McKean-Vlasov processes of bridge type

In this paper, we introduce and study McKean-Vlasov processes of bridge type. Specifically, we examine a stochastic differential equation (SDE) of the form: $$\mathrm{d}{\xi }_t=-\mu (t,\mathbb{E}[{\phi }_1({\xi }_t)])\frac{{\xi }_t}{T-t}\mathrm{d}t+\sigma (t,\mathbb{E}[{\phi }_2({\xi }_t)])\mathrm{d}{W}_t,t<T,$$ where $\mu$ and $\sigma$ are deterministic functions that depend on time $t$ and the expectation of given functions ${\phi }_1$ and ${\phi }_2$ of the process, and $W$ is a Brownian motion. We establish the existence and uniqueness of solutions to this equation and analyze the behavior of the process as $t$ approaches $T$. Furthermore, we provide conditions ensuring the pinned property of the process $\xi$. Finally, we explore explicit solutions in specific cases of interest, including power-weighted expectations and second moments in the drift.