The Feynman-Kac theorem states that for an Ito-process of the form $$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$$ there is a measurable function $g$ such that $$g_t(t,x) + g_x(t, x) \mu(t,x) + \frac{1}{2} g_{xx}(t,x)\sigma(t,x)^2 = 0$$ with an appropriate boundary condition $h$: $g(T,x) = h(x)$. We also know that $g(t,x)$ is of the form $$g(t,x)=\mathbb{E}\left[h(X_T) \big| X_t=x\right].$$
This means that I can price an option with payoff function $h(x)$ at $T$ by solving the differential equation without regard to the stochastic process.
Is there an intuitive explanation how it is possible to model the stochastic behaviour of the Ito-process by a differential equation, even though the differential equation does not have a stochastic component?