摘要翻译：
连续时间随机游动(CTRW)是一种纯跳跃的随机过程，在物理学中有着广泛的应用，在保险、金融、经济等领域也有广泛的应用。给出了一类由CTRW驱动的随机积分的定义，包括Ito和Stratonovich情形。具有零均值跳跃的非耦合CTRW是鞅。根据鞅变换定理，证明了如果CTRW是鞅，则Ito积分也是鞅。本文给出了如何利用随机积分的定义，通过蒙特卡罗模拟来方便地计算它们。数值计算表明，当CTRW的空间跳跃服从对称Levy alpha稳定分布，等待时间服从单参数Mittag-Leffler分布时，CTRW与其二次变分、Stratonovich积分和Ito积分之间的关系。值得注意的是，这些分布具有胖尾和无界二次变异。在尺度参数消失的扩散极限下，这类CTRW的概率密度满足时空分数阶扩散方程(FDE)或更一般的分数阶Fokker-Planck方程，推广了用Wiener过程的概率密度求解的标准扩散方程，从而提供了反常扩散的唯象模型。本文还给出了用FDE描述的随机过程的二次变分的解析表达式，并用蒙特卡罗方法对其进行了检验。
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英文标题：
《Stochastic calculus for uncoupled continuous-time random walks》
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作者：
Guido Germano, Mauro Politi, Enrico Scalas, Ren\'e L. Schilling
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最新提交年份：
2009
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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英文摘要：
  The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Ito integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its Ito integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Levy alpha-stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, that generalize the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE, and check it by Monte Carlo.
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PDF链接：
https://arxiv.org/pdf/0802.3769