Existence and uniqueness conjecture for the random-horizon BSDE

Let $(\mathrm{H1})$ and $(\mathrm{H2}')$ hold, and suppose that $g^b(s,0,0)$ is non-negative for every $s\in[0,T]$. Under Assumption $g$, consider the BSDE

$$Y_t^{b}=\xi^a_{T-}-\int_t^T f^b(s,Y_s^{b},Z_s^{b},\xi^a_s-Y_s^{b})\,ds-\int_t^T Z_s^{b}\cdot dW_s,\qquad t\in[0,T],$$

where

$$f^b(s,y,z,u):=g^b(s,y,z)+\lambda_s\frac{1-e^{\alpha u}}{\alpha}.$$

Existence and uniqueness conjecture. The BSDE admits a unique solution such that $Y^b$ is bounded and $\int_0^{\cdot}Z_s^{b}\,dW_s$ is a $\operatorname{BMO}(\mathbb{G})$-martingale.

This conjecture proposes that the stated assumptions suffice for existence and uniqueness without requiring the stronger regularity assumption on $\xi^a$ used earlier. The source leaves the result for future research; the preceding discussion shows that some left-continuity at the terminal time is necessary in the associated linear BSDE.