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

Strong convergence of a class of adaptive numerical methods for SDEs with jumps