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Geometric bounds on certain sublinear functionals of geometric Brownian motion

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Per Hörfelt
Per Hörfelt*
Affiliation:
Chalmers University of Technology
*
∗Postal address: Department of Mathematics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden. Email address: [email protected]
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Abstract

Suppose that {Xs, 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝm → [0,∞) is a (weighted) lq(ℝm)-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the Lp(μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(Xs), 0 ≤ s ≤ T. 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.

Keywords

Geometric Brownian motionsublinear functionallog-normal distributiongeometric inequalitylog-concave distributiontail probabilitymoment inequalitymoment problemoption pricingAsian basket option

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 893 - 905
Copyright
Copyright © Applied Probability Trust 2003 

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