| Abstract: | It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important role in hybrid modeling and computations. Being constructed on various kinds of hybrid grids, that is, time scales, dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important nat-ural processes with hard-to-predict singularities, such as the epidemic growth with un-predictable jump sizes and option market changes with high uncertainties, as com-pared with conventional derivatives. In this article, we shall review the novel new concepts, explore delicate relations between the most frequently used second-order dy-namic derivatives and conventional derivatives. We shall investigate necessary condi-tions for guaranteeing the consistency between the two derivatives. We will show that such a consistency may never exist in general. This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hy-brid grids. Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications. Numerical experiments will also be given. |
