Numerical solutions of regime-switching jump diffusions

This work develops numerical algorithms for approximating the solutions of stochastic differential equations that involve switching jump diffusion processes, in which the switching is a random process that depends on the jump diffusion. Being non-standard due to the jump diffusion dependent switching makes the problem far more difficult to deal with. Using decreasing step sizes, we construct the algorithm, which is in the spirit of Euler–Maruyama method. To prove the convergence, we first derive the tightness of the algorithm. Then we establish the strong convergence. The strong convergence is in the sense of usual numerical consideration for solutions of stochastic differential equations. That is, we consider uniform mean-square convergence in a finite interval. Finally, numerical examples are provided for demonstration.