Convergence of jump-diffusion models to the Black-Scholes model

We consider a jump-diffusion model for asset price which is described as a solution of a linear stochastic differential equation driven by a Lévy process. Such a market is incomplete and there are many equivalent martingale measures. We price a contingent claim with respect to the minimal martingale measure and construct a hedging strategy for the contingent claim in the locally risk-minimizing sense. We study the problem of convergence of option prices jointly with the costs from the locally risk-minimizing strategies when the jump-diffusion models converge to the Black-Scholes model.