Levy process simulation by stochastic step functions

We study a Monte Carlo algorithm for simulation of probability distributions based on stochastic step functions, and compare to the traditional Metropolis/Hastings method. Unlike the latter, the step function algorithm can produce an uncorrelated Markov chain. We apply this method to the simulation of Levy processes, for which simulation of uncorrelated jumps are essential. We perform numerical tests consisting of simulation from probability distributions, as well as simulation of Levy process paths. The Levy processes include a jump-diffusion with a Gaussian Levy measure, as well as jump-diffusion approximations of the infinite activity NIG and CGMY processes. To increase efficiency of the step function method, and to decrease correlations in the Metropolis/Hastings method, we introduce adaptive hybrid algorithms which employ uncorrelated draws from an adaptive discrete distribution defined on a space of subdivisions of the Levy measure space. The nonzero correlations in Metropolis/Hastings simulations result in heavy tails for the Levy process distribution at any fixed time. This problem is eliminated in the step function approach. In each case of the Gaussian, NIG and CGMY processes, we compare the distribution at t=1 with exact results and note the superiority of the step function approach.Comment: 20 pages, 18 figure

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionaly, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of $\beta$.Comment: 20 pages, 6 figure

Simplicial gauge theory on spacetime

We define a discrete gauge-invariant Yang-Mills-Higgs action on spacetime simplicial meshes. The formulation is a generalization of classical lattice gauge theory, and we prove consistency of the action in the sense of approximation theory. In addition, we perform numerical tests of convergence towards exact continuum results for several choices of gauge fields in pure gauge theory.Comment: 18 pages, 2 figure

The Ekpyrotic Universe

The ekpyrotic universe is a brane cosmology theory with an alternative explanation of the big bang as a collision between two hyperplanes. We first introduce the reader to brane objects in the context of string theory, and then give an introduction to the standard hot big bang model and general properties of brane cosmology models. Thereafter follows a study of the ekpyrotic universe ending with a presentation and discussion of some new numerical results