In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations. DLRA aims at approximating the solution as a linear combination of a small number of basis vectors with random coefficients (low rank format) with the peculiarity that both the basis vectors and the random coefficients vary in time. While the formulation and properties of DLRA are now well understood for random/parametric equations, the same cannot be said for SDEs and this work aims to fill this gap. We start by rigorously formulating a Dynamically Orthogonal (DO) approximation (an instance of DLRA successfully used in applications) for SDEs, which we then generalize to define a parametrization independent DLRA for SDEs. We show local well-posedness of the DO equations and their equivalence with the DLRA formulation. We also characterize the explosion time of the DO solution by a loss of linear independence of the random coefficients defining the solution expansion and give sufficient conditions for global existence.

翻译：本文建立了随机微分方程中动态低秩近似（DLRA）方法的数学基础。DLRA旨在将解近似为少数基向量与随机系数的线性组合（低秩形式），其特殊之处在于基向量与随机系数均随时间变化。尽管DLRA在随机/参数方程中的表述和性质已得到充分理解，但对随机微分方程（SDE）而言情况并非如此，本文旨在填补这一空白。我们首先为SDE严格定义动态正交（DO）近似（一种在应用中取得成功的DLRA实例），随后将其推广，建立与参数化无关的SDE的DLRA。我们证明了DO方程的局部适定性及其与DLRA表述的等价性。此外，我们通过解展开中定义解的随机系数失去线性独立性来刻画DO解的发散时间，并给出了全局存在的充分条件。