Quantization of stochastic processes with applications on Euler-Maruyama schemes

Abstract: This thesis investigates so called quantizations of continuous random variables. A quantization of a continuous random variables is a discrete random variable that approximates the continuous one by having similar properties, often by sharing weak convergence. A measure on how well the quantization approximates the continuous variable is introduced and methods for generating quantizations are developed. The connection between quantization of the normal distribution and the Hermite polynomials is discussed and methods for generating optimal quantizations are suggested. An observed connection between finite differences and quantization is examined and is identified to just be a special case. Finally a method of turning an Euler-Maruyama scheme into a deterministic problem by quantization is presented along with a convergence analysis. The method is reminiscent of standard tree methods, that can be used for e.g. option pricing, but has a few advantages such that there are no requirements on the grid point placements, it can even change for each time step.