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In this paper, we introduce an extension of a Brownian bridge with a randomlength by including uncertainty also in the pinning level of the bridge. Themain result of this work is that unlike for deterministic pinning point, thebridge process fails to be Markovian if the pining point distribution isabsolutely continuous with respect to the Lebesgue measure. Further resultsinclude the derivation of formulae to calculate the conditional expectation ofvarious functions of the random pinning time, the random pinning location, andthe future value of the Brownian bridge, given an observation of the underlyingprocess. For the specific case that the pining point has a two-pointdistribution, we state further properties of the Brownian bridge, e.g., theright continuity of its natural filtration and its semi-martingaledecomposition. The newly introduced process can be used to model the flow ofinformation about the behaviour of a gas storage contract holder; concerningwhether to inject or withdraw gas at some random future time.

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