Clark-Ocone Formula for computing SDE

Question

If we know $X_t$ is an $\mathbb{R}^d$-valued process solving the SDE $$ dY_t = a(t,Y_t)dt +b(t,Y_t)dW_t, $$ then does the Clark-Ocone Formula and the fundamental theorem of Malliavin Calculus imply that: \begin{align} X_T = & E[X_T] + \int_0^{T}E[D_tX_t|\mathfrak{F}_t]dW_t\\ \therefore & \int_0^{T} E[D_tX_t|\mathfrak{F}_t]dW_t = X_T - E[X_T] \\ \therefore & E[D_tX_t|\mathfrak{F}_t] = DX_T - DE[X_T] \\ \therefore & E[D_tX_t|\mathfrak{F}_t] = DX_T - E[DX_T] \end{align} In particular if we set $Y_t = DX_t$ then can we conclude that \begin{align} E[Y_t|\mathfrak{F}_t] & = Y_T - DE[\int_0^TY_tdW_t]? \end{align}