  This paper is devoted to proposing a new asymmetric risk-sensitive criterion
involving different risk attitudes toward varying risk sources. The criterion
can only be defined through the initial value of the minimal solutions of
quadratic backward stochastic differential equations (BSDEs). Before uncovering
the mean-variance representation for the introduced criterion by the
variational approach, some axioms are given for the first time to characterize
a variance decomposition of square integrable random variables. The stochastic
control problems under this criterion are described as a kind of stochastic
recursive control problems that includes controlled quadratic BSDEs. An
asymmetric risk-sensitive global stochastic maximum principle is derived when
the quadratic BSDEs are equipped with bounded data. A closed-form solution of a
stochastic linear-quadratic risk-sensitive control problem is obtained by
introducing a novel completion-of-squares technique for controlled quadratic
BSDEs. In addition, a dynamic portfolio optimization problem featuring a
stochastic return rate is provided as an application of the asymmetric
risk-sensitive control.
