Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T14:07:00.129Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic properties of integral functionals of geometric stochastic processes

Part of: Limit theorems Special processes

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acharya, S. K., and Mishra, M. N. (1988). Upper and lower functions for diffusion processes. Indian J. Pure Appl. Math. 19, 1035–1042.Google Scholar
Bertoin, J., and Werner, W. (1994). Asymptotic windings of planar Brownian motion revisited via the Ornstein–Uhlenbeck process. In Séminaire de Probabilités XXVIII, eds Azéma, J., Meyer, P.-A. and Yor, M. (Lecture Notes in Math. 1532). Springer, Berlin, pp. 138–152.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Econ. 81, 637–654.CrossRefGoogle Scholar
Csáki, E. and Csörgõ, M. (1992). Inequalities for increments of stochastic processes and moduli of continuity. Ann. Prob. 20, 1031–1052.CrossRefGoogle Scholar
Csáki, E., Csörgõ, M., Földes, A. and Révész, P. (1983). How big are the increments of the local time of a Wiener process? Ann. Prob. 11, 593–608.CrossRefGoogle Scholar
Csáki, E., Csörgõ, M., Lin, Z. Y. and Révész, P. (1991). On infinite series of independent Ornstein–Uhlenbeck processes. Stoch. Proc. Appl. 39, 25–44.CrossRefGoogle Scholar
Csörgõ, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.Google Scholar
Csörgõ, M., and Shao, Q.-M. (1993). Strong limit theorems for large and small increments of ellp-valued Gaussian processes. Ann. Prob. 21, 1958–1990.CrossRefGoogle Scholar
De Schepper, A., and Goovaerts, M. (1992). Some further results on annuities certain with random interest. Ins. Math. Econ. 11, 283–290.CrossRefGoogle Scholar
De Schepper, A., Goovaerts, M., and Delbaen, F. (1992). The Laplace transform of annuities certain with exponential time distribution. Ins. Math. Econ. 11, 291–294.CrossRefGoogle Scholar
Dufresne, D. (1989). Weak convergence of random growth processes with applications to insurance. Ins. Math. Econ. 8, 187–201.CrossRefGoogle Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 39–79.CrossRefGoogle Scholar
Goovaerts, M., and Dhaene, J. (1997). Actuarial applications of financial models. CWI Quarterly 10, 55–64.Google Scholar
Gruet, J.-C., and Shi, Z. (1995). Some asymptotic results for exponential functionals of Brownian motion. J. Appl. Prob. 32, 930–940.CrossRefGoogle Scholar
Karatzas, I., and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.CrossRefGoogle Scholar
Keprta, S. (1997). Integral Tests for Brownian Motion and Some Related Processes. Ph.D. Thesis, Carleton University, Ottawa.CrossRefGoogle Scholar
Mishra, M. N., and Acharya, S. K. (1983). On normalization in the law of the iterated logarithm for diffusion processes. Indian J. Pure Appl. Math. 14, 1335–1342.Google Scholar
Mishra, M. N., and Steinebach, J. (1994). Asymptotic laws for a class of diffusion processes. Mathematika Balkanica 8, 21–33.Google Scholar
Ortega, J. (1984). On the size of the increments of nonstationary Gaussian processes. Stoch. Proc. Appl. 18, 47–56.CrossRefGoogle Scholar
Rogers, L. C. G., and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088.CrossRefGoogle Scholar
Yor, M. (1992a). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.CrossRefGoogle Scholar
Yor, M. (1992b). Sur certaines fonctionelles exponentielles du mouvement brownien réel. J. Appl. Prob. 29, 202–208.CrossRefGoogle Scholar