马尔可夫跃迁半群的Wasserstein摄动

IF 1.5 Q2 PHYSICS, MATHEMATICAL
Sven Fuhrmann, M. Kupper, M. Nendel
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引用次数: 7

摘要

本文研究了一类转移概率具有非参数不确定性的时间齐次连续马尔可夫过程。不确定性通过考虑Wasserstein距离内邻近的转移概率的扰动来建模。作为一个在越来越细的时间段上的极限,在这个时间段上,不确定性水平按比例缩放,我们得到了一个满足粘性意义上的非线性偏微分方程的凸半群。一个值得注意的观察是,在标准情况下,由非参数不确定性引起的非线性转移算子与与参数漂移不确定性有关的非线性转移算子重合。在生成器的层面上,不确定性反映为一阶导数的凸泛函形式的加性扰动。我们还提供了凸半群相对于参考模型的灵敏度界。结果用L′evy过程的Wasserstein摄动、无限维Ornstein-Uhlenbeck过程、几何布朗运动和Koopman半群来说明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Wasserstein perturbations of Markovian transition semigroups
In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of L\'evy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
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