Brownian bridge with random length and pinning point for modelling of financial information

Abstract

In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. Further results include the derivation of formulae to calculate the conditional expectation of various functions of the random pinning time, the random pinning location, and the future value of the Brownian bridge, given an observation of the underlying process. For the specific case that the pining point has a two-point distribution, we state further properties of the Brownian bridge, e.g. the right continuity of its natural filtration and its semi-martingale decomposition. The newly introduced process can be used to model the flow of information about the behaviour of a gas storage contract holder; concerning whether to inject or withdraw gas at some random future time.

Keywords:

• Bayes theorem
• Brownian bridges
• stopping times
• Markov processes
• semi-martingale decomposition

MSC 2010:

• 60G15
• 60G40
• 60J25
• 60F99

Acknowledgements

I would like to express particular thanks to the anonymous referees for the constructive comments that greatly improved the manuscript. I would also like to express my deep gratitude to Professor Mohamed Erraoui and Professor Astrid Hilbert for their guidance, enthusiastic encouragement and for many stimulating conversations. My grateful thanks are also extended to Professor Monique Jeanblanc for the many helpful comments during the preparation of the paper. I gratefully acknowledge the financial support by an Erasmus+ International Credit Mobility exchange project coordinated by Linnaeus University.

Disclosure statement

No potential conflict of interest was reported by the author(s).