Short-time behavior of solutions to L\'evy-driven SDEs

We consider solutions of L'evy-driven stochastic differential equations of the form $\mathrm{d}{X}_t=\sigma (X_{t-})\mathrm{d}{L}_t$, $X_0=x$ where the function $\sigma$ is twice continuously differentiable and maximal of linear growth and the driving L'evy process $L=(L_t{)}_{t\ge 0}$ is either vector or matrix-valued. While the almost sure short-time behavior of L'evy 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 $t^{-p}{\int }_{0+}^t\sigma (X_{t-})\mathrm{d}{L}_t$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow 0$ almost surely mirrors the behavior of $t^{-p}{L}_t$. Generalizing $t^p$ to a suitable function $f:[0,\mathrm{\infty })\to \mathbb{R}$ then yields a tool to derive explicit LIL-type results for the solution from the behavior of the driving L'evy process.