In this paper, we set the mathematical foundations of the Dynamical Low Rank Approximation (DLRA) method for high-dimensional 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：本文为高维随机微分方程的动态低秩逼近（DLRA）方法建立数学基础。DLRA旨在将解近似表示为少量具有随机系数的基向量的线性组合（低秩格式），其独特之处在于基向量和随机系数均随时间变化。尽管DLRA的公式化表达和性质在随机/参数方程中已得到充分理解，但在随机微分方程（SDE）领域仍存在空白，本文旨在填补这一空白。我们首先严格构建了SDE的动态正交（DO）逼近（一种在应用中取得成功的DLRA实例），进而将其推广为SDE的参数化无关DLRA。我们证明了DO方程的局部适定性及其与DLRA公式的等价性。同时，我们通过定义解展开的随机系数线性相关性丧失来表征DO解的爆炸时间，并给出了全局存在的充分条件。