  We study two-armed Levy bandits in continuous-time, which have one safe arm
that yields a constant payoff s, and one risky arm that can be either of type
High or Low; both types yield stochastic payoffs generated by a Levy process.
The expectation of the Levy process when the arm is High is greater than s, and
lower than s if the arm is Low.
  The decision maker (DM) has to choose, at any given time t, the fraction of
resource to be allocated to each arm over the time interval [t,t+dt). We show
that under proper conditions on the Levy processes, there is a unique optimal
strategy, which is a cut-off strategy, and we provide an explicit formula for
the cut-off and the optimal payoff, as a function of the data of the problem.
We also examine the case where the DM has incorrect prior over the type of the
risky arm, and we calculate the expected payoff gained by a DM who plays the
optimal strategy that corresponds to the incorrect prior.
  In addition, we study two applications of the results: (a) we show how to
price information in two-armed Levy bandit problem, and (b) we investigate who
fares better in two-armed bandit problems: an optimist who assigns to High a
probability higher than the true probability, or a pessimist who assigns to
High a probability lower than the true probability.
