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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(More)
A class of implicit methods is introduced for Ito stochastic differential equations with Poisson-driven jumps. A convergence proof shows that these implicit methods share the same strong finite-time convergence rate as the explicit Euler–Maruyama scheme. A mean-square linear stability analysis shows that implicitness offers benefits, and a natural analogue(More)
We present an error analysis for a general semilinear stochastic evolution equation in d dimensions based on pathwise approximation. We discretize in space by a Fourier Galerkin method and in time by a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the(More)
The concept of an attractor for autonomous systems is generally too restrictive in the nonautonomous context. An appropriate generalization is the cocycle attractor which consists of a family of equivariant sets. Here the cocycle description of a nonautonomous system and the concept of a cocycle attractor are reviewed in the context of nonautonomous(More)
Various types of attractors are considered and compared for non-autonomous dynamical systems involving a cocycle state space mapping that is driven by an autonomous dynamical system on a compact metric space. In particular, conditions are given for a uniform pullback attractor of the cocycle mapping to form a global attractor of the associated autonomous(More)
Stochastic differential equations (SDE's) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here. High-order numerical methods are developed for the integration of(More)
Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional(More)
The occurrence of shocks in the financial market is well known and, since the 1976 paper of the Noble Prize laureate R.C. Merton, there have been numerous attempts to incorporated them into financial models. Such models often result in jump–diffusion stochastic differential equations. This chapter describes the use of maple for such equations, in particular(More)
Differential female longevity is so far unexplained in evolutionary terms. The theory of evolutionarily necessary aging which goes back to Wallace appears to be up to the task. In this theory, aging minimizes competition between forebear and offspring. The aging equation which is implicit contains the well-known empirical Gompertz law as a special case.(More)