Optimal prediction problems and the last zero of spectrally negative Lévy processes

In recent years the study of Levy processes has received considerable attention in the literature. In particular, spectrally negative Levy processes have applications in insurance, finance, reliability and risk theory. For instance, in risk theory, the capital of an insurance company over time is studied. A key quantity of interest is the moment of ruin, which is classically defined as the first passage time below zero. Consider instead the situation where after the moment of ruin the company may have funds to endure a negative capital for some time. In that case, the last time below zero becomes an important quantity to be studied. An important characteristic of last passage times is that they are random times which are not stopping times. This means that the information available at any time is not enough to determine its value and only with the whole realisation of the process that it can be determined. On the other hand, stopping times are random times such that its realisation can be derived only with the past information. Suppose that at any time period there is a need to know the value of a last passage time for some appropriate actions to be taken. It is then clear that an alternative to this problem is to approximate the last passage time with a stopping time such that they are close in some sense. In this work, we consider the optimal prediction to the last zero of a spectrally negative Levy process. This is equivalent to find a stopping time that minimises its distance with respect to the last time the process goes below zero. In order to fulfil this goal, we also study the last zero before at any fixed time and its dynamics as a process. Moreover, having in mind some applications in the insurance sector, we study the joint distribution of the number of downcrossings by jump and the local time before an exponential time