Particle tracking using stochastic differential equation driven by pure jump Levy processes

Stochastic diffusion driven by a pure jump Levy process is an important core concept for particle tracking methods used in stochastic hydrology and for tempered anomalous diffusion models used in (Geo) Physics. In this work we discuss the jump Levy diffusion in terms of stochastic differential equations (SDEs). We examine the existence and uniqueness of solutions of stochastic differential equations of the form dYt=aYt dt+bYtdXt where {Xt} is a pure jump Levy process. Further, we rigorously derive the infinitesimal generator and the backward equation. It can be shown that the infinitesimal generator is a pseudo differential operator. Using this form with the backward equation, we derive the forward equation by an involution type technique. The forward equation associated with the transition density of the solution process is analogous to the governing advection-dispersion equation used in particle tracking of heavy tailed flows and tempered anomalous diffusion models.