A new type distribution-dependent SDE for singular nonlinear PDE (Q6042118)

The author studies a new type SDE depending on the future distributions with all initial values as follows: \[ dX_{s,t}^{x}=\sigma_{t}(X_{s,t}^{x})dW_{t}+\{b_{T-t}+F_{T-t}(\cdot,\psi_{t},\nabla \psi_{t})\}(X_{s,t}^{x})dt, \] \[ \psi_{t}(y)=\mathbb{E}[u_{0}(X_{t,T}^{y}){\mathrm{e}}^{\int_{t}^{T}V_{T-r}(X_{t,r}^{y})dr}+\int_{t}^{T}g_{T-r}(X_{t,r}^{y}){\mathrm{e}}^{\int_{t}^{T}V_{T-r'}(X_{t,r'}^{y})dr'}dr],\quad y\in\mathbb{R}^{d}. \] The author establishes the correspondence between this equation and the associated singular nonlinear PDE. Well-posedness and regularities are investigated.