I understand that the probability of absorbing eventually is one in a simple random walk with a single absorbing boundary.
However, if we depart from this simple random and consider biased random walks with unequal step sizes, the absorbing probability may not always be one. Specifically, let $S_n = X_1 + \cdots + X_n$, where $X_i = 1$ with probability $p$ and $X_i = -a$ with probability $1-p$, and $a > 1$. If the single boundary is $b > 0$, $p \leq 1/2$, and the process stops once $S_n \geq b$, the probability of the process stopping would not be zero.
Let $ R(a, b, p) $ be the probability of stopping with the parameters $a,b$, and $p$. My question is whether this probability is monotonic in these parameters. Would $R(a, b, p)$ always (weakly) decrease as $a$ increases? Also, would $R(a,b,p)$ (weakly) increase as $p$ increases?
If $E(e^{sX_1})=\varphi(s)$ and $T=\inf \{n; S_n=b\}$ then $M_n=e^{sS_n-n\varphi (s)}$ is a martingale and if $s>0$ we have $$1=E(M_T)=E(e^{sb-T\varphi (s)}1_{T<\infty})+\Pr(T=\infty)$$ which leads to the calculation of the distribution of $T.$