Small-time Expansions for Local Jump-diffusion Models with Infinite Jump Activity

We consider a Markov process X which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W . Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of Z outside any neighborhood of the origin, we obtain a small-time secondorder polynomial expansion for the tail distribution and the transition density of the process X. Our method of proof combines a recent approach for deriving the analog small-time expansions for a Lévy process based on a regularization technique of the Lévy density near the origin with classical tail and density estimates for jump-diffusion processes with small jumps based on Malliavin calculus. As an application, the leading term for out-of-the-money option prices in short maturity under a local jump-diffusion model is also derived.