A Comparison Theorem for Stochastic Differential Equations Under the Novikov Condition

作者：Lanconelli Alberto^*

来源：Potential Analysis, 2014, 41(4): 1065-1077.

DOI：10.1007/s11118-014-9413-x

• Stochastic differential equations
• Girsanov theorem
• Novikov condition
• Convex envelope
• Comparison theorems

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.

• 出版日期2014-11