Optimal Time-Dependent Jump Truncation for Time-Singular L\'evy Processes

Abstract

We study optimal jump truncation for the additive time-singular pure-jump model of the form $X_T=\int_0^T t^{-\sigma} dZ_t$, where $Z$ is a L\'evy 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 L\'evy 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 $0<\sigma<1/2$. Finally, for symmetric L\'evy measures and cutoffs satisfying the corresponding $L^p$-integrability assumptions, we derive the weak-error bound $W_p(r)\le C_p \mathcal{E}^{p/2}(r)$, $0<p<2$, demonstrating that any cutoff which minimizes the residual variance also minimizes the corresponding upper bound for the weak error.