Necessary and sufficient condition for the comparison theorem of multidimensional anticipated backward stochastic differential equations (Q547354)

A necessary and sufficient condition is given under which the comparison theorem holds for multidimensional anticipated backward stochastic differential equations (ABSDEs). The main results are given in Theorem 3.1, Theorem 3.2 and Theorem 3.3. Their results extend some similar results for multidimensional backward stochastic equations given by \textit{Y. Hu} and \textit{S. Peng} [C. R., Math., Acad. Sci. Paris 343, No.~2, 135--140 (2006; Zbl 1098.60052)] or \textit{R. Buckdahn, M. Quincampoix} and \textit{A. Răşcanu} [Probab. Theory Relat. Fields 116, No.~4, 485--504 (2000; Zbl 0969.60061)] and for the case of 1-dimensional ABSDEs given by \textit{X. Xu} [Sci. China, Math. 54, No.~2, 301--310 (2011; Zbl 1235.60071)].