  This paper is mainly a survey of recent research developments regarding
methods for risk minimization in financial markets modeled by It\^o-L\'evy
processes, but it also contains some new results on the underlying stochastic
maximum principle.
  The concept of a convex risk measure is introduced, and two representations
of such measures are given, namely: (i) the dual representation and (ii) the
representation by means of backward stochastic differential equations (BSDEs)
with jumps. Depending on the representation, the corresponding risk minimal
portfolio problem is studied, either in the context of stochastic differential
games or optimal control of forward-backward SDEs.
  The related concept of recursive utility is also introduced, and
corresponding recursive utility maximization problems are studied.
  In either case the maximum principle for optimal stochastic control plays a
crucial role, and in the paper we prove a version of this principle which is
stronger than what was previously known.
  The theory is illustrated by examples, showing explicitly the risk minimizing
portfolio in some cases.
