Enlarged filtrations and indistinguishable processes

Abstract

We study modification properties of stochastic processes under different probability measures in an initially enlarged filtration setup. For this purpose, we consider several pure-jump Lévy processes under two equivalent probability measures and derive the associated martingale compensators with respect to different enlarged filtrations. As our main result, we prove that the obtained martingale processes under different probability measures in our enlarged filtration approach are indistinguishable. In addition, we provide a condition under which the pure-jump result can be carried over to the Brownian motion case. In this context, we show how indistinguishable Brownian motions under different probability measures can be constructed in an enlarged filtration framework. We finally apply our theoretical results to precipitation derivatives pricing under weather forecasts.

Keywords:

• Enlarged filtration
• indistinguishable processes
• Lévy process
• Poisson random measure
• martingale compensator
• Radon–Nikodym density
• Doléans–Dade exponential
• stochastic differential equation
• weather derivative
• precipitation swap

Mathematics Subject Classification (2010):

• Primary: 60G44
• 60G51
• 60H10
• 60G57
• Secondary: 91B44
• 91B70
• 91G10
• 93E20

JEL Classification:

• C02
• C53
• D52
• D80
• G11
• G13
• G14

Acknowledgments

The author would like to thank Jean Jacod (Paris) for interesting e-mail conversation on some aspects of this article.