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A Comparison Theorem for Stochastic Differential Equations Under the Novikov Condition

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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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  1. Dipartimento di Matematica, Università degli Studi di Bari, via Orabona 4, 70125, Bari, Italy

    Alberto Lanconelli

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  1. Alberto Lanconelli
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Correspondence to Alberto Lanconelli.

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Lanconelli, A. A Comparison Theorem for Stochastic Differential Equations Under the Novikov Condition. Potential Anal 41, 1065–1077 (2014). https://doi.org/10.1007/s11118-014-9413-x

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  • DOI: https://doi.org/10.1007/s11118-014-9413-x

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