Silting and tilting objects in cleft extensions of abelian categories

We establish connections between silting and tilting objects in an abelian category $\mathcal{B}$ and those in a cleft extension $\mathcal{A}$ of $\mathcal{B}$, which provides a method for constructing more silting and tilting objects. Then we apply our results to the cleft extensions of module categories, and characterize silting and tilting modules over $\theta$-extension of rings. Some known results over trivial extension of rings are extended and strengthened.