A comparison theorem for stochastic differential equations under a Novikov-type condition

摘要

We consider a system of stochastic differential equations driven by astandard n-dimensional Brownian motion where the drift coefficient satisfies aNovikov-type condition while the diffusion coefficient is the identity matrix.We define a vector Z of square integrable stochastic processes with thefollowing property: if the filtration of the translated Brownian motionobtained from the Girsanov transform coincides with the one of the drivingnoise then Z coincides with the unique strong solution of the equation;otherwise the process Z solves in the strong sense a related stochasticdifferential inequality. This fact together with an additional assumption willprovide a comparison result similar to well known theorems obtained in thepresence of strong solutions.