Kolmogorov equations for evaluating the boundary hitting of degenerate diffusion with unsteady drift

TL;DR

This paper analyzes degenerate Kolmogorov equations for Jacobi diffusions with unsteady drift, focusing on boundary hitting probabilities, computational methods, and a mean-field model in tourism management.

Contribution

It introduces a new boundary hitting evaluation for Jacobi diffusions with unsteady drift and proposes a finite difference method for related Kolmogorov equations.

Findings

The finite difference method provides a unique numerical solution under discrete ellipticity.

Boundary regularity affects the accuracy of the finite difference method.

The mean field model demonstrates the impact of drift dependence on boundary hitting probabilities.

Abstract

Jacobi diffusion is a representative diffusion process whose solution is bounded in a domain under certain drift and diffusion coefficient conditions. However, the process without such conditions has not been thoroughly investigated. We explore a Jacobi diffusion whose drift coefficient is affected by another deterministic process, causing the process to hit the boundary of a domain in finite time. The Kolmogorov equation (a degenerate elliptic partial differential equation) for evaluating the boundary hitting of the proposed Jacobi diffusion is then presented and analyzed, with several conditional arguments, some of which are addressed computationally. We also investigate a related mean-field-type (McKean-Vlasov) self-consistent model arising in tourism management, where the drift depends on the index for sensor boundary hitting, thereby confining the process to a domain with higher probability. We propose a finite difference method for the linear and nonlinear Kolmogorov equations, which yields a unique numerical solution because of discrete ellipticity if the discount is positive. The accuracy of the finite difference method critically depends on the regularity of the boundary condition, and the use of high-order discretization is not always effective. Finally, we computationally investigate the mean field effect.