arXiv 0711.01632007-11-01math.PR

等视野和粗糙路径正则性

Isoperimetry and Rough Path Regularity

Peter Friz, Harald Oberhauser

摘要：

随机过程的最优样本路径性质通常涉及广义Hölder范数或变分范数。根据Taylor的一个经典结果，在$0+$附近用$ψ(x) \equiv $ $x^{2}/\log \log (1/x) $度量布朗运动的精确变分.这样的$ψ$ -变差结果扩展到抽象度量空间中的值的过程类。(No假设高斯或马尔可夫性质。）为了建立$ψ$ -变分的可积性，我们转向一大类高斯粗糙路径（例如，布朗运动和被视为李群中的过程的Lévy面积），并使用抽象Wiener空间上的Borell不等式证明高斯可积性。对这些结果的兴趣在于它们与粗糙路径理论相容，并产生某些尖锐的正则性和可积性性质（例如，对于迭代Stratonovich积分），这是难以获得的。最后，证明了$ψ$ -变差是（随机）粗微分方程在半鞅之外解的鲁棒正则性.

英文摘要：

Optimal sample path properties of stochastic processes often involve generalized Hölder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $ψ(x) \equiv $ $x^{2}/\log \log (1/x) $ near $0+$. Such $ψ$-variation results extend to classes of processes with values in abstract metric spaces. (No Gaussian or Markovian properties are assumed.) To establish integrability properties of the $ψ$-variation we turn to a large class of Gaussian rough paths (e.g. Brownian motion and Lévy's area viewed as a process in a Lie group) and prove Gaussian integrability properties using Borell's inequality on abstract Wiener spaces. The interest in such results is that they are compatible with rough path theory and yield certain sharp regularity and integrability properties (for iterated Stratonovich integrals, for example) which would be difficult to obtain otherwise. At last, $ψ$-variation is identified as robust regularity property of solutions to (random) rough differential equations beyond semimartingales.

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摘要：

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