Non-negative diffusion bridge of the McKean-Vlasov type: analysis of singular diffusion and application to fish migration

Abstract

The objective of this paper is to provide a new mathematical tool for fish migration that has not been studied well. McKean-Vlasov stochastic differential equations (MVSDEs) have broad potential applications in science and engineering, but remain insufficiently explored. We consider a non-negative McKean-Vlasov diffusion bridge, a diffusion process pinned at both initial and terminal times, motivated by diurnal fish migration phenomena. This type of MVSDEs has not been previously studied. Our particular focus is on a singular diffusion coefficient that blows up at the terminal time, which plays a role in applications of the proposed MVSDE to real fish migration data. We prove that the well-posedness of the MVSDE depends critically on the strength of the singularity in the diffusion coefficient. We present a sufficient condition under which the MVSDE admits a unique strong solution that is continuous and non-negative. We also apply the MVSDE to the latest fine fish count data with a 10-min time interval collected from 2023 to 2025 and computationally investigate these models. Thus, this study contributes to the formulation of a new non-negative diffusion bridge along with an application study.