scholarly journals MULTIFRACTAL MEASURES CHARACTERIZED BY THE ITERATIVE MAP WITH TWO CONTROL PARAMETERS

Fractals ◽  
2000 ◽  
Vol 08 (02) ◽  
pp. 181-187 ◽  
Author(s):  
KYUNGSIK KIM ◽  
G. H. KIM ◽  
Y. S. KONG
Keyword(s):  
First Passage Time ◽  
Logistic Models ◽  
Passage Time ◽  
Two Dimensional ◽  
Control Parameters ◽  
First Passage ◽  
One Dimensional ◽  
Numerical Result ◽  
Multifractal Spectra ◽  
Multifractal Behavior

A one-dimensional iterative map with two control parameters — the Kim–Kong map — is proposed. Our purpose is to investigate the characteristic properties of this map, and to discuss numerically the multifractal behavior of the normalized first passage time. Based especially on the Monte Carlo simulation, the normalized first passage time to arrive at the absorbing barrier after starting from an arbitrary site is mainly obtained in the presence of both absorption and reflection on a two-dimensional Sierpinski gasket. We also discuss the multifractal spectra of the normalized first passage time, and the numerical result of the Kim–Kong model presented will be compared with that of the Sinai and logistic models.

First-passage-time densities for time-non-homogeneous diffusion processes

1997 ◽  
Vol 34 (3) ◽  
pp. 623-631 ◽  
Author(s):  
R. Gutiérrez ◽  
L. M. Ricciardi ◽  
P. Román ◽  
F. Torres
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Continuous Functions ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

Hitting-Time Densities of a Two-Dimensional Markov Process

1992 ◽  
Vol 6 (4) ◽  
pp. 561-580
Author(s):  
C. H. Hesse
Keyword(s):  
Global Analysis ◽  
First Passage Time ◽  
Type Equation ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Wide Range ◽  
The Laplace Transform ◽  
Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

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.

scholarly journals First excess levels of vector processes

1994 ◽  
Vol 7 (3) ◽  
pp. 457-464 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Point Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Compound Process ◽  
First Time

This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e. reaches or exceeds) its associated level for the first time. The author targets a joint transformation of the first excess level, first passage time, and the index of the point process which labels the first passage time. The cases when both marks are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic models are discussed.

scholarly journals The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Chuancun Yin ◽  
Huiqing Wang
Keyword(s):  
Boundary Conditions ◽  
Value Function ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
The Laplace Transform ◽  
The Value Function ◽  
Reflecting Barriers

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered.

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.

Moments of the first-passage time of a Wiener process with drift between two elastic barriers

1995 ◽  
Vol 32 (4) ◽  
pp. 1007-1013 ◽  
Author(s):  
Marco Dominé
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
Special Cases ◽  
Backward Equation ◽  
The One ◽  
First Passage Problem ◽  
Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

First-Passage Time for Oscillators With Nonlinear Damping

1978 ◽  
Vol 45 (1) ◽  
pp. 175-180 ◽  
Author(s):  
J. B. Roberts
Keyword(s):  
First Passage Time ◽  
Digital Simulation ◽  
Planck Equation ◽  
Passage Time ◽  
Nonlinear Damping ◽  
First Passage ◽  
One Dimensional ◽  
Fokker Planck Equation ◽  
First Passage Failure ◽  
Fokker Planck

A simple numerical scheme is proposed for computing the probability of first passage failure, P(T), in an interval O-T, for oscillators with nonlinear damping. The method depends on the fact that, when the damping is light, the amplitude envelope, A(t), can be accurately approximated as a one-dimensional Markov process. Hence, estimates of P(T) are found, for both single and double-sided barriers, by solving the Fokker-Planck equation for A(t) with an appropriate absorbing barrier. The numerical solution of the Fokker-Planck equation is greatly simplified by using a discrete time random walk analog of A(t), with appropriate statistical properties. Results obtained by this method are compared with corresponding digital simulation estimates, in typical cases.

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