Peter E. Kloeden

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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)
Abstract. 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(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)
A systematic method for the derivation of high order schemes for affinely controlled nonlinear systems is developed. Using an adaptation of the stochastic Taylor expansion for control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-Kutta type schemes can be obtained. Furthermore an approximation technique(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)
We propose a mathematical model of the memory retrieval process based on dynamical systems over a metric space of p-adic numbers representing a configuration 'space of ideas' in which two ideas are close if they have a sufficiently long common root. Our aim is to suggest a new way of conceptualizing human memory retrieval that might be useful for simulation(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)