Bridge-Type Processes Associated with L\'evy Processes and Their Decompositions

We study a class of stochastic bridge-type processes whose terminal pinning value is random and is generated by an underlying stochastic process.

In contrast with classical bridges, the construction depends not only on the terminal value of the driving process but also on its evolution before the terminal time.

This dynamic stochastic input breaks some of the classical Markovian structure and requires a separate analysis of the semimartingale decomposition in the natural filtration.

We first analyze the Brownian case, which provides a Gaussian reference model, and show that the corresponding process is not Markovian in its natural filtration.

We then extend the study to non-Gaussian Lévy drivers, focusing on finite variation jump processes and on Lévy processes with both Gaussian and jump components.

In each case, we study the Doob--Meyer decomposition in the natural filtration.