We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipsch...

Suppose we are given a black-box evaluator (an oracle that returns the function value at a given point) for a Lipschitz function with a known Lipschitz constant. We consider queries that can be answered about the function by using a finite number of black-box evaluations. Specifically, we study the problems of approximating a Lipschitz function, approximately integrating a Lipschitz function, a...

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit $\mathcal{C}^{1,1}$-parametrization and they solve the geodesic equation in sense of Filippov this parametrization. Our proof shows are either everywhere lightlike or timelike. Furthermore, demonstrates an $\alpha$-H\"older $\mathcal{C}^{1,\frac{\alpha}{4}}$-parametrization.

Nonlinear programming problems are analyzed for Lipschitz and upper-Lipschitz behavior of their solutions and stationary points under general perturbations. Facts from a diversity of sources are put together to obtain new characterizations of several local stability properties.

We show that James tree space JT can be renormed to be Lipschitz separated. It negatively answers the question of J. Borwein, J. Giles and J. Vanderwerff whether every Lipschitz separated Banach space is an Asplund space.

We construct Lipschitz functions such that for all s > 0 they are s{HH older, and so proximally, subdiierentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed HH older and approximate subderivatives.

We prove generalizations of the relative Schoennies extension theorem for topo-logical, quasiconformal, or bi-Lipschitz embeddings due to Gauld and VV aiss all a, and show that maximal dilatations and bi-Lipschitz constants of the extensions can be controlled.