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A Diffusion Limit for Generalized Correlated Random Walks

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Urs Gruber  and
Martin Schweizer
Urs Gruber*
Affiliation:
Allianz AG
Martin Schweizer*
Affiliation:
ETH Zürich
*
∗ Postal address: Allianz AG, Allianz Global Risks, Königinstrasse 28, D-80802 München, Germany. Email address: ursgruber@gmx.net
∗∗ Postal address: Departement Mathematik, ETH Zürich, ETH-Zentrum, HG G28.2, CH-8092 Zürich, Switzerland. Email address: martin.schweizer@math.ethz.ch
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Abstract

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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.

Keywords

Correlated random walkdiffusion limitweak convergencemathematical financelarge investortransaction costbinomial tree

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 60 - 73
Copyright
© Applied Probability Trust 2006 

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