The correlated random walk

1981 ◽  
Vol 18 (2) ◽  
pp. 403-414 ◽  
Author(s):  
Eric Renshaw ◽  
Robin Henderson
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Correlated Random Walk ◽  
Limiting Distributions ◽  
One Dimensional ◽  
Step Transition

A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 – p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated.

The correlated random walk

1981 ◽  
Vol 18 (02) ◽  
pp. 403-414 ◽  
Author(s):  
Eric Renshaw ◽  
Robin Henderson
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Correlated Random Walk ◽  
Limiting Distributions ◽  
One Dimensional ◽  
Step Transition

A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 – p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated.

A correlated random walk model for two-dimensional diffusion

1984 ◽  
Vol 21 (2) ◽  
pp. 233-246 ◽  
Author(s):  
Robin Henderson ◽  
Eric Renshaw ◽  
David Ford
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Exact Expression ◽  
Dispersion Matrix ◽  
Two Dimensional ◽  
Correlated Random Walk ◽  
Dimensional Diffusion ◽  
Step Transition ◽  
Lattice Random Walk ◽  
Previous Step

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

A correlated random walk model for two-dimensional diffusion

1984 ◽  
Vol 21 (02) ◽  
pp. 233-246 ◽  
Author(s):  
Robin Henderson ◽  
Eric Renshaw ◽  
David Ford
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Exact Expression ◽  
Dispersion Matrix ◽  
Two Dimensional ◽  
Correlated Random Walk ◽  
Dimensional Diffusion ◽  
Step Transition ◽  
Lattice Random Walk ◽  
Previous Step

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

scholarly journals Zero-one-only process: A correlated random walk with a stochastic ratchet

2014 ◽  
Vol 28 (29) ◽  
pp. 1450201
Author(s):  
Seung Ki Baek ◽  
Hawoong Jeong ◽  
Seung-Woo Son ◽  
Beom Jun Kim
Keyword(s):  
Transport Phenomenon ◽  
Random Walk ◽  
Random Walks ◽  
Stochastic Processes ◽  
Function Method ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Persistent Random Walk ◽  
The One ◽  
Previous Step

The investigation of random walks is central to a variety of stochastic processes in physics, chemistry and biology. To describe a transport phenomenon, we study a variant of the one-dimensional persistent random walk, which we call a zero-one-only process. It makes a step in the same direction as the previous step with probability p, and stops to change the direction with 1 − p. By using the generating-function method, we calculate its characteristic quantities such as the statistical moments and probability of the first return.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (X n ) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

scholarly journals Stochastic LU factorizations, Darboux transformations and urn models

2018 ◽  
Vol 55 (3) ◽  
pp. 862-886 ◽  
Author(s):  
F. Alberto Grünbaum ◽  
Manuel D. de la Iglesia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Parameter ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Urn Models ◽  
Triangular Matrices ◽  
New Family ◽  
Step Transition

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

Some problems on a one-dimensional correlated random walk with various types of barrier

1992 ◽  
Vol 29 (01) ◽  
pp. 196-201 ◽  
Author(s):  
Yuan Lin Zhang
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Expected Duration ◽  
Explicit Expressions

In this paper one-dimensional correlated random walks (CRW) with various types of barrier such as elastic barriers, absorbing barriers and so on are defined, and explicit expressions are derived for the ultimate absorbing probability and expected duration. Some numerical examples to illustrate the effects of correlation are also presented.

Some problems on a one-dimensional correlated random walk with various types of barrier

1992 ◽  
Vol 29 (1) ◽  
pp. 196-201 ◽  
Author(s):  
Yuan Lin Zhang
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Expected Duration ◽  
Explicit Expressions

In this paper one-dimensional correlated random walks (CRW) with various types of barrier such as elastic barriers, absorbing barriers and so on are defined, and explicit expressions are derived for the ultimate absorbing probability and expected duration. Some numerical examples to illustrate the effects of correlation are also presented.

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