有漂移的布朗运动的截断变分、向上截断变分和向下截断变分--它们的特征及应用
Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications
摘要：
在论文“On Truncated Variation of布朗运动with Drift”（Bull.Pol.Acad. Sci. Math.56（2008），no.4，267 - 281），我们定义了具有漂移的布朗运动的截断变差$W_t = B_t + μt, t\geq 0,$，其中$(B_t)$是标准布朗运动。截断变分与规则变分的不同之处在于它忽略了小于某个固定$c > 0$的跳跃。证明了截断变差是一个随机变量，对任意复变元具有有限的矩母函数。我们还定义了两个密切相关的量-向上截断变差和向下截断变差。定义的数量可能在金融数学中有一些解释。向上截断变差的指数矩可以解释为当资产价格的动态遵循几何布朗运动过程时，在存在固定佣金的情况下交易金融资产的最大可能回报。计算了上下截断变分的矩母函数关于时间参数的拉普拉斯变换。作为所得公式的一个应用，我们给出了上下截尾变差期望值的一个精确公式。我们也给精确的（普遍常数）估计所提到的数量的期望值。
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, $W_t = B_t + μt, t\geq 0,$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
