The strong predictable representation property in initially enlarged filtrations under the density hypothesis

TL;DR

This paper demonstrates that the strong predictable representation property can be transferred to initially enlarged filtrations under the density hypothesis, broadening martingale representation results without requiring law equivalence.

Contribution

It generalizes existing martingale representation results to initial enlargements under the density hypothesis without assuming law equivalence.

Findings

Representation depends on the density process at zero.

Results apply to hedging with insider information.

Generalizes classical martingale representation theorems.

Abstract

We study the strong predictable representation property in filtrations initially enlarged with a random variable L. We prove that the strong predictable representation property can always be transferred to the enlarged filtration as long as the classical density hypothesis of Jacod (1985) holds. This generalizes the existing martingale representation results and does not rely on the equivalence between the conditional and the unconditional laws of L. Depending on the behavior of the density process at zero, different forms of martingale representation are established. The results are illustrated in the context of hedging contingent claims under insider information.