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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 (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.

Keywords

Correlated random walkdiffusion limitweak convergencemathematical financelarge investortransaction costbinomial tree

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes

Information

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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