McKean-Vlasov processes of bridge type
Wolfgang Bock, Astrid Hilbert, Mohammed Louriki
TL;DR
The paper develops a unified theory for McKean–Vlasov SDEs of bridge type where the drift is driven by time-decay and mean-field functionals, and the diffusion can depend on the law as well. It provides explicit strong solutions in power-weighted drift cases, proves existence/uniqueness, and establishes pinned behavior (convergence to zero at the terminal time) while highlighting Gaussian structure in several solvable instances. It then extends the framework to general expectation-dependent drift and diffusion coefficients, giving a constructive Gaussian representation and showing dissipation of variance at the horizon under mild regularity. The results offer a flexible, analytically tractable toolkit for information-based modeling and applications in finance, physics, and biology.
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: $$\mathrm{d} ξ_t=-μ(t,\mathbb{E}[\varphi_1(ξ_t)]) \frac{ξ_t}{T-t} \mathrm{d} t+σ(t,\mathbb{E}[\varphi_2(ξ_t)]) \mathrm{d} W_t,\,\, t<T,$$ where $μ$ and $σ$ 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 $ξ$. Finally, we explore explicit solutions in specific cases of interest, including power-weighted expectations and second moments in the drift.