  We study the nonlinear operator of mapping the terminal value $\xi$ to the
corresponding minimal supersolution of a backward stochastic differential
equation with the generator being monotone in $y$, convex in $z$, jointly lower
semicontinuous and bounded below by an affine function of the control variable
$z$. We show existence, uniqueness, monotone convergence, Fatou's lemma and
lower semicontinuity of this operator. We provide a comparison principle for
minimal supersolutions of BSDEs.
