Fractional Brownian motion as a weak limit of Poisson shot noise processes-with applications to finance

Claudia Klüppelberg; Christoph Kühn

doi:10.1016/j.spa.2004.03.015

Fractional Brownian motion as a weak limit of Poisson shot noise processes-with applications to finance

Claudia Klüppelberg
, Christoph Kühn

Chair of Mathematical Statistics

Johann Wolfgang Goethe University

Research output: Contribution to journal › Article › peer-review

49 Scopus citations

Abstract

We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise process itself can be chosen arbitrage-free. Using the marked point process skeleton of the shot noise process we construct a corresponding equivalent martingale measure explicitly.

Original language	English
Pages (from-to)	333-351
Number of pages	19
Journal	Stochastic Processes and their Applications
Volume	113
Issue number	2
DOIs	https://doi.org/10.1016/j.spa.2004.03.015
State	Published - Oct 2004

Keywords

Alternative stock price models
Arbitrage
Fractional Brownian motion
Functional limit theorems
Non-explosiveness of point processes
Shot noise process

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10.1016/j.spa.2004.03.015

Cite this

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title = "Fractional Brownian motion as a weak limit of Poisson shot noise processes-with applications to finance",

abstract = "We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise process itself can be chosen arbitrage-free. Using the marked point process skeleton of the shot noise process we construct a corresponding equivalent martingale measure explicitly.",

keywords = "Alternative stock price models, Arbitrage, Fractional Brownian motion, Functional limit theorems, Non-explosiveness of point processes, Shot noise process",

author = "Claudia Kl{\"u}ppelberg and Christoph K{\"u}hn",

note = "Funding Information: The authors thank Yuri Kabanov, Thomas Mikosch, and an anonymous referee for valuable comments on this paper. In addition, financial support by the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg “Angewandte Algorithmische Mathematik” at Munich University of Technology is gratefully acknowledged by the second author.",

year = "2004",

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T1 - Fractional Brownian motion as a weak limit of Poisson shot noise processes-with applications to finance

AU - Klüppelberg, Claudia

AU - Kühn, Christoph

N1 - Funding Information: The authors thank Yuri Kabanov, Thomas Mikosch, and an anonymous referee for valuable comments on this paper. In addition, financial support by the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg “Angewandte Algorithmische Mathematik” at Munich University of Technology is gratefully acknowledged by the second author.

PY - 2004/10

Y1 - 2004/10

N2 - We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise process itself can be chosen arbitrage-free. Using the marked point process skeleton of the shot noise process we construct a corresponding equivalent martingale measure explicitly.

AB - We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise process itself can be chosen arbitrage-free. Using the marked point process skeleton of the shot noise process we construct a corresponding equivalent martingale measure explicitly.

KW - Alternative stock price models

KW - Arbitrage

KW - Fractional Brownian motion

KW - Functional limit theorems

KW - Non-explosiveness of point processes

KW - Shot noise process

UR - https://www.scopus.com/pages/publications/4544374045

U2 - 10.1016/j.spa.2004.03.015

DO - 10.1016/j.spa.2004.03.015

M3 - Article

AN - SCOPUS:4544374045

SN - 0304-4149

VL - 113

SP - 333

EP - 351

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 2

ER -

Fractional Brownian motion as a weak limit of Poisson shot noise processes-with applications to finance

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Keywords

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