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

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.

💡 Research Summary

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The paper introduces a novel stochastic model for intraday fish migration that combines two previously separate ideas: (1) a non‑negative diffusion bridge, i.e., a stochastic differential equation (SDE) whose solution is constrained to start at a prescribed value at sunrise and to hit zero at sunset, and (2) a McKean–Vlasov (mean‑field) interaction, where the drift and diffusion coefficients depend on the expectation of the process itself. The authors start from the classical Cox–Ingersoll–Ross (CIR) dynamics, well‑known in finance for its analytical tractability and non‑negativity, and modify it by adding a singular term (-X_t/(T-t)) in the drift and a diffusion term that blows up as ((T-t)^{-\alpha}) when the terminal time (T) is approached. The resulting McKean–Vlasov SDE (MVSDE) reads

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