VaR and ES via Peaks-Over-Threshold

You are modeling extreme daily losses using Extreme Value Theory. Suppose daily losses $L_t$ above a high threshold $u$ follow a Generalized Pareto Distribution (GPD) with shape parameter $\xi$ and scale parameter $\sigma$: $P(L_t - u > y \mid L_t > u) = \begin{cases} \left(1 + \xi \frac{y}{\sigma}\right)^{-1/\xi} & \xi \neq 0 \\ e^{-y/\sigma} & \xi = 0 \end{cases}$ for $y > 0$ (with $y < -\sigma / \xi$ when $\xi < 0$). You observe $T$ total daily losses, of which $N_u$ exceed the threshold $u$. 1. Derive closed-form expressions for VaR at confidence level $\alpha$ (close to 1) and Expected Shortfall (ES) at level $\alpha$, in terms of $\xi$, $\sigma$, $u$, $N_u$, and $T$. 2. Explain how you would estimate $\xi$ and $\sigma$ from the $N_u$ exceedances, and how those estimates plug into your VaR and ES formulas. 3. State the regularity conditions on the threshold $u$ and the shape parameter $\xi$ that are required for these formulas to be valid.