Kolmogorov equations for evaluating the boundary hitting of degenerate diffusion with unsteady drift
This work addresses a theoretical gap in diffusion processes with applications in fields like tourism management, but it appears incremental as it extends existing methods to a specific case.
The paper tackles the problem of analyzing boundary hitting for a Jacobi diffusion with unsteady drift, presenting and analyzing the Kolmogorov equation for this process and proposing a finite difference method for numerical solutions, with computational investigations showing effects on domain confinement.
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.