A first-passage problem for a two-dimensional controlled random walk

1986 ◽  
Vol 23 (3) ◽  
pp. 670-678
Author(s):  
S. Lalley
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensions ◽  
Main Diagonal ◽  
Two Dimensional ◽  
First Passage ◽  
Explicit Representations ◽  
Mean Vectors ◽  
First Passage Problem

The process of interest is a controlled random walk in two dimensions: whenever the walker is above the main diagonal, the next increment to his position is chosen from a distribution FA; whenever the walker is below the diagonal, the next increment comes from another distribution FB. The two distributions have mean vectors which tend to push the walker back toward the diagonal. We analyze the problem of first passage to the first quadrant, obtaining explicit representations for the limiting first-entry distribution and expected first-passage time.

A first-passage problem for a two-dimensional controlled random walk

1986 ◽  
Vol 23 (03) ◽  
pp. 670-678
Author(s):  
S. Lalley
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensions ◽  
Main Diagonal ◽  
Two Dimensional ◽  
First Passage ◽  
Explicit Representations ◽  
Mean Vectors ◽  
First Passage Problem

The process of interest is a controlled random walk in two dimensions: whenever the walker is above the main diagonal, the next increment to his position is chosen from a distribution FA ; whenever the walker is below the diagonal, the next increment comes from another distribution FB. The two distributions have mean vectors which tend to push the walker back toward the diagonal. We analyze the problem of first passage to the first quadrant, obtaining explicit representations for the limiting first-entry distribution and expected first-passage time.

On first passage times of sticky reflecting diffusion processes with double exponential jumps

2020 ◽  
Vol 57 (1) ◽  
pp. 221-236 ◽  
Author(s):  
Shiyu Song ◽  
Yongjin Wang
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Explicit Solutions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Double Exponential ◽  
Upper Level ◽  
First Passage Problem ◽  
Passage Times

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

scholarly journals Tracking a random walk first-passage time through noisy observations

2012 ◽  
Vol 22 (5) ◽  
pp. 1860-1879 ◽  
Author(s):  
Marat V. Burnashev ◽  
Aslan Tchamkerten
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

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.

scholarly journals Fractional compound Poisson processes with multiple internal states

2018 ◽  
Vol 13 (1) ◽  
pp. 10 ◽  
Author(s):  
Pengbo Xu ◽  
Weihua Deng
Keyword(s):  
Stochastic Process ◽  
Random Walk ◽  
Waiting Time ◽  
Anomalous Diffusion ◽  
First Passage Time ◽  
Passage Time ◽  
Poisson Processes ◽  
First Passage ◽  
Internal States ◽  
Functional Distribution

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals.

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.

The density of interfaces: a new first-passage problem

1993 ◽  
Vol 30 (04) ◽  
pp. 851-862 ◽  
Author(s):  
L. Chayes ◽  
C. Winfield
Keyword(s):  
Correlation Length ◽  
First Passage Time ◽  
Passage Time ◽  
Scaling Behavior ◽  
First Passage Percolation ◽  
First Passage ◽  
Site Percolation ◽  
Percolation Problem ◽  
First Passage Problem

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

Random walk and first passage time on a weighted hierarchical network

2014 ◽  
Vol 25 (09) ◽  
pp. 1450037 ◽  
Author(s):  
Feng Zhu ◽  
Meifeng Dai ◽  
Yujuan Dong ◽  
Jie Liu
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Scale Factor ◽  
Hierarchical Network ◽  
Self Similarity ◽  
First Passage ◽  
Hierarchical Networks ◽  
Average Value ◽  
Peripheral Node

This paper reports a weighted hierarchical network generated on the basis of self-similarity, in which each edge is assigned a different weight in the same scale. We studied two substantial properties of random walk: the first-passage time (FPT) between a hub node and a peripheral node and the FPT from a peripheral node to a local hub node over the network. Meanwhile, an analytical expression of the average sending time (AST) is deduced, which reflects the average value of FPT from a hub node to any other node. Our result shows that the AST from a hub node to any other node is related to the scale factor and the number of modules. We found that the AST grows sublinearly, linearly and superlinearly respectively with the network order, depending on the range of the scale factor. Our work may shed some light on revealing the diffusion process in hierarchical networks.

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