  This paper builds a core-satellite model of semi-static Kelly betting and
log-optimal investment. We study the problem of a saver whose core portfolio
consists in unlevered (1x) retirement plans with no access to margin debt.
However, the agent has a satellite investment account with recourse to
significant, but not unlimited, leverage; accordingly, we study optimal
controllers for the satellite gearing ratio. On a very short time horizon, the
best policy is to overbet the satellite, whereby the overriding objective is to
raise the aggregate beta toward a growth-optimal level. On an infinite horizon,
by contrast, the correct behavior is to blithely ignore the core and optimize
the exponential growth rate of the satellite, which will anyways come to
dominate the entire bankroll in the limit. For time horizons strictly between
zero and infinity, the optimal strategy is not so simple: there is a key
trade-off between the instantaneous growth rate of the composite bankroll, and
that of the satellite itself, which suffers ongoing volatility drag from the
overbetting. Thus, a very perspicacious policy is called for, since any losses
in the satellite will constrain the agent's access to leverage in the
continuation problem. We characterize the optimal feedback controller, and
compute it in earnest by solving the corresponding HJB equation recursively and
backward in time. This solution is then compared to the best open-loop
controller, which, in spite of its relative simplicity, is expected to perform
similarly in practical situations.
