Consider a random walk $X_t$ in continuous time in $\mathbb{Z}$. Suppose that the walker expects an exponential time of rate $\alpha$ to jump to the right and an exponential time of rate $\beta$ to jump to the left.
Suppose $X_0 = 0$ and that $\alpha >\beta$. I want to show that
$$\mathbb{P}\left(X_t < \frac{(\alpha -\beta)}{2}t\right)\leq Ce^{-Kt}$$
where $K$ and $C$ are constants.
My only question is:
During any time interval $t$, the displacement of the walk can be limited by a random variable $Y_t \sim \text{Poisson} ((\alpha-\beta) t)$?
I think so, but I can't justify it. If the answer is yes, I think I can solve it.