In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ using a family of estimators $\xi^\delta_\alpha(T)$, $T>0$, where $\alpha\in(0,2]$ is the Hurst parameter, and $\delta\geq0$ is the step size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_\alpha^0 - \mathcal{H}_\alpha^\delta$, whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case $\alpha\in(0,1]$ and agrees up to logarithmic terms for $\alpha\in(1,2)$. Moreover, we show that all moments of $\xi_\alpha^\delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^{\alpha}\}$, as T becomes large.

Transform-based algorithms have wide applications in applied probability, but rarely provide computable error bounds to guarantee the accuracy. We propose an inversion algorithm for two-sided Laplace transforms with computable error bounds. The algorithm involves a discretization parameter C and a truncation parameter N. By choosing C and N using the error bounds, the algorithm can achieve any desired accuracy. In many cases, the bounds decay exponentially, leading to fast computation. Therefore, the algorithm is especially suitable to provide benchmarks. Examples from financial engineering, including valuation of cumulative distribution functions of asset returns and pricing of European and exotic options, show that our algorithm is fast and easy to implement.