  In this paper, we establish an analytic framework for studying set-valued
backward stochastic differential equations (set-valued BSDE), motivated largely
by the current studies of dynamic set-valued risk measures for multi-asset or
network-based financial models. Our framework will make use of the notion of
Hukuhara difference between sets, in order to compensate the lack of "inverse"
operation of the traditional Minkowski addition, whence the vector space
structure in set-valued analysis. While proving the well-posedness of a class
of set-valued BSDEs, we shall also address some fundamental issues regarding
generalized Aumann-It\^o integrals, especially when it is connected to the
martingale representation theorem. In particular, we propose some necessary
extensions of the integral that can be used to represent set-valued martingales
with non-singleton initial values. This extension turns out to be essential for
the study of set-valued BSDEs.
