Application of the Discrete Moment Problem for Numerical Integration and Solution of a Special Type of Moment Problems

OF THE DISSERTATION Application of the discrete moment problem for numerical integration and solution of a special type of moment problems by Mariya Naumova Dissertation Director: András Prékopa We present a brief survey of some of the basic results related to the classical continuous moment problems (CMP) and the recently developed discrete moment problems (DMP), clarifying their relationship. We also introduce a new numerical integration method, based on DMP and termed Discrete Moment Method (DMM), that can be used for univariate piecewise higher order convex functions. This means that the interval where the function is defined can be subdivided into non-overlapping subintervals such that in each interval all divided differences of given orders, do not change the sign. The new method uses piecewise polynomial lower and upper bounds on the function, created in connection with suitable dual feasible bases in the univariate discrete moment problem and the integral of the function is approximated by tight lower and upper bounds on them. Numerical illustrations are presented for the cases of the normal, exponential, gamma and Weibull probability density functions. We show how a similar approach can be applied for solving the problems of a special structure, namely the discrete conditional moment problems and present the corresponding numerical results. Finally, we present novel applications to valuations of financial instruments.