Are all semimartingale Gaussian processes weak solutions to a diffusion SDE?

Question

Let $(X_t)$ be a Gaussian semimartingale process with mean $\mu(t)$ and covariance function $U(s,t)$. Is it true that we can always find functions $\lambda$ and $\sigma$ such that $(X_t)$ is solution in law to the SDE

$$dY_t= \lambda(t,Y_t) dt + \sigma(t,Y_t) dB_t, $$

i.e $X_t = Y_t$ in law, where $(B_t)$ is a standard Brownian motion ?