Geometric bounds on certain sublinear functionals of geometric Brownian motion

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.

The moment problem for some Wiener functionals: corrections to previous proofs (with an appendix by H. L. Pedersen)

“…In [16], Yor studied the question of whether the random variable is 'determinate by its moments', that is, if the law of the random variable is uniquely determined by its moments. Nikeghbali, and later on also the author, showed that a result of Pakes [12] implied that the integral of a geometric Brownian motion is 'indeterminate by its moments' (see Nikeghbali [11] and Hörfelt [8]). However, the result by Pakes turned out to be false, as recently shown in [3] and, thus, the proofs of the results about indeterminacy in [11] and [8] have to be re-examined.…”

“…Nikeghbali, and later on also the author, showed that a result of Pakes [12] implied that the integral of a geometric Brownian motion is 'indeterminate by its moments' (see Nikeghbali [11] and Hörfelt [8]). However, the result by Pakes turned out to be false, as recently shown in [3] and, thus, the proofs of the results about indeterminacy in [11] and [8] have to be re-examined. This paper will prove that the integral of a geometric Brownian motion is indeed indeterminate by its moments.…”