Let $(M_t)_{0 \leq t \leq 1} be a continuous martingale with
initial law $M_0 \sim \mu_0$, and terminal law $M_1 \sim \mu_1$, and let $S =
\sup_{0 \leq t \leq 1} M_t$. In this paper we prove that there exists a
greatest lower bound with respect to stochastic ordering of probability
measures, on the law of $S$. We give an explicit construction of this
bound. Furthermore a martingale is constructed which attains this minimum by
solving a Skorokhod embedding problem. The form of this martingale is motivated
by a simple picture. The result is applied to the robust hedging of a forward
start digital option.
@article{1023481014,
author = {Hobson, David G. and Pedersen, J. L.},
title = {The minimum maximum of a continuous martingale with given
initial and terminal laws},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 978-999},
language = {en},
url = {http://dml.mathdoc.fr/item/1023481014}
}
Hobson, David G.; Pedersen, J. L. The minimum maximum of a continuous martingale with given
initial and terminal laws. Ann. Probab., Tome 30 (2002) no. 1, pp. 978-999. http://gdmltest.u-ga.fr/item/1023481014/