample

Let X(t), t≥0, be a real Gaussian process with mean 0, stationary increments, and a2(t) = E|X(t) - X(0)|2. Here dH(λ), for some bounded monotone H. We summarize the main results.

An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments

1985 ◽  
Vol 22 (2) ◽  
pp. 454-460 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Asymptotic Formula ◽  
Gaussian Process ◽  
Variance Function ◽  
Stationary Increments ◽  
Distribution Of The Maximum

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ2(t) is convex and σ2(t) = o(t) for t → 0, then P(max[o,t]X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

scholarly journals 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.

scholarly journals Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (2) ◽  
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.

A Stochastic Model for the Inheritance of the Cancer Proneness Phenotype

1987 ◽  
Vol 26 (03) ◽  
pp. 117-123
Author(s):  
P. Tautu ◽  
G. Wagner
Keyword(s):  
Stochastic Model ◽  
Gaussian Process ◽  
Covariance Function ◽  
Neoplastic Disease ◽  
Disease Phenotype ◽  
Probabilistic Representation ◽  
Temporal Behaviour ◽  
Continuous Trait ◽  
Continuous Parameter ◽  
Level Zero

SummaryA continuous parameter, stationary Gaussian process is introduced as a first approach to the probabilistic representation of the phenotype inheritance process. With some specific assumptions about the components of the covariance function, it may describe the temporal behaviour of the “cancer-proneness phenotype” (CPF) as a quantitative continuous trait. Upcrossing a fixed level (“threshold”) u and reaching level zero are the extremes of the Gaussian process considered; it is assumed that they might be interpreted as the transformation of CPF into a “neoplastic disease phenotype” or as the non-proneness to cancer, respectively.

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