A Survey of Some Applications of Probability and Stochastic Control Theory to Finite Difference Methods for Degenerate Elliptic and Parabolic Equations

Finite difference methods for degenerate linear and (certain) nonlinear elliptic and parabolic equations are analyzed from a probabilistic point of view, and probabilistic methods are used to show convergence to the correct weak or strong sense solution. The techniques generalize results in numerical analysis, and the probabilistic approach allows some additional physical insight to be brought to bear on the problem of selecting suitable approximations and methods of solution. The equations which are discussed all have probabilistic interpretations. In each case, the finite difference equations (with appropriately chosen finite difference approximations) are also equations which are satisfied by certain functionals of certain Markov chains, whose transition functions are given directly by the coefficients in the finite difference equation. Suitable continuous time interpolations of the chains converge to various diffusion processes which are connected with the original partial differential operators, and ...