Strong solutions of a class of hybrid diffusion processes with state-dependent regime-switching

This work focuses on the strong solutions of a class of hybrid diffusion processes with state-dependent regime-switching. This important class of processes originates from the purpose of modeling the interest rate in finance. They have no any explicit solutions. Moreover, state-dependent regime-switching and non-Lipschitz diffusion coefficient pose a challenge to our analysis. To overcome all of these, we consider the Euler numerical schemes rather than the traditional Picard iterations in the existing results of solutions of stochastic differential equations. The weak convergence of numerical algorithms is first established by a martingale problem formulation. Using this result, we can also obtain the strong convergence of the algorithms. The existence of strong solutions is then confirmed. In addition, decreasing stepsize iterative algorithms and their weak convergence are presented. Several numerical experiments are also provided for illustration.