Radon-Nikodym derivatives, passages and maxima for a Gaussian process with particular covariance and mean

1975 ◽  
Vol 12 (4) ◽  
pp. 724-733 ◽  
Author(s):  
Israel Bar-David
Keyword(s):  
Parameter Estimation ◽  
Gaussian Process ◽  
First Passage Time ◽  
Passage Time ◽  
Time Interval ◽  
Multiple Reflections ◽  
Process Mean ◽  
First Passage ◽  
Estimation Problems ◽  
Rectangular Signal

We find expressions for the R–N derivative of the stationary Gaussian process with the particular covariance and mean, respectively, R(t, s) = max(1 – |t – s|, 0) and m(t)= aR(t, D), 0 ≦ D ≦ 1, within the time interval [0, 1]. We use these results, and a lemma on multiple reflections of the Wiener process, to find formulae for the probabilities of first passage time and maxima in [0, 1], and bounds on the former within [– 1, 1]. While previous work dealt extensively with the zero mean process, mean functions, as defined here, appear in signal detection and parameter estimation problems under the hypothesis that a rectangular signal centered at t = D is present in an observed process.

Radon-Nikodym derivatives, passages and maxima for a Gaussian process with particular covariance and mean

1975 ◽  
Vol 12 (04) ◽  
pp. 724-733 ◽  
Author(s):  
Israel Bar-David
Keyword(s):  
Parameter Estimation ◽  
Gaussian Process ◽  
First Passage Time ◽  
Passage Time ◽  
Time Interval ◽  
Multiple Reflections ◽  
Process Mean ◽  
First Passage ◽  
Estimation Problems ◽  
Rectangular Signal

We find expressions for the R–N derivative of the stationary Gaussian process with the particular covariance and mean, respectively, R(t, s) = max(1 – |t – s|, 0) and m(t)= aR(t, D), 0 ≦ D ≦ 1, within the time interval [0, 1]. We use these results, and a lemma on multiple reflections of the Wiener process, to find formulae for the probabilities of first passage time and maxima in [0, 1], and bounds on the former within [– 1, 1]. While previous work dealt extensively with the zero mean process, mean functions, as defined here, appear in signal detection and parameter estimation problems under the hypothesis that a rectangular signal centered at t = D is present in an observed process.

First-passage time for a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (01) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x 0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

First-passage time for a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (1) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

scholarly journals LQG Homing in a Finite Time Interval

2011 ◽  
Vol 2011 ◽  
pp. 1-3 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Optimal Control ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Fixed Constant

LetX(t)be a controlled one-dimensional diffusion process having constant infinitesimal variance. We consider the problem of optimally controllingX(t)until timeT(x)=min{T1(x),t1}, whereT1(x)is the first-passage time of the process to a given boundary andt1is a fixed constant. The optimal control is obtained explicitly in the particular case whenX(t)is a controlled Wiener process.

CROSSREG — A Technique for First Passage and Wave Density Analysis

1993 ◽  
Vol 7 (1) ◽  
pp. 125-148 ◽  
Author(s):  
Igor Rychlik ◽  
Georg Lindgren
Keyword(s):  
Gaussian Process ◽  
First Passage Time ◽  
Computer Time ◽  
Passage Time ◽  
First Passage ◽  
Special Cases ◽  
Crossing Time ◽  
General Bound ◽  
Bivariate Density ◽  
Main Routine

The density of the first passage time in a nonstationary Gaussian process with random mean function can be approximated with arbitrary accuracy from a regression-type expansion. CROSSREG is a package of FORTRAN subroutines that perform intelligent transformations and numerical integrations to produce high-accuracy approximations with a minimum of computer time. The basic routines, collected in the unit ONEREG, give the density of the crossing time of a general bound. An additional set of routines make up the unit TWOREG, which also gives the bivariate density of the crossing time and the value of an accompanying process at the time of the crossing. These routines can be used to find the wavelength and amplitude density in any stationary Gaussian process. ONEREG and TWOREG are special cases of a routine MREG, which is the main routine in CROSSREG.

First passage time for the state of exact chemical equilibrium

1980 ◽  
Vol 45 (3) ◽  
pp. 777-782 ◽  
Author(s):  
Milan Šolc
Keyword(s):  
Chemical Equilibrium ◽  
First Passage Time ◽  
Passage Time ◽  
The State ◽  
Recurrence Time ◽  
First Passage ◽  
First Order ◽  
The Mean ◽  
Passage Times ◽  
Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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