scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer

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.

2006 ◽  
Vol 43 (1) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (Xn) 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 Yn. Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.


1982 ◽  
Vol 19 (02) ◽  
pp. 430-432
Author(s):  
A. J. Woods

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.


1982 ◽  
Vol 19 (2) ◽  
pp. 430-432 ◽  
Author(s):  
A. J. Woods

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.


2004 ◽  
Vol 41 (2) ◽  
pp. 483-496 ◽  
Author(s):  
Pieter Allaart

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.


1986 ◽  
Vol 38 (2) ◽  
pp. 397-415 ◽  
Author(s):  
Jairo Charris ◽  
Mourad E. H. Ismail

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by


1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen

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.


1986 ◽  
Vol 23 (1) ◽  
pp. 201-207
Author(s):  
Gillian Iossif

A correlated random walk on a d-dimensional integer lattice is studied in which, at any stage, the probabilities of the next step being in the various possible directions depend on the direction of the previous step. Using a renewal argument, asymptotic estimates are obtained for the probability of return to the origin after n steps.


1988 ◽  
Vol 25 (A) ◽  
pp. 335-346
Author(s):  
J. Gani

This paper considers a bivariate random walk modelon a rectangular lattice for a particle injected into a fluid flowing in a tank. The numbers of jumps of the particle in thexandydirections in this particular model are correlated. It is shown that when the random walk forms a bivariate Markov chain in continuous time, it is possible to obtain the state probabilitiespxy(t) through their Laplace transforms. Two exit rules are considered and results for both of them derived.


Author(s):  
H. D. Miller

SummaryThis paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.


1972 ◽  
Vol 9 (3) ◽  
pp. 572-579 ◽  
Author(s):  
D. J. Emery

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.


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