A new first order numerical scheme for nonlinear jump-diffusion problems and its strong convergence analysis

In this study, based on jump-adapted time partition, we propose a novel time-stepping scheme, called jump-adapted split-step Milstein method, to solve a class of nonlinear jump-diffusion problems. Comparing with the classic type Milstein methods for jump-diffusion problems, the proposed one is simpler in form and easier to implement on the computer. Under non-global Lipschitz conditions, by overcoming the main difficulties caused by strong nonlinear coefficients, weaker temporal regularity and the Poisson path dependent time partition in numerical analysis, we rigorously establish strong error estimates for the proposed numerical method, and obtain the explicit optimal mean square convergence rate of order one. Finally, numerical examples are provided to validate the theoretical results.