  The overarching goal of this paper is to establish a set-valued It\^{o}'s
formula. As an application, we obtain the existence and uniqueness of solutions
for the general set-valued backward stochastic differential equation which
gives an answer to an open question proposed by Ararat et al. (C. Ararat, J. Ma
and W.Q. Wu, Set-valued backward stochastic differential equation,
arXiv:2007.15073).
