论文标题
莱维驱动SDE的解决方案的短期行为
Short-time behavior of solutions to Lévy-driven SDEs
论文作者
论文摘要
我们考虑了$ \ mathrm {d} x_t =σ（x_ {t - }）\ Mathrm {d} l_t} l_t $，$ x_0 = x $的解决方案$ \ mathrm {d} x_t =σ（x__ {t - }）的解决方案$ l =（l_t）_ {t \ geq0} $是向量或矩阵值。尽管Lévy过程的短时间行为几乎是众所周知的，并且可以通过特征性三重态来表征，但该过程的行为$ x $没有完整的表征。使用随机演算的方法，我们得出了从$ \ smash {t^{ - p} \ int_ {0+}^tσ（x__ {t - }）\ \ mathrm {d} l_t} $的限制结果几乎可以肯定地反映了$ t^{ - p} l_t $的行为。将$ t^p $概括为合适的函数$ f：[0，\ infty）\ rightarrow \ mathbb {r} $，然后产生一种工具，以从驾驶lévy流程的行为中得出明确的lil-type结果。
We consider solutions of Lévy-driven stochastic differential equations of the form $\mathrm{d} X_t=σ(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $σ$ is twice continuously differentiable and maximal of linear growth and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix-valued. While the almost sure short-time behavior of Lévy processes is well-known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the process $X$. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the from $\smash{t^{-p}\int_{0+}^tσ(X_{t-})\mathrm{d} L_t}$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely mirrors the behavior of $t^{-p}L_t$. Generalizing $t^p$ to a suitable function $f:[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit LIL-type results for the solution from the behavior of the driving Lévy process.
