For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, where $X_t$ takes values in some suitable Banach space $\mathcal X$. Namely, $$dX_t = b(X_t)dt + a(X_t)dW_t,$$

where $b, a: \mathcal X\to \mathcal X$ are measurable maps. Assume that there exists a unique weak solution to the above equation. Under which conditions (on $b,a$ and $(\mathcal F_t)$) there exists a unique strong solution?

We distinguish $\mathcal X=\mathbb R^n$ and $\mathcal X$ being a Hilbert space. I especially look for references for the second context.