We develop adaptive time-stepping strategies for It\^o-type stochastic differential equations (SDEs) with jump perturbations. Our approach builds on adaptive strategies for SDEs. Adaptive methods can ensure strong convergence of nonlinear SDEs with drift and diffusion coefficients that violate global Lipschitz bounds by adjusting the stepsize dynamically on each trajectory to prevent spurious growth that can lead to loss of convergence if it occurs with sufficiently high probability. In this article we demonstrate the use of a jump-adapted mesh that incorporates jump times into the adaptive time-stepping strategy. We prove that any adaptive scheme satisfying a particular mean-square consistency bound for a nonlinear SDE in the non-jump case may be extended to a strongly convergent scheme in the Poisson jump case where jump and diffusion perturbations are mutually independent and the jump coefficient satisfies a global Lipschitz condition.

翻译：我们针对带有跳跃扰动的Itô型随机微分方程开发了自适应时间步进策略。该方法建立在针对随机微分方程的自适应策略基础上。自适应方法通过动态调整每条轨迹上的步长，能够确保漂移系数和扩散系数违反全局Lipschitz条件的非线性随机微分方程的强收敛性，从而防止可能导致以较高概率出现收敛性丧失的虚假增长。本文论证了采用跳跃自适应网格将跳跃时间纳入自适应时间步进策略的方案。我们证明，在非跳跃情形下满足特定均方一致性界线的任意自适应格式，可扩展为泊松跳跃情形下具有强收敛性的格式，其中跳跃与扩散扰动相互独立，且跳跃系数满足全局Lipschitz条件。