We consider a system of stochastic differential equations driven by a standard n-dimensional Brownian motion where the drift function b is bounded and the diffusion coefficient is the identity matrix.We define via a duality relation a vector Z (which depends on b) of square integrable stochastic processes which is shown to coincide with the unique strong solution of the previously mentioned equation. We show that the process Z is well defined independently of the boundedness of b and that it makes sense under the more general Novikov condition, which is known to guarantee only the existence of a weak solution. We then prove that under this mild assumption 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. Our framework is also suitable to treat path-dependent stochastic differential equations and an application to the famous Tsirelson equation is presented.

A comparison theorem for stochastic differential equations under the Novikov condition

LANCONELLI, ALBERTO
2014

Abstract

We consider a system of stochastic differential equations driven by a standard n-dimensional Brownian motion where the drift function b is bounded and the diffusion coefficient is the identity matrix.We define via a duality relation a vector Z (which depends on b) of square integrable stochastic processes which is shown to coincide with the unique strong solution of the previously mentioned equation. We show that the process Z is well defined independently of the boundedness of b and that it makes sense under the more general Novikov condition, which is known to guarantee only the existence of a weak solution. We then prove that under this mild assumption 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. Our framework is also suitable to treat path-dependent stochastic differential equations and an application to the famous Tsirelson equation is presented.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/65716
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