Approach to Solving Stochastic Differential Equations with Jumps Using Adaptive Time-Stepping and High-Order Milstein Schemes

Abstract

This paper presents a novel and efficient numerical method for solving Stochastic Differential Equations with Jumps (SDEJs). We introduce an adaptive time-stepping scheme coupled with a high-order Milstein approximation to enhance the accuracy and stability of the solution. The adaptive time-stepping is designed to dynamically adjust the step size based on the local behavior of the solution, thereby improving computational efficiency. The high-order Milstein scheme is tailored to handle the jump component effectively, particularly when the jump sizes are significant. We provide a rigorous convergence analysis of the proposed method and demonstrate its superior performance through numerical experiments. The results show that the adaptive Milstein scheme offers a significant improvement in accuracy and efficiency compared to traditional fixed-step methods, especially for SDEJs with large jump intensities.