Optimal Time-Dependent Jump Truncation for Time-Singular Lévy Processes

We study optimal jump truncation for the additive time-singular pure-jump model of the form $X_T={\int }_0^T{t}^{-\sigma }d{Z}_t$, where $Z$ is a Lévy process and $\sigma \ge 0$. For a fixed expected jump cost, we minimize the residual small-jump variance over measurable time-dependent cutoffs. Under regularity and tail assumptions on the Lévy measure, we prove that the problem admits an optimal cutoff, unique up to a.e. equality, of the form $r^{\ast }(t)=(c^{\ast }{t}^{\sigma })\wedge 1$. In the symmetric $\alpha$-stable case, we obtain explicit matched-cost comparisons with the classical fixed cutoff and show that within the admissible non-truncated regime the Dynamic Cutting family is strictly better than the classical fixed cutoff whenever $0<\sigma <1\mathrm{/}2$. Finally, for symmetric Lévy measures and cutoffs satisfying the corresponding $L^p$-integrability assumptions, we derive the weak-error bound $W_p(r)\le C_p{\mathcal{E}}^{p\mathrm{/}2}(r)$, $0<p<2$, demonstrating that any cutoff which minimizes the residual variance also minimizes the corresponding upper bound for the weak error.