Brownian bridge with random length and pinning point for modelling of financial information
Developed countries are increasingly relying on gas storage to ensure security of supply. In this article we consider an approach to gas storage valuation in which the information about the time at which the holder of a gas storage contract should choose to inject or withdraw gas is modelled using a Brownian bridge that starts at zero and is conditioned to equal a constant x in the time of injection and a constant y in the time of withdrawal. This enables to catch some empirical facts on the behavior of gas storage markets: when the Brownian bridge process is away from the boundaries x and y, the holder of the gas storage contract can be relatively sure that the best decision is to do nothing. However, when the bridge information process absorbs y, the holder of the contract should take the decision of withdrawing gas on the other hand, they should take the decision to inject gas when the process absorbs x. In this sense the Brownian bridge information process leaks information concerning the time at which the holder of a storage contract can choose to inject gas, do nothing, or withdraw gas. The issue of storage valuation is not limited to gas markets, storages also plays a significant, balancing role in, for example, oil markets, soft commodity markets and even electricity. The principle of our approach is applicable to those markets as well. In this paper we define and study the Brownian bridge with random length and pinning point. Its main objectives is to see if the properties of Brownian bridges with deterministic length and pinning point remain valid in case its length and pinning point are random. Amongst other we prove that the bridge fails to be Markovian for pinning points having a law, which is absolutely continuous with respect to the Lebesgue measure.