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 $L=(L_t)_{ t\geq0}$ 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 $\smash{ t^{ -p}\int_{ 0+}^t\sigma(X_{ t-})\mathrm{ d} L_t}$ to show that the behavior of the quantity $t^{ -p}(X_t-X_0)$ for $t\downarrow0$ 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 L\'evy process.
Categories: math.PR