Problems in applied mathematics: Analysis of Bermudan options, and selected topics in the analysis of quantum field-theoretical perturbative series

The development of high speed computers adds several new tools to the applied mathematics tool box. Computers may be used to do high speed numerical computations (for example for quadrature and for root finding). More recently symbolic manipulation software such as Maple has been developed to enable computers to assist with analytic calculation as well.; In this thesis I present two problems, drawn from the very different areas of financial options pricing and elementary partical physics. I have tackled these problems using a mixture of computer algebra and numerical analysis techniques.; Financial Mathematics describes how to value and optimally exercise a security known as a Perpetual Bermudan option (see [1] and [2]). This problem is formulated as a coupled system of an integral equation involving a paramer theta for a function V(S) and a nonlinear algebraic equation relating this parameter to the function V( S). Here V(S) is the value of the option. S is stock price and theta is the optimal exercise boundary. I solve this system both numerically and via an iterative sequence of analytic approximations.; In my thesis, I also address several series issues within the context of perturbative quantum field theory. I first examined the renormalization-group summation (RGSigma) of leading and nonleading logarithms within the QCD expression for the Higgs → two gluons (gg) decay rate to demonstrate how RGSigma reduces the dependence of this decay rate on the unphysical renormalization mass parameter mu. I then addressed the relationship between interaction coupling constants in different renormalization schemes, expanding on previous work of Elias pertinent to supersymmetric gluodynamics. I found the perturbative relationship between the DRED-scheme and NSVZ-scheme coupling constants for the more general case of supersymmetric QCD with arbitrary numbers for color and flavour. For ordinary (non-supersymmetric) QCD, I also examined the relationship between coupling constants in two different renormalization schemes, the MS scheme and the 't Hooft's scheme.; Finally, I expanded on Elias's Pade-approximant procedures for determining whether or not the DRED beta-function in supersymmetric gluodynamics has a pole or an infrared stable fixed point. I found [2/2], [1/3] and [3/1] approximants in which the (presently-unknown) next-order coefficient R 4 in the DRED beta function is incorporated as an arbitrary parameter. For all three approximants, a positive approximant-zero is always preceded by a positive approximant-pole, regardless of the value of R 4, consistent with the existence of a DRED beta-function pole.