  We characterize a stochastic dynamical system with tempered stable noise, by
examining its probability density evolution. This probability density function
satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition
principle that the probability measure-valued solution to this nonlocal
Fokker-Planck equation is equivalent to the martingale solution composed with
the inverse stochastic flow. This result together with a Schauder estimate
leads to the existence and uniqueness of strong solution for the nonlocal
Fokker-Planck equation. Second, we devise a convergent finite difference method
to simulate the probability density function by solving the nonlocal
Fokker-Planck equation. Finally, we apply our aforementioned theoretical and
numerical results to a nonlinear filtering system by simulating a nonlocal
Zakai equation.
