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.

Long-term risk stratification for survivors of acute coronary syndromes - Results from the long-term intervention with pravastatin in ischemic disease (LIPID) study

OBJECTIVES We developed a prognostic strategy for quantifying the long-term risk of coronary heart disease (CHD) events in survivors of acute coronary syndromes (ACS). BACKGROUND Strategies for quantifying long-term risk of CHD events have generally been confined to primary prevention settings. The Long-term Intervention with Pravastatin in Ischemic Disease (LIPID) study, which demonstrated that pravastatin reduces CHD events in ACS survivors with a broad range of cholesterol levels, enabled assessment of long-term prognosis in a secondary prevention setting. METHODS Based on outcomes in 8,557 patients in the LIPID study, a multivariate risk factor model was developed for prediction of CHD death or nonfatal myocardial infarction. Prognostic indexes were developed based on the model, and low-, medium-, high- and very high-risk groups were defined by categorizing the prognostic indexes. RESULTS In addition to pravastatin treatment, the independently significant risk factors included: total and high density lipoprotein cholesterol, age, gender, smoking status, qualifying ACS, prior coronary revascularization, diabetes mellitus, hypertension and prior stroke. Pravastatin reduced coronary event rates in each risk level, and the relative risk reduction did not vary significantly between risk levels. The predicted five-year coronary event rates ranged from 5% to 19% for those assigned pravastatin and from 6.4% to 23.6% fur those assigned placebo. CONCLUSIONS Long-term prognosis of ACS survivors varied substantially according to conventional risk factor profile. Pravastatin reduced coronary risk within all risk levels; however, absolute risk remained high in treated patients with unfavorable profiles. Our risk stratification strategy enables identification of ACS survivors who remain at very high risk despite statin therapy. CT Am Coil Cardiol 2001;38:56-63) (C) 2001 by the American College of Cardiology.