Modern Stochastics: Theory and Applications

The notion of the transportation distance on the set of the Lévy measures on $\mathbb{R}$ is introduced. A Lévy-type process with a given symbol (state dependent analogue of the characteristic triplet) is proved to be well defined as a strong solution to a stochastic differential equation (SDE) under the assumption of Lipschitz continuity of the Lévy kernel in the symbol w.r.t. the state space variable in the transportation distance. As examples, we construct Gamma-type process and α-stable like process as strong solutions to SDEs.