发表机构
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明粗糙随机微分方程系统在无界系数下的弱存在性，并建立 Yamada--Watanabe 定理：弱存在性与路径唯一性蕴含强解及联合分布唯一性。
AI 中文摘要
对于 Friz--Hocquet--Lê 意义下的粗糙随机微分方程系统，在固定确定性粗糙驱动下，我们证明两个结果。第一个结果是具有无界系数的弱存在性。Itô 数据被假设为连续且线性增长，不要求 Hölder 或 Lipschitz 连续性，也不要求椭圆性。粗糙向量场是一个具有有界导数的线性增长受控对，满足范数-曲率衰减或对角性且具有坐标方向曲率衰减；在对角情形下，额外要求扩散矩阵的坐标方向上有上界。已有结果要求 Itô 数据和粗糙场有界。我们去除该有界性，作为交换，要求粗糙场满足曲率假设、其导数有界，以及更窄的指标配置。结论在两方面较弱：定义解的有界性是在均值意义下而非在概率空间上一致地施加，且解的矩的阶数不超过初始分布所携带的阶数。第二个结果是粗糙方程的 Yamada--Watanabe 蕴含关系：弱存在性连同路径唯一性，推出强解和联合分布唯一性。这里的解不是由路径恒等式定义，而是由对余项的两种有界性定义，其中一种是有条件的，并且对滤流的任意扩大不一定保持该有界性。然而，Yamada--Watanabe 构造所产生的扩大是浸入，沿着浸入，两种有界性均不变地传递。
英文摘要
For systems of rough stochastic differential equations in the sense of Friz--Hocquet--Lê, with a fixed deterministic rough driver, we prove two results. The first is weak existence with unbounded coefficients. The Itô data is assumed continuous and of linear growth, with no Hölder or Lipschitz continuity and no ellipticity. The rough vector field is a controlled pair of linear growth with bounded derivatives, subject either to a norm-curvature decay or to diagonality with coordinatewise curvature decay; in the diagonal case a coordinatewise upper bound on the diffusion matrix is added. The existing results ask the Itô data and the rough field to be bounded. We remove that boundedness, and ask in exchange for a curvature hypothesis on the rough field, bounds on its derivatives, and a narrower index configuration. The conclusion is weaker on two counts: the bounds that define a solution are imposed in the mean rather than uniformly on the probability space, and the solution's moments stop at the order the initial law carries. The second is the Yamada--Watanabe implication for rough equations: weak existence, together with pathwise uniqueness, yields a strong solution and joint uniqueness in law. A solution here is not defined by a pathwise identity but by two bounds on a remainder, one of them conditional, and an arbitrary enlargement of the filtration need not preserve that one. The enlargements the Yamada--Watanabe construction produces are immersions, however, and along an immersion both bounds transfer unchanged.
CommentsThis is an extended version of the manuscript, containing additional remarks
