On the integral of geometric Brownian motion

By Michael Schroder. 2002

94 citations recorded by the source corpus.

. This paper studies the law of any power of the integral of geometric Brownian motion over any ﬁnite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is the main methodological contribution of the paper. The one of our integrals has a similar structure to that obtained in [Y] , while the other is in terms of Hermite functions as those of [Du01] . Performing or not performing a certain Girsanov transformation is identiﬁed as the source of these two forms of the laws. For exponents equal to 1 our results specialize to those obtained in [Y] , but for exponents equal to minus 1 they give representations for the laws which are markedly diﬀerent from those obtained in [Du01] .

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/5d20a2cb3e8cd383f45f2bc3beaea32e4958a15b

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Citations in the corpus

Exponential functionals of Brownian motion, I: Probability laws at fixed time (2005, methodology): “See also [8],[14], [44], [50], [52] and [61].”
Exponential functionals of Brownian motion, II: Some related diffusion processes (2005, background): “(5.2) The identity in law for a fixed t > 0 has been obtained by Dufresne [19] and the extension (5.2) at the process level has been given in [47].”
The Wiener disorder problem with finite horizon (2006, background): “and we set φ(x) = (1/ √ 2π)e−x 2/2 for x ∈ R (for related expressions in terms of Hermite functions see [3] and [20]).”
Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates (2004, background): “Among them are averages and exponential Brownian functionals (e.g., Dufresne 2000, 2001; Geman and Yor 1993; Linetsky 2001; Yor 2001) local times (e.g., Carr and Jarrow 1990), and occupation times (e.g., Chesney et al. 1997; Davydov and Linetsky 2002; Linetsky 1998, 1999).”
Brownian analogues of Burke's theorem (2001, background): “In fact, r(0) has the same law as −log Zm, where Zm is gamma-distributed with parameter m: this is Dufresne’s identity (Dufresne, 2001).”