Let $S_N=\sum_{i=1}^N x_i$.
For each $x_i$, we have $x_i=1$ with probability $p$ and $x_i=-\lambda$ with probability $1-p$ with $\lambda>0$. Assume i.i.d.
Consider the stopping time $T$,
$T=inf\{N;S_N>0\ \text{or}\ S_N<K\}$ where $K<0$.
How to calculate $P(S_T>0)$ and $P(S_T<K)$?
Thanks