Let $\mathcal P(\mathbb R)$ be the space of probability measures and $(W_t)_{t\ge 0}$ be a standard Brownian motion.

For given functions $b, \sigma, \beta: \mathbb R_+\times \mathbb R\times \mathbb R\times \mathcal P(\mathbb R)\to \mathbb R$, consider the stochastic differential equation (SDE) below :

$$X_t = X_0+\int_0^{t\wedge \tau} b(s,X_s,\hat X_s, \mu_s)ds+\int_0^{t\wedge \tau} \sigma(s,X_s,\hat X_s, \mu_s)dW_s+\mathbb E\left[\int_0^{t\wedge \tau}{\bf 1}_{\{\hat\tau>t\}}b(s,X_s,\hat X_s, \mu_s)d\hat X_s\right] - \int_0^{t\wedge \tau}{\bf 1}_{\{\hat\tau>t\}}b(s,\hat X_s,X_s, \mu_s)dX_s,\quad \mbox{for all } t\ge 0,$$

where $\tau:=\inf\{t\ge 0: X_t\le 0\}$, $\hat\tau:=\inf\{t\ge 0: \hat X_t\le 0\}$, $\mu_t=\mathcal L(X_t)$, i.e. the law of $X_t$ is $\mu_t$, and $(\hat X_t)_{t\ge 0}$ is a copy of $(X_t)_{t\ge 0}$, i.e. $(\hat X_t)_{t\ge 0}$ and $(X_t)_{t\ge 0}$ are independent and identically distributed. My questions are as follows :

Under which conditions (on $b$, $\sigma$ and $\beta$), the existence of solutions is ensured?

If $X_t{\bf 1}_{\{\tau>t\}}$ admits a probability density, denoted by $p_t(x)$, then what PDE/integral PDE is satisfied by $p$?