Geometric bounds on certain sublinear functionals of geometric Brownian motion

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