2013•arXiv (Cornell University)Open access
A comparison theorem for stochastic differential equations under a\n Novikov-type condition
Alberto Lanconelli
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Abstract
We consider a system of stochastic differential equations driven by a\nstandard n-dimensional Brownian motion where the drift coefficient satisfies a\nNovikov-type condition while the diffusion coefficient is the identity matrix.\nWe define a vector Z of square integrable stochastic processes with the\nfollowing property: if the filtration of the translated Brownian motion\nobtained from the Girsanov transform coincides with the one of the driving\nnoise then Z coincides with the unique strong solution of the equation;\notherwise the process Z solves in the strong sense a related stochastic\ndifferential inequality. This fact together with an additional assumption will\nprovide a comparison result similar to well known theorems obtained in the\npresence of strong solutions.\n
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