Processes that can be embedded in a geometric Brownian motion

Abstract

The main result is a counterpart of the theorem of Monroe [\emph{Ann. Probability} \textbf{6} (1978) 42--56] for a geometric Brownian motion: A process is equivalent to a time change of a geometric Brownian motion if and only if it is a nonnegative supermartingale. We also provide a link between our main result and Monroe [\emph{Ann. Math. Statist.} \textbf{43} (1972) 1293--1311]. This is based on the concept of a \emph{minimal} stopping time, which is characterised in Monroe [\emph{Ann. Math. Statist.} \textbf{43} (1972) 1293--1311] and Cox and Hobson [\emph{Probab. Theory Related Fields} \textbf{135} (2006) 395--414] in the Brownian case. We finally suggest a sufficient condition for minimality (for the processes other than a Brownian motion) complementing the discussion in the aforementioned papers.