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Conditional Distributions of Processes Related to Fractional Brownian Motion

Part of: Mathematical finance Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Holger Fink ,
Claudia Klüppelberg  and
Martina Zähle
Holger Fink*
Affiliation:
Technische Universität München
Claudia Klüppelberg*
Affiliation:
Technische Universität München
Martina Zähle*
Affiliation:
University of Jena
*
∗ Postal address: Center for Mathematical Sciences, Technische Universität München, 85748 Garching, Germany. Email address: [email protected]
∗∗ Postal address: Center for Mathematical Sciences, and Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany. Email address: [email protected]
∗∗∗ Postal address: Mathematical Institute, University of Jena, 07740 Jena, Germany. Email address: [email protected]
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Abstract

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Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

Keywords

Affine processconditional characteristic functionfractional affine processshort ratemacroeconomic variables processfractional Brownian motionlong-range dependenceinterest ratezero-coupon bondfractional Vasicek modelprediction

MSC classification

Primary: 60G15: Gaussian processes 60G22: Fractional processes, including fractional Brownian motion 60H10: Stochastic ordinary differential equations 60H20: Stochastic integral equations 91G30: Interest rates (stochastic models)
Secondary: 60G10: Stationary processes 91G60: Numerical methods (including Monte Carlo methods)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 166 - 183
Copyright
Copyright © Applied Probability Trust 2013 

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