Wald's Fundamental Identity for Drifted Random Walks

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Martingales / Wald Identities.

Your Task
Let $X_1, X_2, \dots$ be i.i.d. random variables with mean $\mu = \mathbb{E}[X_i]$ and variance $\sigma^2 = \text{Var}(X_i) < \infty$. Let $S_n = \sum_{i=1}^n X_i$. Let $N$ be a stopping time with respect to the filtration generated by $\{X_i\}$ such that $\mathbb{E}[N] < \infty$.
(1) State and prove Wald's First Identity for $\mathbb{E}[S_N]$.
(2) State and prove Wald's Second Identity for $\text{Var}(S_N)$ when $\mu = 0$.

Constraints

• Transitions and increments satisfy the Markov or martingale property with respect to natural filtration.
• Stopping times are almost surely finite.