${\mathsf{F}}= \left\{ f(x) = x x^{\dagger} \right\}$ be a singleton set, consisting only one test function (which maps to ${\mathbb{C}}^{n\times n}$). Assume that the time evolution equation $\partial_t \mu_t  = {\mathcal{A}}_{t} \mu_t$ is the Fokker-Planck equation of a SDE of the form $$\label{eqn::sde}
{\,\mathrm{d}}X_t = a(X_t, t){\,\mathrm{d}}t + \sum_{j=1}^{N} b_j(X_t, t) {\,\mathrm{d}}W_j,$$ where $X_t\in {{\mathbb{C}}^{{n}}}$, $a$ and $b_j$ are functions ${{\mathbb{C}}^{{n}}}\times {[0,T]}\rightarrow {{\mathbb{C}}^{{n}}}$ and $W_j$ are independent real-valued standard Brownian motions. With some additional assumptions [[and restrictions]{}]{}, one could obtain the low-rank dynamics given by [Eq. ]{} (or equivalently [Eq. ]{}). The details are given in [Theorem \[thm::fx=xx\]]{}, which is one of the main results in this paper.

As already mentioned above, the stochastic dynamical low-rank approximation is also motivated by developing efficient methods for the Lindblad quantum master equations. In that context, the deterministic low-rank approximation has been studied by Le Bris and Rouchon to find low-rank approximation of [Lindblad]{} equation [@LeBris13]. In the subsequent work [@LeBris15], Le Bris, Rouchon and Roussel also introduced an unraveling scheme for the low-rank quantum master equation obtained in [@LeBris13]. The unraveling of Lindblad equations and its connection with the low-rank approximation will be discussed in Section \[sec::discuss::unravel\]. In particular, as another main 