  We consider the problem of utility maximization with exponential preferences
in a market where the traded stock/risky asset price is modelled as a
L\'evy-driven pure jump process (i.e. the driving L\'evy process has no
Brownian component). In this setting, we study the terminal utility
optimization problem in the presence of a European contingent claim. We
consider in detail the BSDE (backward stochastic differential equation)
characterising the value function when using an exponential utility function.
First we analyse the well-definedness of the generator. This leads to some
conditions on the market model related to conditions for the market to admit no
free lunches. Then we give bounds on the candidate optimal strategy.
  Thereafter, we discuss the example of a cross-hedging problem and, under
severe assumptions on the structure of the claim, we give explicit solutions.
Finally, we establish an explicit solution for a related BSDE with a suitable
terminal condition but a simpler generator.
