Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (2) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (02) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (02) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (2) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

scholarly journals Stochastic Analysis of Gaussian Processes via Fredholm Representation

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Tommi Sottinen ◽  
Lauri Viitasaari
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Differential Equations ◽  
Gaussian Process ◽  
Stochastic Differential Equations ◽  
Gaussian Processes ◽  
Stochastic Analysis ◽  
Series Expansions ◽  
Transfer Principle ◽  
Maximum Likelihood Estimations

We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.

scholarly journals THE GENERALIZED BIFRACTIONAL BROWNIAN MOTION

2018 ◽  
Vol 14 (4) ◽  
pp. 81-89 ◽  
Author(s):  
Charles El-Nouty
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Bifractional Brownian Motion ◽  
Class A ◽  
Convexity Properties

To extend several known centered Gaussian processes, we introduce a new centered Gaussian process, named the generalized bifractional Brownian motion. This process depends on several parameters, namely  α > 0 , β>0 ,  0<H<1  and  0<K≤1 . When  K=1, we investigate its convexity properties. Then, when  2HK≤ 1, we prove that this process is an element of the QHASI class, a class of centered Gaussian processes, which was introduced in  2015

The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

1989 ◽  
Vol 21 (02) ◽  
pp. 315-333 ◽  
Author(s):  
H. E. Daniels
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Covariance Function ◽  
Brownian Bridge ◽  
Breaking Load ◽  
Plastic Yield ◽  
Analogous Process ◽  
The Mean

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t 1(1 + t 2), .

The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres

1989 ◽  
Vol 21 (2) ◽  
pp. 315-333 ◽  
Author(s):  
H. E. Daniels
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Covariance Function ◽  
Brownian Bridge ◽  
Breaking Load ◽  
Plastic Yield ◽  
Analogous Process ◽  
The Mean

Daniels and Skyrme (1985) derived the joint distribution of the maximum, and the time at which it is attained, of a Brownian path superimposed on a parabolic curve near its maximum. In the present paper the results are extended to include Gaussian processes which behave locally like Brownian motion, or a process transformable to it, near the maximum of the mean path. This enables a wider class of practical problems to be dealt with. The results are used to obtain the asymptotic distribution of breaking load and extension of a bundle of fibres which can admit random slack or plastic yield, as suggested by Phoenix and Taylor (1973). Simulations confirm the approximations reasonably well. The method requires consideration not only of a Brownian bridge but also of an analogous process with covariance function t1(1 + t2), .

scholarly journals Optimalities for random functions Lee-Wiener’s network and non-canonical representation of stationary Gaussian processes

1998 ◽  
Vol 149 ◽  
pp. 9-17 ◽  
Author(s):  
Win Win Htay
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Theoretical Approach ◽  
Canonical Representation ◽  
Prediction Theory ◽  
Random Functions ◽  
Canonical Representations ◽  
Stationary Gaussian Processes ◽  
Information Theoretical Approach

Abstract.Representation of a Gaussian process in terms of a Brownian motion is a powerful tool in the investigation of its structure. Among various representations is the canonical representation which is viewed as the best one from the viewpoint of the prediction theory. We have discovered some significance of non-canonical representations and discuss their optimality in an information theoretical approach.

scholarly journals On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 991 ◽  
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Brownian Motion ◽  
Markov Process ◽  
Gaussian Process ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Statistical Parameters ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Over Time

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

scholarly journals On the structure of splitting fields of stationary Gaussian processes with finite multiple Markovian property

1974 ◽  
Vol 54 ◽  
pp. 191-213 ◽  
Author(s):  
Yasunori Okabe
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Markovian Property ◽  
Stationary Gaussian Processes ◽  
Continuous Gaussian Process

Let X = (X(t); t ∈ R) be a real stationary mean continuous Gaussian process with expectation zero which is purely nondeterministic. In this paper we shall investigate the structure of splitting fields of X having finite multiple Markovian property using the results in [6]. We follow the notations and terminologies in [6].

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