截断变异，向上截断变异，向下截断 布朗运动随漂移的变化--它们的特征和 应用程序

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摘要翻译：
在《关于带漂移的布朗运动的截断变分》(Bull.pol.Acad.Sci.Math.56(2008),No.4,267-281）中，我们定义了带漂移的布朗运动的截断变分，$w_t=B_t+\mu t,t\geq0,$其中$(B_t)$是标准布朗运动。截断变异与常规变异不同，因为忽略了小于某些固定$C>0$的跳跃。我们证明了截断变分对于任何复变元都是具有有限矩母函数的随机变量。我们还定义了两个密切相关的量--向上截断变异量和向下截断变异量。定义的数量可能在金融数学中有一定的解释。向上截断变化的指数矩可以解释为当一项金融资产的价格动态服从几何布朗运动过程时，在固定佣金存在的情况下交易该资产的最大可能收益。我们计算了向上和向下截断变化的矩母函数关于时间参数的拉普拉斯变换。作为所得公式的应用，我们给出了向上和向下截断变化期望值的精确公式。我们也给出了这些量的期望值的精确估计（直到普遍常数）。
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英文标题：
《Truncated Variation, Upward Truncated Variation and Downward Truncated
  Variation of Brownian Motion with Drift - their Characteristics and
  Applications》
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作者：
Rafa{\l} {\L}ochowski
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最新提交年份：
2011
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
--

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英文摘要：
  In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, $W_t = B_t + \mu t, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. Truncated variation differs from regular variation by neglecting jumps smaller than some fixed $c > 0$. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument. We also define two closely related quantities - upward truncated variation and downward truncated variation. The defined quantities may have some interpretation in financial mathematics. Exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process. We calculate the Laplace transform with respect to time parameter of the moment-generating functions of the upward and downward truncated variations. As an application of the obtained formula we give an exact formula for expected value of upward and downward truncated variations. We give also exact (up to universal constants) estimates of the expected values of the mentioned quantities.
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PDF链接：
https://arxiv.org/pdf/0912.4533

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关键词：应用程序 布朗运动 特征和 Applications Econophysics 情况 特征 Drift motion 矩母