AI 中文总结
该研究针对时间非齐次跳扩散过程运行最大值的分布性质，运用Malliavin分析等工具，证明两种情形下的绝对连续性，还揭示静默初始区间的原子效应并给出分解。
AI 中文摘要
我们证明了由布朗运动和独立的非截断纯跳Lévy过程驱动的一维时间非齐次Lévy–Itô扩散过程的运行最大值$X^{\u0026}_T=\sup_{0\leq s\leq T}X_s$的绝对连续性。我们在Wiener–Poisson空间上运用Bismut方向Malliavin分析，结合Song–Xie与Nakagawa–Suzuki的运行最大值判据，将问题归约为构造一个容许方向$Θ$，核心是确保对所有$t\in(0,T]$，方向导数$D_ΘX_t$严格为正。我们在两种情形下给出了显式方向：在带时间非齐次系数的一致椭圆情形中，纯布朗扰动给出了$D_ΘX_t$的显式正积分表示，因此$X^{\u0026}_T$无需截断跳分量就存在密度；在布朗退化的纯跳模型中，其带界确定性时变跳权重$κ(t)$可在子区间上为零，我们在“对每个$t\u003e0$有$\int_0^t κ(s)^2\\,ds\u003e0$”的最小时域非退化条件及Lévy测度的无限活性条件下，证明了绝对连续性，一个加权Poisson正性引理是关键的新工具。最后我们证明静默初始区间会产生原子，并推导了显式的原子-密度分解。
英文摘要
We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous Lévy--Itô diffusions driven by a Brownian motion and an independent non-truncated pure-jump Lévy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space together with the running-maximum criteria of Song--Xie and Nakagawa--Suzuki, we reduce the problem to constructing an admissible direction $Θ$. The key is to ensure that the directional derivative $D_ΘX_t$ is strictly positive for all $t\in(0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_ΘX_t$, and hence $X^{\ast}_T$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $κ(t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition $\int_0^t κ(s)^2\,ds>0$ for every $t>0$ and infinite activity of the Lévy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom--density decomposition.
