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

TL;DR

This paper establishes a necessary and sufficient condition for the comparison theorem to hold in multidimensional anticipated backward stochastic differential equations with generators independent of the anticipated Z-term.

Contribution

It provides a precise condition under which the comparison theorem is valid for a class of multidimensional anticipated backward stochastic differential equations.

Findings

Derived a necessary and sufficient condition for the comparison theorem.

Extended the comparison theorem to multidimensional anticipated BSDEs.

Focused on equations with generators independent of the anticipated Z-term.

Abstract

Anticipated backward stochastic differential equations, studied the first time in 2007, are equations of the following type: {tabular}{rlll} $-dY_t$ &=& $f(t, Y_t, Z_t, Y_{t+\delta(t)}, Z_{t+\zeta(t)})dt-Z_tdB_t, $ & $ t\in[0, T];$ $Y_t$ &=& $\xi_t, $ & $t\in[T, T+K];$ $Z_t$ &=& $\eta_t, $ & $t\in[T, T+K].$ In this paper, we give a necessary and sufficient condition under which the comparison theorem holds for multidimensional anticipated backward stochastic differential equations with generators independent of the anticipated term of $Z$.