Skip to main content Accessibility help

Geometric bounds on certain sublinear functionals of geometric Brownian motion

  • Per Hörfelt (a1)


Suppose that {X s , 0 ≤ sT} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ m → [0,∞) is a (weighted) l q (ℝ m )-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L p (μ)-norm, 1 ≤ p ≤ ∞, of the function sϕ(X s ), 0 ≤ sT. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.


Corresponding author

Postal address: Department of Mathematics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden. Email address:


Hide All
[1] Alili, L. (1995). Fonctionnelles exponentielles et valeurs principales du mouvement Brownien. Doctoral Thesis, Université Paris VI.
[2] Barouch, E., Kaufman, G. M., and Glasser, M. L. (1986). On sums of lognormal random variables. Stud. Appl. Math. 75, 3755.
[3] Bhattacharya, R., Thomann, E., and Waymire, E. (2001). A note on the distribution of integrals of geometric Brownian motion. Statist. Prob. Lett. 55, 187192.
[4] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12, 239252.
[5] Borell, C. (1975). The Brunn—Minkowski inequality in Gauss space. Invent. Math. 30, 207216.
[6] Borell, C. (1975). Convex set functions in d-space. Period. Math. Hungar. 6, 111136.
[7] Brigo, D., Mercurio, F., Rapisarda, F., and Scotti, R. (2001). Approximated moment-matching dynamics for basket-options simulation. Working paper, Banca IMI.
[8] Comtet, A., and Monthus, C. (1996). Diffusion in one-dimensional random medium and hyperbolic Brownian motion. J. Phys. A 29, 13311345.
[9] Donati-Martin, C., Matsumoto, H., and Yor, M. (2000). On positive and negative moments of the integral of geometric Brownian motion. Statist. Prob. Lett. 49, 4552.
[10] Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407428.
[11] Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223241.
[12] Heyde, C. C. (1963). On a property of the lognormal distribution. J. R. Statist. Soc. B 25, 392393.
[13] Hoffmann-Jörgensen, J., Shepp, L. A., and Dudley, R. M. (1979). On the lower tail of Gaussian seminorms. Ann. Prob. 7, 319342.
[14] Hörfelt, P. (2002). On the error in the Monte Carlo pricing of some familiar European path-dependent options. Working paper, Chalmers University of Technology.
[15] Janos, W. (1970). Tail of the distribution of sums of log-normal variates. IEEE Trans. Inf. Theory 3, 299302.
[16] Kuelbs, J., and Li, W. (1998). Some shift inequalities for Gaussian measures. In High Dimensional Probability (Progress Prob. 43), Birkhäuser, Basel, pp. 233243.
[17] Linetsky, V. (2001). Spectral expansions for Asian (average price) options. Working paper, Northwestern University.
[18] Nikeghbali, A. (2002). Moment problem for some convex functionals of Brownian motion and related problems. Prépublication Probabilités et Modèles Aléatoires 706, Université Paris VI.
[19] Pakes, A. G. (2001). Remarks on converse Carleman and Krein criteria for the classical moment problem. J. Austral. Math. Soc. 71, 81104.
[20] Rogers, L. C. G., and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 10771088.
[21] Sudakov, V. N. and Cirelśon, B. S. (1974). Extremal properties of half-spaces for spherically invariant measures. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 41, 14–24 (in Russian). English translation: (1978) J. Sov. Math. 9, 918.
[22] Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed