Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions
Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions
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DOI：
10.1016/j.apm.2007.02.020
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发表时间：
2008-05
影响因子：
5
通讯作者：
G. Jumarie
中科院分区：
文献类型：
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作者：
G. Jumarie
分数阶随机动力学通常被建模为分数布朗运动驱动的非随机微分方程。在这里，我们建议，而不是使用（标准）布朗运动驱动的非随机分数动力学。关键是分数阶泰勒级数f（x+h）=Eα（hαDxα）f（x）其中Eα（·）表示Mittag-Leffler函数，Dxα是我们最近引入的所谓修正的Riemann-Liouville分数阶导数，以消除所考虑的函数的非零初值的影响。阐明了这两种模型的等价性，并说明了如何从一种模型切换到另一种模型。给出了定义分数阶有色噪声的随机微分方程和分数阶指数增长的随机微分方程两个例子。
Stochastic dynamics of fractional order are usually modeled as non-random differential equation driven by fractional Brownian motion. Here we propose rather to use a non-random fractional dynamics driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order f(x+h)=Eα(hαDxα)f(x) where Eα(·) denotes the Mittag–Leffler function, and Dxαis the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. The equivalence of the two models is clarified, and one shows how to switch from one of them to the other one. Two illustrative examples are displayed, which are the stochastic differential equations defining fractional coloured noises on the other hand, and fractional exponential growth on the other hand.