  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.
