The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence
The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence
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DOI：
10.1112/blms.12420
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发表时间：
2019-05
影响因子：
0.9
通讯作者：
H. Boedihardjo;J. Diehl;M. Mezzarobba;H. Ni
中科院分区：
文献类型：
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作者：
H. Boedihardjo;J. Diehl;M. Mezzarobba;H. Ni
期望的签名是粗糙路径上的概率测度的拉普拉斯变换的模拟。这方面的一个关键问题是确定一个一般条件，以确保预期的签名唯一地确定措施。Chevyrev和里昂最近给出了一个充分条件，要求期望签名有一个强上界。虽然上界在许多已知过程中得到了验证，直到确定性时间，但不知道所需的界是否适用于随机时间。事实上，即使是最简单的布朗运动的情况下，平面盘的出口时间是开放的。对于这种特殊情况，我们使用期望签名的适当双曲投影来回答这个问题。该投影满足一个三维线性偏微分方程系统，该系统（令人惊讶地）可以显式求解，并且允许我们证明预期签名的上限不满足。
The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied.