SHORT-TIME BEHAVIOR OF SOLUTIONS TO LEVY-DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS

We consider solutions of Levy-driven stochastic differential equations of the form dX(t)=sigma(Xt-)dL(t),X-0=x, where the function sigma is twice continuously differentiable and thedriving Levy processL=(L-t)(t >= 0)is either vector or matrix valued. While the almostsure short-time behavior of Levy processes is well known and can be characterized interms of the characteristic triplet, there is no complete characterization of the behaviorof the solution X. Using methods from stochastic calculus, we derive limiting resultsfor stochastic integrals of the form t(-p) integral(t)(0+)sigma(Xt-)dL(t) to show that the behavior of the quantity t(-p)(X-t-X-0)for t down arrow 0 almost surely reflects the behavior oft(-p)L(t). Generalizing t(p)to a suitable function f:[0,infinity)-> R then yields a tool to derive explicit law of the iterated logarithm type results for the solution from the behavior of the driving Levy process