  We present a detailed analysis and implementation of a splitting strategy to
identify simultaneously the local-volatility surface and the jump-size
distribution from quoted European prices. The underlying model consists of a
jump-diffusion driven asset with time and price dependent volatility. Our
approach uses a forward Dupire-type partial-integro-differential equations for
the option prices to produce a parameter-to-solution map. The ill-posed inverse
problem for such map is then solved by means of a Tikhonov-type convex
regularization. The proofs of convergence and stability of the algorithm are
provided together with numerical examples that substantiate the robustness of
the method both for synthetic and real data.
