The sign and the co-monotonicity of Z for a class of decoupled FBSDEs: Theory and applications

In this paper, we investigate the properties of the control process $\left(z\right)$ in a class of backward stochastic differential equation (BSDE) with Markovian terminal data driven by a forward stochastic differential equation (SDE). We focus on determining the sign of ($z$), and find that it is determined via certain information about the BSDE driver and the terminal data. Notably, we explore the co-monotonicity property of ($z$) when the terminal value of the BSDE within the forward–backward SDE (FBSDE) system is non-monotonic. Three applications are also showed in the paper. First, we provide a sufficient condition ensuring that the nonlinear $g$-expectation is additive. Second, we obtain the explicit solution for a class of nonlinear BSDEs. Third, we offer a closed form representation for the standard aggregator utility under ambiguity, as discussed in Chen and Epstein (2002).