If I have a process $X(t)\in \mathbb{R}$ with the SDE $$dX(t) = F(X(t))dt + \sigma dW(t)$$
where $W(t)$ is Brownian motion, using Ito's rule, am I correct to say that the SDE from a linear transformation $$X'(t) = X(t) + a$$ where $a$ is constant would be $$dX'(t) = dX(t) + a$$ ?
By linearity, we have $$ dX'(t) = dX(t) + da = dX(t) = F(X(t))\,dt + \sigma\,dW_t = F(X'(t) - a)\,dt + \sigma\,dW_t.$$ Note that there is no need to for Ito's formula.