Correlated random walks in heterogeneous landscapes: Derivation, homogenization, and invasion fronts

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.

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DISCRETE MODELS OF HEAT FLOW IN LAYERED MATERIALS USING SIMPLE AND CORRELATED RANDOM WALKS

International Journal of Numerical Modelling Electronic Networks Devices and Fields ◽

10.1002/(sici)1099-1204(199611)9:6<445::aid-jnm251>3.0.co;2-e ◽

1996 ◽

Vol 9 (6) ◽

pp. 445-457 ◽

Cited By ~ 3

Author(s):

P. ENDERS ◽

D. DE COGAN

Keyword(s):

Heat Flow ◽

Random Walks ◽

Layered Materials ◽

Discrete Models ◽

Correlated Random Walks

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Transition probability matrices for correlated random walks

Journal of Applied Probability ◽

10.1017/s0021900200046994 ◽

1980 ◽

Vol 17 (01) ◽

pp. 253-258 ◽

Cited By ~ 3

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.

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Return probabilities for correlated random walks

Journal of Applied Probability ◽

10.2307/3214129 ◽

1986 ◽

Vol 23 (1) ◽

pp. 201-207

Author(s):

Gillian Iossif

Keyword(s):

Random Walk ◽

Random Walks ◽

Integer Lattice ◽

Asymptotic Estimates ◽

Correlated Random Walk ◽

Correlated Random Walks ◽

Previous Step

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.

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Optimal Buy/Sell Rules for Correlated Random Walks

Journal of Applied Probability ◽

10.1017/s0021900200003934 ◽

2008 ◽

Vol 45 (01) ◽

pp. 33-44 ◽

Cited By ~ 1

Author(s):

Pieter Allaart ◽

Michael Monticino

Keyword(s):

Random Walks ◽

Negative Reinforcement ◽

Optimal Strategies ◽

Trading Strategies ◽

Stock Index ◽

Elementary Model ◽

Correlated Random Walks ◽

Multiple Stopping ◽

Stop Rules ◽

Random Step

Correlated random walks provide an elementary model for processes that exhibit directional reinforcement behavior. This paper develops optimal multiple stopping strategies - buy/sell rules - for correlated random walks. The work extends previous results given in Allaart and Monticino (2001) by considering random step sizes and allowing possibly negative reinforcement of the walk's current direction. The optimal strategies fall into two general classes - cases where conservative buy-and-hold type strategies are optimal and cases for which it is optimal to follow aggressive trading strategies of successively buying and selling the commodity depending on whether the price goes up or down. Simulation examples are given based on a stock index fund to illustrate the variation in return possible using the theoretically optimal stop rules compared to simpler buy-and-hold strategies.

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