Consider an asymmetric random walk $(X_n)$ in which the initial point is zero ($X_0 = 1$). It increases by $a$ with a probability of 0.3, remains the same with a probability of 0.4, and decreases by $b$ with a probability of 0.3, where $a<b$ and $a,b \in \mathbb{R}$. Also, suppose this process stops once $X_n \geq \lambda$.
To be more specific, say $a=0.3$, $b=0.5$, and $\lambda=1.4$. How can I calculate the probability of this process stopping? That is, what would be the probability of $X_n$ ever getting bigger than $\lambda=1.4$?
I found other problems with asymmetric random walks with equal integer step sizes. However, I am unsure if I can still use the same ways in the simple random walks on integer space.
Also, would it be a completely different problem in an environment where the step sizes are irrational numbers?