  We present a probabilistic formulation of risk aware optimal control problems
for stochastic differential equations. Risk awareness is in our framework
captured by objective functions in which the risk neutral expectation is
replaced by a risk function, a nonlinear functional of random variables that
account for the controller's risk preferences. We state and prove a risk aware
minimum principle that is a parsimonious generalization of the well-known risk
neutral, stochastic Pontryagin's minimum principle. As our main results we give
necessary and also sufficient conditions for optimality of control processes
taking values on probability measures defined on a given action space. We show
that remarkably, going from the risk neutral to the risk aware case, the
minimum principle is simply modified by the introduction of one additional
real-valued stochastic process that acts as a risk adjustment factor for given
cost rate and terminal cost functions. This adjustment process is explicitly
given as the expectation, conditional on the filtration at the given time, of
an appropriately defined functional derivative of the risk function evaluated
at the random total cost. For our results we rely on the Fr\'echet
differentiability of the risk function, and for completeness, we prove under
mild assumptions the existence of Fr\'echet derivatives of some common risk
functions. We give a simple application of the results for a portfolio
allocation problem and show that the risk awareness of the objective function
gives rise to a risk premium term that is characterized by the risk adjustment
process described above. This suggests uses of our results in e.g. pricing of
risk modeled by generic risk functions in financial applications.
