How would one go about showing that there exists a solution $C(t)$ and $D(t)$ to the following pair of functional equations and that this pair of solutions is a unique solution?
$$C(t) = a \vee \sup_{s \in [0,t]} (B_s^\mu +D(t))$$ $$-D(t) = a \wedge \inf_{s \in [0,t]} (B_s^\mu +1 - C(t))$$
We have that $B=(B_t^\mu)_{t \geq 0}$ is a Brownian motion with drift $\mu \in \mathbb{R}$ and also have that $a$ is a constant between $0$ and $1$. How would one show this existence and uniqueness and are there any additional conditions required on $C$ and $D$?