# Existence of solutions to some Mckean-Vlasov SDE

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 :

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

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