Implicit Taylor methods for stiff stochastic differential equations

In this paper we discuss implicit Taylor methods for stiff Ito stochastic differential equations. Based on the relationship between Ito stochastic integrals and backward stochastic integrals, we introduce three implicit Taylor methods: the implicit Euler-Taylor method with strong order 0.5, the implicit Milstein-Taylor method with strong order 1.0 and the implicit Taylor method with strong order 1.5. The mean-square stability properties of the implicit Euler-Taylor and Milstein-Taylor methods are much better than those of the corresponding semi-implicit Euler and Milstein methods and these two implicit methods can be used to solve stochastic differential equations which are stiff in both the deterministic and the stochastic components. Numerical results are reported to show the convergence properties and the stability properties of these three implicit Taylor methods. The stability analysis and numerical results show that the implicit Euler-Taylor and Milstein-Taylor methods are very promising methods for stiff stochastic differential equations.

Topological methods for some boundary value problems

We establish existence of solutions for a finite difference approximation to y = f(x, y, y ') on [0, 1], subject to nonlinear two-point Sturm-Liouville boundary conditions of the form g(i)(y(i),y ' (i)) = 0, i = 0, 1, assuming S satisfies one-sided growth bounds with respect to y '. (C) 2001 Elsevier Science Ltd. All rights reserved.

The effects of normalisation on the ability of business end-users to detect data anomalies: An experimental evaluation

With the proliferation of relational database programs for PC's and other platforms, many business end-users are creating, maintaining, and querying their own databases. More importantly, business end-users use the output of these queries as the basis for operational, tactical, and strategic decisions. Inaccurate data reduce the expected quality of these decisions. Implementing various input validation controls, including higher levels of normalisation, can reduce the number of data anomalies entering the databases. Even in well-maintained databases, however, data anomalies will still accumulate. To improve the quality of data, databases can be queried periodically to locate and correct anomalies. This paper reports the results of two experiments that investigated the effects of different data structures on business end-users' abilities to detect data anomalies in a relational database. The results demonstrate that both unnormalised and higher levels of normalisation lower the effectiveness and efficiency of queries relative to the first normal form. First normal form databases appear to provide the most effective and efficient data structure for business end-users formulating queries to detect data anomalies.

Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations

In this paper we discuss implicit methods based on stiffly accurate Runge-Kutta methods and splitting techniques for solving Stratonovich stochastic differential equations (SDEs). Two splitting techniques: the balanced splitting technique and the deterministic splitting technique, are used in this paper. We construct a two-stage implicit Runge-Kutta method with strong order 1.0 which is corrected twice and no update is needed. The stability properties and numerical results show that this approach is suitable for solving stiff SDEs. (C) 2001 Elsevier Science B.V. All rights reserved.