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.


1. Has this equation been studied (I don't even know how to call such equation)?
2. Do we have the existence/uniqueness result?
3. If $X\_t$ has a density function, denoted by $f(t,x)$, is there any analytic equation for $f$?
4. 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 $a\in \mathbb R$.



Has this "stochastic differential equation" been studied?
Updates : This is an answer to the simplest question above, existence and uniqueness.


If the existence holds, then any solution must be a strong solution. Let $X, Y$ be two arbitrary solution, it appears that $|X\_t-Y\_t|$ is deterministic and satisfies further


$$\mathbb E|X\_t-Y\_t| = |X\_t-Y\_t| \le \mathbb E\left[\frac{1}{\epsilon}\int\_0^t |X^-\_s-Y^-\_s|ds\right] \le \frac{1}{\epsilon}\int\_0^t \mathbb E[|X\_s-Y\_s|]ds,\quad \forall t\ge 0,$$


which yields the uniqueness by Gronwall's inequality.


In the following, let us show the existence. To do so, we adopt the argument of fixed point. Let $C$ be the space of continous functions on $\mathbb R\_+$ and $C\_{\preceq}\subset C$ be the subset of non-increasing functions $f$ s.t. $f(0)=1$ and $\inf\_{t\ge 0} f(t)\ge 0$. Define further the operator $\Gamma: C\_{\preceq}\to C\_{\preceq}$ by


$$\Gamma[f](t):=\mathbb E\left[\exp\left(-\frac{1}{\epsilon}\int\_0^t\big(X^f\_s\big)^-ds\right)\right],\quad \forall t\ge 0,$$


where $X^f\_t:=X\_0+W\_t-1+f(t)$. Clearly $\Gamma[f]\in C\_{\preceq}$ for every $f\in C\_{\preceq}$. Then it suffices to show $\Gamma$ has a fixed point in $C\_{\preceq}$. First, we observe that $\Gamma$ is monotone in the following sense: if $f\preceq g$, then $\Gamma[f]\preceq \Gamma[g]$. Here we say $f\preceq g$ iff $f(t)\le g(t)$ for all $t\ge 0$. Next, set $f\_0\equiv 1\in C\_{\preceq}$ and define $f\_{n+1}:=\Gamma[f\_n]$ for every $n\ge 0$. By induction, one has that $n\mapsto f\_n(t)$ is non-increasing for all $t\ge 0$. Thus the pointwise $f(t):=\lim\_{n\to\infty}f\_n(t)$ exists. Finally, using the equality


$$f\_{n+1}(t)=\mathbb E\left[\exp\left(-\frac{1}{\epsilon}\int\_0^t\big(X\_0+W\_s-1+f\_n(s)\big)^-ds\right)\right]$$


and the dominated convergence theorem, we may conclude the desired existence.


