Lower bound for stochastic process

Question

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the equilibrium distribution exists, can we conclude that $\lim_{t\rightarrow \infty} E[X_t | Y_t \in A] \geq Z(A) E[T_A]/2$ ?