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

TL;DR

This paper establishes a comparison theorem for stochastic differential equations with a Novikov-type drift condition, linking solutions to stochastic inequalities and strong solutions under certain filtration conditions.

Contribution

It introduces a novel comparison result for SDEs with Novikov-type conditions, extending classical theorems to cases involving stochastic inequalities and filtration considerations.

Findings

Z coincides with the strong solution when filtrations match

Z solves a stochastic differential inequality otherwise

Comparison theorem similar to classical results is established

Abstract

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