论文标题
非线性随机微分方程的明确数值方案的强收敛和渐近稳定性
Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations
论文作者
论文摘要
在本文中，我们介绍了几种易于实现的明确方案，这些方案适合Khasminski的技术，并且特别适合高度非线性随机的微分方程（SDE）。我们表明，没有其他限制条件，除了保证确切解决方案在某些Lyapunov函数方面具有界限的条件，数值解决方案在有限的时间内强烈融合到确切的解决方案。此外，基于非负半明星收敛定理，在这里证明了关于显式数值近似重现SDE的众所周知的Lasalle型定理的能力的积极结果，我们从中推断出数值溶液的渐近稳定性。提供了一些示例和仿真来支持理论结果并证明该方法的有效性。
In this article we introduce several kinds of easily implementable explicit schemes, which are amenable to Khasminski's techniques and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that without additional restriction conditions except those which guarantee the exact solutions possess their boundedness in expectation with respect to certain Lyapunov functions, the numerical solutions converge strongly to the exact solutions in finite-time. Moreover, based on the nonnegative semimartingale convergence theorem, positive results about the ability of explicit numerical approximation to reproduce the well-known LaSalle-type theorem of SDEs are proved here, from which we deduce the asymptotic stability of numerical solutions. Some examples and simulations are provided to support the theoretical results and to demonstrate the validity of the approach.
