Let $\xi_i$ be i.i.d $L^2(\mathbb{P})$ random variables in a probability space $(\Omega, \mathcal{F},\mathbb{P})$, with $\mathbb{E}[\xi_1] = 0, \mathbb{E}[\xi_1^2]=1$, and let $S_n = \sum_{i=1}^n \xi_i$.
Define the one-sided hitting time $\tau:$ $$\tau := \inf \{ k \geq 0 : S_k < 0 \}.$$ My question is: can we find an expression for $\mathbb{E}[S_\tau]$ and $\mathbb{E}[S_\tau^2]$ and/or determine when these are finite?
I am aware of expressions which make use of optional stopping under some assumptions on the distribution of $\tau$; for example, if $T$ is a two-sided hitting time, since $(S_{n \wedge T})$ is a bounded martingale, we have $\mathbb{E}[S_T] = 0$ and $\mathbb{E}[S_T^2] = \mathbb{E}[T]$.
My issue is that many of these common analysis tools don't seem to work; since $\tau$ is only one-sided, $\mathbb{E}[\tau] = \infty$ which violates a finite expectation assumption made by many of these tools.
In the case of the simple symmetric random walk, for example, where $\mathbb{P}(\xi_1 = -1) = \mathbb{P}(\xi_1 = 1) = 1/2$, it is quite immediate that $\mathbb{E}[S_\tau] = -1$ and $\mathbb{E}[S_\tau^2] = 1$. This exploits the discrete state space for $(S_n)$. But can anything be said more generally? I am particularly intrested in the case where $\xi_1 \sim \mathcal{N}(0,1)$.