  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.
