McKean-Vlasov processes of bridge type

TL;DR

This paper introduces a new class of McKean-Vlasov processes of bridge type, establishing their existence, uniqueness, and behavior near the terminal time, with explicit solutions in special cases.

Contribution

It defines and analyzes McKean-Vlasov bridge processes, providing existence, uniqueness, and explicit solutions under certain conditions.

Findings

Proved existence and uniqueness of solutions.

Analyzed process behavior as t approaches T.

Derived explicit solutions for specific cases.

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: $$\mathrm{d} \xi_t=-\mu(t,\mathbb{E}[\varphi_1(\xi_t)]) \frac{\xi_t}{T-t} \mathrm{d} t+\sigma(t,\mathbb{E}[\varphi_2(\xi_t)]) \mathrm{d} W_t,\,\, t<T,$$ where $\mu$ and $\sigma$ are deterministic functions that depend on time $t$ and the expectation of given functions $\varphi_1$ and $\varphi_2$ 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.