An example of martingale representation in progressive enlargement by an accessible random time

When a filtration represented by a martingale M is enlarged by the observation of a martingale N it can happens that the cross variation process  [M,N]  enters in the martingale representation of the enlarged filtration.... [ view full abstract ]

When a filtration represented by a martingale M is enlarged by the observation of a martingale N it can happens that the cross variation process  [M,N]  enters in the martingale representation of the enlarged filtration. This fact on one hand influences the multiplicity of the enlarged filtration and on the other hand it is linked to the behavior of the sharp brackets of the martingales. Here we illustrate these arguments presenting  an elementary example of martingale representation in the context of progressive enlargement by an accessible random time.