时间奇异Lévy过程的最优时间依赖跳跃截断
Optimal Time-Dependent Jump Truncation for Time-Singular Lévy Processes
AI总结：
本文针对时间奇异Lévy过程，提出并证明了最优时间依赖跳跃截断的存在性与唯一性，并在对称稳定情形下优于经典固定截断，同时给出弱误差界。
AI中文摘要：
我们研究了形如$X_T=\int_0^T t^{-\sigma} dZ_t$的加性时间奇异纯跳模型的最优跳跃截断问题，其中$Z$是一个Lévy过程且$\sigma\ge 0$。对于固定的期望跳跃成本，我们在可测的时间依赖截断函数上最小化残差小跳跃方差。在Lévy测度的正则性和尾部假设下，我们证明了该问题存在一个最优截断，且在几乎处处相等的意义下唯一，其形式为$r^\ast(t)=(c^\ast t^\sigma)\wedge 1$。在对称$\alpha$-稳定情形下，我们获得了与经典固定截断的显式匹配成本比较，并表明在允许的非截断区域内，当$0<\sigma<1/2$时，动态切割族严格优于经典固定截断。最后，对于满足相应$L^p$-可积性假设的对称Lévy测度和截断，我们推导出弱误差界$W_p(r)\le C_p \mathcal{E}^{p/2}(r)$，$0<p<2$，证明任何最小化残差方差的截断也最小化相应的弱误差上界。
英文摘要：
We study optimal jump truncation for the additive time-singular pure-jump model of the form $X_T=\int_0^T t^{-σ} dZ_t$, where $Z$ is a Lévy process and $σ\ge 0$. 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 $r^\ast(t)=(c^\ast t^σ)\wedge 1$. In the symmetric $α$-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 $0<σ<1/2$. Finally, for symmetric Lévy measures and cutoffs satisfying the corresponding $L^p$-integrability assumptions, we derive the weak-error bound $W_p(r)\le C_p \mathcal{E}^{p/2}(r)$, $0<p<2$, demonstrating that any cutoff which minimizes the residual variance also minimizes the corresponding upper bound for the weak error.
