McKean-Vlasov processes of bridge type

Abstract

In this paper, we introduce and study McKean-Vlasov processes of bridge type. Specifically, we examine a stochastic differential equation (SDE) of the form: $$\mathrm{d}\hat{I}{¾}_t=-\hat{I}¼(t,\mathbb{E}[\ddot{I}{†}_1(\hat{I}{¾}_t)])\frac{\hat{I}{¾}_t}{T-t}\mathrm{d}t+\ddot{I}ƒ(t,\mathbb{E}[\ddot{I}{†}_2(\hat{I}{¾}_t)])\mathrm{d}{W}_t,t<T,$$ where $\hat{I}¼$ and $\ddot{I}ƒ$ are deterministic functions that depend on time $t$ and the expectation of given functions $\ddot{I}{†}_1$ and $\ddot{I}{†}_2$ 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 $\hat{I}¾$. Finally, we explore explicit solutions in specific cases of interest, including power-weighted expectations and second moments in the drift.

23 pages