arXiv 1407.35532014-07math.PR

通过测度集中求小偏差的一般方法

A general approach to small deviation via concentration of measures

Ehsan Azmoodeh, Lauri Viitasaari

摘要：

我们提供了一个通用的方法来获得小偏差$ \mathbb{P}(\Vert y \Vert \le ε)$在不同的规范，即上确界和$β$-Hölder规范的上界。所考虑的大类过程$y$采用$y_t= X_t + \int_0^t a_s d s$的形式，其中$X$和$a$是两个可能相关的随机过程。我们的方法提供了一个小偏差的上限，每当上限的\textit{concentration of measures}的$L^p$-范数的随机向量的增量的过程$X$和\textit{large deviation}估计过程$a$是可用的。使用我们的方法，除其他外，我们获得了具有Hurst参数$H\le\ \frac{1}{2}$的分数布朗运动的上确界和$β$-Hölder范数的最佳小偏差率。作为一个应用，我们讨论了我们的有用的小偏差的随机变量的路径随机积分表示的对冲问题在数学金融。

英文摘要：

We provide a general approach to obtain upper bounds for small deviations $ \mathbb{P}(\Vert y \Vert \le ε)$ in different norms, namely the supremum and $β$- Hölder norms. The large class of processes $y$ under consideration takes the form $y_t= X_t + \int_0^t a_s d s$, where $X$ and $a$ are two possibly dependent stochastic processes. Our approach provides an upper bound for small deviations whenever upper bounds for the \textit{concentration of measures} of $L^p$- norm of random vectors built from increments of the process $X$ and \textit{large deviation} estimates for the process $a$ are available. Using our method, among others, we obtain the optimal rates of small deviations in supremum and $β$- Hölder norms for fractional Brownian motion with Hurst parameter $H\le\ \frac{1}{2}$. As an application, we discuss the usefulness of our upper bounds for small deviations in pathwise stochastic integral representation of random variables motivated by the hedging problem in mathematical finance.

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摘要：

英文摘要：

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