A note on the distribution of integrals of geometric Brownian motion

Rabi Bhattacharya, Enrique Thomann, Edward Waymire

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At := ∫t0 exp{Zs} ds, t ≥ 0, where {Zs: s ≥ 0} is a one-dimensional Brownian motion with drift coefficient μ and diffusion coefficient σ2. In particular, both expected values of the form v(t, x) := Ef(x+At), f homogeneous, as well as the probability density a(t, y) dy := P(At ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.

Original languageEnglish (US)
Pages (from-to)187-192
Number of pages6
JournalStatistics and Probability Letters
Volume55
Issue number2
DOIs
StatePublished - Nov 15 2001

Keywords

  • Asian options
  • Geometric Brownian motion
  • Turbulence

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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