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.

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.

PS : More precisely, $X$ is said to be a strong solution if it is continuous, $(\mathcal F_t)-$adapted and almost surely $$X_t=X_0+\int_0^t b(X_s)ds + \int_0^t a(X_s)dW_s,\quad \forall t>0.$$ A weak solution corresponds to a continuous process $Y=(Y_t)$ and a filtered probability space $\big(\Sigma, \mathcal G, (\mathcal G_t),\mathbb Q\big)$ which supports a Brownian motion $B$ such that $\mathbb Q-$almost surely $$Y_t=Y_0+\int_0^t b(Y_s)ds + \int_0^t a(Y_s)dB_s,\quad \forall t>0.$$