Let $(W_t)_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}_t$ be the natural filtration. Consider a BSDE $$ dX_t=f(t,X_t)dt+\sigma(t,X_t) dW_t $$ with terminal condition $X_1=x$, where $f(t,\cdot)$ and $\sigma(t, \cdot)$ are $\mathcal{F}_t$-adapted square integrable processes.

My question: is it possible for the BSDE to be well defined if $\sigma(t,X)=0$ for all $t\in [0,1]$ and all $X$? Also, it seems unlikely to me that I can treat such a case as an ODE since reversing time would screw with the progressive measurability of $f$. Am I wrong?