In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). 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：本文为随机微分方程(SDEs)的动态低秩逼近(DLRA)方法奠定了数学基础。DLRA旨在将解近似为少量基向量与随机系数的线性组合（低秩格式），其特点在于基向量与随机系数均随时间变化。尽管DLRA在随机/参数方程中的表述与性质已得到充分理解，但在SDEs领域尚未形成系统理论，本研究旨在填补这一空白。我们首先为SDEs严格构建了动态正交(DO)逼近（一种已在应用中成功使用的DLRA实例），进而将其推广为一种参数化无关的SDEs-DLRA框架。我们证明了DO方程的局部适定性及其与DLRA表述的等价性。同时，通过解展开式中随机系数线性独立性的丧失来刻画DO解的爆破时间，并给出了全局解存在的充分条件。