Discretely sampled signals and the rough Hoff process
Discretely sampled signals and the rough Hoff process
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DOI：
10.1016/j.spa.2016.02.011
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发表时间：
2016-09-01
影响因子：
1.4
通讯作者：
Lyons, Terry
中科院分区：
文献类型：
--
作者：
Flint, Guy;Hambly, Ben;Lyons, Terry
我们介绍了一种将离散顺序数据集转换为由超前-滞后增量组成的相关粗糙路径的规范方法。特别地，通过在一组时间D = {t(i)}处对一个D维连续半鞅X: [0,1] -> R-d进行采样，我们构造了一个由过去和未来分量组成的分段线性、轴向过程X: [0,1] -> R-2d。我们称这样的对象为与离散数据{X-t}相关的霍夫过程（ti是D的一个元素）。霍夫过程可以提升到其自然粗糙路径增强，我们考虑了随着采样频率增加的收敛问题。证明了由X-D分量驱动的随机ode序列可以恢复Ito积分。这与经典的Wong-Zakai定理（Wong and Zakai, 1965）提出的通常的Stratonovich积分极限形成对比。这种随机ode在数学金融的背景下有一个自然的解释。(C) 2016 Elsevier B.V.版权所有
We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a d-dimensional continuous semimartingale X : [0, 1] -> R-d at a set of times D = {t(i)}, we construct a piecewise linear, axis-directed process X-D : [0, 1] -> R-2d comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data {X-t}(ti is an element of D). The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the Ito integral can be recovered from a sequence of random ODEs driven by the components of X-D. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong-Zakai Theorem (Wong and Zakai, 1965). Such random ODEs have a natural interpretation in the context of mathematical finance. (C) 2016 Elsevier B.V. All rights reserved.