# stochastic processes – Existence of solutions to some Mckean-Vlasov SDE

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 :

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 $$p$$?

Posted on Categories Articles