Let $(S_n)_n$ be an assymetric random walk on $\mathbb{Z}$. A player starts with $a$ dollars capital and plays until he either wins $b$ dollars or lost his entire capital $a$.
We know that $$T_{a,b} := \min \{ n \ge 1 \mid S_n = b \text{ or } S_n = -a \}$$ is a stopping time.
Apply the optional stopping theorem to the bounded stopping time $T_{a,b} \land N$ and the exponential martingale $Z^{(u)} := (Z_n^{(u)})_{n \ge 0}$ with $Z_n^{(u)} = \exp(iuS_n)\varphi(u)^{-n}$, where $\varphi (u) = \mathbb{E}(e^{iuX_n}) = 1$. If $n \rightarrow \infty$ this gives us an equation for the ruin probability $\mathbb{P}(S_{T_{a,b}} = -a)$. Determine this probability.
First of all, I do not understand what is meant with $\varphi(u) = 1$; does that mean that we can drop the $\varphi$ in $Z^{(u)}$? Could you please explain this to me?