Let $mathcal P(mathbb R)$ be the space of probability measures and $(W_t)_{tge 0}$ be a standard Brownian motion.
For given functions $b, sigma, beta: mathbb R_+times mathbb Rtimes mathbb Rtimes mathcal P(mathbb R)to mathbb R$, consider the stochastic differential equation (SDE) below :
$$X_t = X_0+int_0^{twedge tau} b(s,X_s,hat X_s, mu_s)ds+int_0^{twedge tau} sigma(s,X_s,hat X_s, mu_s)dW_s+mathbb Eleft(int_0^{twedge tau}{bf 1}_{{hattau>t}}b(s,X_s,hat X_s, mu_s)dhat X_sright) – int_0^{twedge tau}{bf 1}_{{hattau>t}}b(s,hat X_s,X_s, mu_s)dX_s,quad mbox{for all } tge 0,$$
where $tau:=inf{tge 0: X_tle 0}$, $hattau:=inf{tge 0: hat X_tle 0}$, $mu_t=mathcal L(X_t)$, i.e. the law of $X_t$ is $mu_t$, and $(hat X_t)_{tge 0}$ is a copy of $(X_t)_{tge 0}$, i.e. $(hat X_t)_{tge 0}$ and $(X_t)_{tge 0}$ 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 $p$?