I have two questions on a rather basic topic on martingales and brownian motion. Now I can't seem to find a proof of this on the internet nor textbook, since they seem to skip this part of the martingale condition:
Let $X=\left(X_n\right)_{n\in N}$ be a random walk given by: $$X_n=\sum_{i=1}^{n}\varepsilon_i$$ where $\varepsilon_i$ are independent with $P\left(\varepsilon_i=1\right)=p$ and $P\left(\varepsilon_i=-1\right)=1-p$
Show that $S_n=S_0e^{\sigma X_n}$ is a martingale if $p=\frac{1-e^{-\sigma}}{e^\sigma-e^{-\sigma}}$
Now the conditional expectation conditions of martingales is simple, but I have problems showing that: $E|S_n|<\infty$.
And furtheremore, if one has a brownian motion and want to proof it is a martingale, is it possible to use the cauchy-schwarz inequality in the following way? Or is there an easier way to show this?
$$E|B_t|\leq \sqrt{E\left(B_t^2\right)} = \sigma \sqrt(t) < \infty$$
Thanks in advance!
Obviously, $|X_n| \leq n$ and therefore $|S_n| \leq |S_0| e^{\sigma n}$. This implies $\mathbb{E}(|S_n|)<\infty$ (assuming that $S_0 \in L^1$). Concerning your 2nd question: Since $B_t \sim N(0,t)$, we have $$\mathbb{E}(|B_t|) = \frac{1}{\sqrt{2\pi t}} \int_{\mathbb{R}} |x| \exp \left(- \frac{|x|^2}{2t} \right) \, dx.$$ Since $x \mapsto |x| \exp \left(- \frac{|x|^2}{2t} \right)$ is integrable (the exponential decay gobbles up the $|x|$-term), this proves $\mathbb{E}(|B_t|)<\infty$. The same argumentation shows $\mathbb{E}(|B_t|^n)<\infty$ for any $n \geq 1$. However, your proof is also correct.