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This talk was Dedicated to Einstein's miracle year at his 26

以此次讲座，致敬当年爱因斯坦26岁时的几篇牛文之一，对布朗运动的研究。

对随机游走的研究，已经取得了很深入的进展，本次讲座从布朗运动模型入手，逐步深入，引入分数阶布朗运动，levy随机飞行等概念

这些模型在各种复杂系统中非常常见，比如金融市场，网络交通流量等等，

会简略介绍这些模型在金融系统的应用，以及分析基于布朗运动随机游走的金融模型的弊端

给大家一个随机游走世界的全景

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- 1. Above and Under Brownian MotionBrownian Motion , Fractional Brownian Motion , Levy Flight, and beyond Seminar Talk at Beijing Normal University Xiong Wang 王雄 Centre for Chaos and Complex Networks City University of Hong Kong 1
- 2. Outline Discrete Time Random walks Ordinary random walks Lévy flights Generalized central limit theorem Stable distribution Continuous time random walks Ordinary Diffusion Lévy Flights Fractional Brownian motion (subdiffusion) Ambivalent processes 2
- 3. Part 1Discrete Time Random walks 3
- 4. Ordinary random walks 4
- 5. Central limit theorem 5
- 6. Lévy ﬂights
- 7. Lévy ﬂight scalessuperdiffusively
- 8. Part 2Generalized central limittheorem 9
- 9. Generalized central limittheorem A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | α + 1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution f(x;α,0,c,0) as the number of variables grows. 10
- 10. Stable distribution In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha- stable distribution. 11
- 11. Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters β and α, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures). C:chaosTalklevyStableDensityFunction.cdf
- 12. Characteristic function ofStable distribution A random variable X is called stable if its characteristic function is given by 13
- 13. Symmetric α-stable distributionswith unit scale factor 14
- 14. Skewed centered stabledistributions with different β 15
- 15. Unified normal and power law For α = 2 the distribution reduces to a Gaussian distribution with variance σ2 = 2c2 and mean μ; the skewness parameter β has no effect The asymptotic behavior is described, for α < 2 16
- 16. Log-log plot of skewed centered stable distribution PDFs showing thepower law behavior for large x. Again the slope of the linear portionsis equal to -(α+1)
- 17. Part 1Continuous time random walks 18
- 18. spatial displacement ∆x and atemporal increment ∆t
- 19. Ordinary Diffusion
- 20. Lévy Flights
- 21. Fractional Brownian motion(subdiffusion)
- 22. 1d Fractional Brownian motion
- 23. 2d Fractional Brownian motion
- 24. Ambivalent processes
- 25. Concluding RemarksThe ratio of the exponents α/β resembles theinterplay between sub- and superdiffusion.For β < 2α the ambivalent CTRW is effectivelysuperdiffusive,for β > 2α effectively subdiffusive.For β = 2α the process exhibits the samescaling as ordinary Brownian motion, despitethe crucial difference of inﬁnite moments and anon-Gaussian shape of the pdf W(x, t). 28
- 26. Xiong Wang 王雄Centre for Chaos and Complex NetworksCity University of Hong KongEmail: wangxiong8686@gmail.com 29

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