DISCRETE-TIME MARTINGALES WITH SPATIAL PARAMETERS
Our analysis of a certain stochastic difference equation driven by a martingale k↦M(x,k) that depends on a spatial parameter x∈R d requires some regularity properties of the underlying martingale be satisfied. Because of their independent interest, we present these regularity properties in this article. We study first the continuity and Lipschitz continuity properties under corresponding conditions on the quadratic covariation of the martingale. We follow this with differentiability and integrability properties. Our analysis of the stochastic difference equation requires a discrete-time version of Ito's formula. The discrete-time Ito formula we have derived involves a martingale transform term. The purpose of the final section is to introduce linear and nonlinear martingale transforms and analyze their properties.
Abstract We present a convenient analytical parametrization, in both configuration and momentum spaces, of the deuteron wave-function calculated with the Paris potential.