Strong and weak convergence rates of logarithmic transformed truncated EM methods for SDEs with positive solutions
Strong and weak convergence rates of logarithmic transformed truncated EM methods for SDEs with positive solutions
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DOI：
10.1016/j.cam.2022.114758
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发表时间：
2022-08
期刊：
影响因子：
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通讯作者：
Ziyi Lei;S. Gan;Ziheng Chen
中科院分区：
文献类型：
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作者：
Ziyi Lei;S. Gan;Ziheng Chen
为了在数值上继承具有非全局Lipschitz系数的随机微分方程的正性，我们设计了一种新的显式方法，称为对数变换截断Euler-Maruyama方法.然而，保持正性是要付出代价的，即对数变换会导致变换后的SDES的系数超线性甚至指数增长，这使得强收敛和弱收敛分析更加复杂。基于指数可积性、截断技巧和其他一些论证，我们证明了该数值方法的强收敛速度为1/2，弱收敛速度可以任意接近1.据我们所知，这是第一个结果，建立了弱收敛速度的数值方法的一般微分方程的正解。最后通过数值实验验证了我们的理论结果。
To inherit numerically the positivity of stochastic differential equations (SDEs) with non-globally Lipschitz coefficients, we devise a novel explicit method, called logarithmic transformed truncated Euler–Maruyama method. There is however a price to be paid for the preserving positivity, namely that the logarithmic transformation would cause the coefficients of the transformed SDEs growing super-linearly or even exponentially, which makes the strong and weak convergence analysis more complicated. Based on the exponential integrability, truncation techniques and some other arguments, we show that the strong convergence rate of the underlying numerical method is 1/2, and the weak convergence rate can be arbitrarily close to 1. To the best of our knowledge, this is the first result establishing the weak convergence rate of numerical methods for the general SDEs with positive solutions. Numerical experiments are finally reported to confirm our theoretical results.