Existence and Uniqueness for Stochastic Dynamic Equations
The theory of stochastic dynamic equations extends and unifies the theories of stochastic difference and differential equations. In this paper, we prove the existence and uniqueness of the strong solution of a certain class of stochastic dynamic equations. As a principal tool in the proof, we define and develop the properties of stochastic dynamic integrals with respect to a Brownian motion indexed by a time scale. Finally we illustrate our theory with the examples of stochastic exponential and geometric Brownian motion.
We describe preliminary results from an effort to quantify the uncertainties in parton distribution functions and the resulting uncertainties in predicted physical quantities. The production cross section of the $W$ boson is given as a first example. Constraints due to the full data sets of the CTEQ global analysis are used in this study. Two complementary approaches, based on the Hessian and the Lagrange multiplier method respectively, are outlined. We discuss issues on obtaining meaningful uncertainty estimates that include the effect of correlated experimental systematic uncertainties and illustrate them with detailed calculations using one set of precision DIS data.