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

Risk minimization in financial markets modeled by It\^o-L\'evy processes