Stochastic Integrals and SDE Driven by Nonlinear Lévy Noise

Abstract

We develop the theory of SDE driven by nonlinear Levy noise, aiming at applications to Markov processes. It is shown that a conditionally positive integro-differential operator (of the Levy–Khintchine type) with variable coefficients (diffusion, drift and Levy measure) depending Lipschitz continuously on its parameters generates a Markov semigroup, where the measures are metricized by the Wasserstein–Kantorovich metrics W p . The analysis of SDE driven by nonlinear Levy noise was initiated by the author in (“Kolokoltsov, Probability Theory Related Fields, DOI: 10.1007/s00440-010-0293-8, 2009”) (inspired partially by “Carmona and Nualart, Nonlinear Stochastic Integrators, Equations and Flows, Stochatic Monographs, v. 6, Gordon and Breach, 1990”), see also (“Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, Monograph. To appear in CUP, 2010”). Here, we suggest an alternative (seemingly more straightforward) approach based on the path-wise interpretation of these integrals as nonhomogeneous Levy processes. Moreover, we are working with more general W p -distances rather than with W 2.