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Asymptotic properties of integral functionals of geometric stochastic processes

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Endre Csáki ,
Miklós Csörgő ,
Antónia Földes  and
Pál Révész
Endre Csáki*
Affiliation:
Rényi Institute, Budapest
Miklós Csörgő*
Affiliation:
Carleton University
Antónia Földes*
Affiliation:
City University of New York
Pál Révész*
Affiliation:
Technische Universität Wien
*
∗Postal address: A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary, P.O.B. 127, H-1364. Email address: csaki@math-inst.hu
∗∗Postal address: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
∗∗∗Postal address: City University of New York, 2800 Victory Blvd., Staten Island, NY 10314, USA
∗∗∗∗Postal address: Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8–10/107, A-1040 Vienna, Austria
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Abstract

We study strong asymptotic properties of two types of integral functionals of geometric stochastic processes. These integral functionals are of interest in financial modelling, yielding various option pricings, annuities, etc., by appropriate selection of the processes in their respective integrands. We show that under fairly general conditions on the latter processes the logs of the integral functionals themselves asymptotically behave like appropriate sup functionals of the processes in the exponents of their respective integrands. We illustrate the possible use and applications of these strong invariance theorems by listing and elaborating on several examples.

Keywords

Integral and sup functionalsgeometric processesstrong invariancestrong theoremsLIL and strong incrementsgeometric Brownian motionBlack-Scholes financial modelgeometric fractional Brownian motiongeometric Gaussian and diffusion processes

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 480 - 493
Copyright
Copyright © by the Applied Probability Trust 2000 

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