Update: Thanks to GJC20's answer on the existence and uniqueness. Let me reformulate my questions 3/4 as follows: There exists a unique non-increasing and continuously differentiable function $f:\mathbb R \to [0,1]$ s.t.

$$X_t=X_0+W_t-1+f(t) \mbox{ and } f(t)= \mathbb E \left[ \exp\left(-\frac 1 \epsilon \int_0^t X_s^-ds\right)\right],\quad \forall t\ge 0.$$

What is a characterization for $f$?

Consider

$$X_t=X_0+ W_t-1 + \mathbb E \left[ \exp\left(-\frac 1 \epsilon \int_0^t X_s^-ds\right)\right],\quad \forall t\ge 0,\quad\quad\quad(\ast)$$

where $X_0>0$ is a random variable as nice as possible, $(W_t)_{t\ge 0}$ is an independent Brownian motion, $\epsilon>0$ is fixed and can be as small as possible.

- Has this equation been studied (I don't even know how to call such equation)?
- Do we have the existence/uniqueness result?
- If $X_t$ has a density function, denoted by $f(t,x)$, is there any analytic equation for $f$?
- Let $\tau:=\inf\{t\ge 0: X_t \le 0\}$. If $X_t{\bf 1}_{\{\tau>t\}}$ has a density function, denoted by $g(t,x)$, is there any analytic equation for $g$?

Many thanks for the answer, comments and references.

PS : Here I adopt the notation $a^-:=\max(-a, 0)$ for all $a\in \mathbb R$.