Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: Convergence and its rate
Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: Convergence and its rate
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DOI：
10.1016/j.cam.2020.112771
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发表时间：
2020-08
期刊：
影响因子：
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通讯作者：
Xiaoyue Li;G. Yin
中科院分区：
文献类型：
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作者：
Xiaoyue Li;G. Yin
虽然已经发展了一些隐式数值方法来处理高度非线性，但是否可以使用显式格式来获得类似于Milstein方法的收敛速度的问题仍然悬而未决。这将我们带到目前的工作，集中在使用显式格式的随机微分方程的数值解上。我们的主要目标是在有限时间区间内获得一阶收敛于二阶矩。与隐式格式相比，显式格式具有优势，易于实现，并且计算量较小。为了克服系数的超线性增长带来的困难，我们在算法中使用了截断装置。在分析部分，除了达到上述目标外，还提供了数值算例来说明我们的结果。
Although some implicit numerical procedures have been developed to treat high nonlinearity, the question whether one can use explicit schemes to achieve convergence rate similar to that of Milstein’s procedure remained open. This brings us to the current work that focuses on numerical solutions of stochastic differential equations using explicit schemes. Our main goals are to obtain order one convergence in the second moment in a finite-time interval. In contrast to the implicit schemes, explicit schemes are advantageous, easily implementable, and computationally less intensive. To overcome the difficulties due to super-linear growth of the coefficients, a truncation device is used in our algorithm. In addition to reaching aforementioned goals in the analysis part, numerical examples are provided to demonstrate our results.