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

Jacobi diffusion is a representative diffusion process whose solution is bounded in a domain under certain conditions on its drift and diffusion coefficients. However, the process without such conditions has been far less investigated. We explore a Jacobi diffusion whose drift coefficient is modulated by another process, which causes 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 mathematically analyzed. We also investigate a related mean field game arising in tourism management, where the drift depends on the index for sensor boundary hitting, thereby confining the process in a domain with a higher probability. In this case, the Kolmogorov equation becomes nonlinear. We propose a finite difference method applicable to both the linear and nonlinear Kolmogorov equations that yields a unique numerical solution due to discrete ellipticity. The accuracy of the finite difference method critically depends on the regularity of the boundary condition, and the use of a high-order discretization method is not always effective. Finally, we computationally investigate the mean field effect.