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：我们针对含有跳扰动的Itô型随机微分方程（SDEs）开发了自适应时间步长策略。该方法建立在SDEs的自适应策略基础之上。通过动态调整每条轨迹上的步长以防止可能导致收敛性丧失的异常增长（若该增长以足够高的概率发生），自适应方法可确保具有违反全局Lipschitz条件的漂移系数与扩散系数的非线性SDEs实现强收敛。本文展示了将跳适应网格（即把跳时刻融入自适应步长策略）的运用方法。我们证明：在无跳情况下满足非线性SDE特定均方相容性界的任意自适应格式，可推广至泊松跳情形下的强收敛格式，其中跳扰动与扩散扰动相互独立，且跳系数满足全局Lipschitz条件。