  There are two major streams of literature on the modeling of financial
bubbles: the strict local martingale framework and the Johansen-Ledoit-Sornette
(JLS) financial bubble model. Based on a class of models that embeds the JLS
model and can exhibit strict local martingale behavior, we clarify the
connection between these previously disconnected approaches. While the original
JLS model is never a strict local martingale, there are relaxations which can
be strict local martingales and which preserve the key assumption of a
log-periodic power law for the hazard rate of the time of the crash. We then
study the optimal investment problem for an investor with constant relative
risk aversion in this model. We show that for positive instantaneous expected
returns, investors with relative risk aversion above one always ride the
bubble.
