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

## TL;DR

This paper extends the Brownian bridge model by incorporating randomness in both the pinning time and level, revealing non-Markovian properties and providing formulas for conditional expectations relevant to financial modeling.

## Contribution

It introduces a novel Brownian bridge with random length and pinning point, analyzing its properties and potential applications in modeling gas storage decisions.

## Key findings

- The process is non-Markovian with absolutely continuous pinning point distribution.
- Formulas for conditional expectations of the process are derived.
- Properties like right continuity and semi-martingale decomposition are established for specific cases.

## 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.