  The Fokker-Planck equations (FPEs) for stochastic systems driven by additive
symmetric $\alpha$-stable noises may not adequately describe the time evolution
for the probability densities of solution paths in some practical applications,
such as hydrodynamical systems, porous media, and composite materials. As a
continuation of previous works on additive case, the FPEs for stochastic
dynamical systems with multiplicative symmetric $\alpha$-stable noises are
derived by the adjoint operator method, which satisfy the nonlocal partial
differential equations. A finite difference method for solving the nonlocal
Fokker-Planck equation (FPE) is constructed, which is shown to satisfy the
discrete maximum principle and to be convergent. Moreover, an example is given
to illustrate this method. For asymmetric case, general finite difference
schemes are proposed, and some analyses of the corresponding numerical schemes
are given. Furthermore, the corresponding result is successfully applied to the
nonlinear filtering problem.
