Bridges with Random Length and Pinning Point for Modelling the Financial Information
Authors:   ,   Linnaeus University, Louriki Mohammed 
Published: October 2022

The impact of the information concerning an event of interest occurring at a future random time is the main topic of this work. The event can massively influence financial markets and the problem of modelling the information on the time at which it occurs is of crucial importance in financial modelling. To give an explicit description of the flow of information concerning this time we define a bridge process starting from zero and conditioned to be equal to a preset level when the event occurs. In this sense the bridge process leaks information concerning the event before it occurs. However, in order to compile some facts on this event it is very important to study the properties of the bridge process. Specifically, the Markov property, the right continuity of its natural filtration and its semi-martingale decomposition. Therefore, introducing a wider class of information processes and studying their properties is the focus of this thesis. From a theoretical point of view, we generalize the concept of bridge process in the sense that the length and the pinning point will both be random. The main objective is to check if certain basic properties of bridges with deterministic length and pinning point remain valid also for bridges with random length and pinning point. In this thesis, we provide several contributions to the field of Markov bridges. In the first chapter, the theoretical foundations of this work are given, the definition and the properties of the conditional expectation, are recalled before introducing the Bayes formula and the notion of a Markov process. In the second chapter, we introduce and study Bridges with random length associated with Gamma process. Here the bridge process is defined by using the randomization approach since Gamma bridges have explicit pathwise representations. However, it is not the case for a general Lévy process. A natural question arises: How to define a Lévy process pinned at random time? the third chapter aims to provide an answer to this question. We study the mathematical properties of the resulting process, and we present certain examples. In the fourth chapter, we introduce an extension of a Brownian bridge with a random pinning time to one, for which the pinning point is also given by a random variable. The main result of this work is that, unlike for deterministic pinning point, the bridge fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. For the specific case where the pining point has a two-point distribution, we state further properties for the Brownian bridge, namely 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. The fifth chapter aims to extend the information-based asset-pricing framework of Brody-Hughston-Macrina to a more general set-up. We include a wider class of models for market information. In addition, we consider a model in which a credit risky asset is modelled in the presence of a default time. Instead of using only a Brownian bridge as a noise, we consider another important type of noise. We model the flow of information about a default bond with given random repayments at a predetermined maturity date by the so-called market information process, this process is the sum of two terms, namely the cash flow induced by the repayment at maturity and a noise, a stochastic process set up by adding a Brownian bridge with length equal to the maturity date to a drift (linear in time) multiplied by a time changed Lévy process. In this model the information concerning the random cash-flow is modelled explicitly but the default time of the company is not, since the payment is contractually set to take place at maturity only. We suggest a model, in which the cash flow and the time of bankruptcy are both modelled. From a theoretical point of view, this chapter deals with conditions, which allow to keep the properties of bridges with deterministic length and pinning point, when replacing the pinning point in the Brownian bridge by a process. For this purpose, we first study the basic mathematical properties of a bridge between two Brownian motions.