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

摘要

Finite difference methods for degenerate linear and (certain) non-linear elliptic and parabolic equations are analyzed from a probabilistic point of view, and probabilistic methods 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.