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

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative $\alpha$-stable noises