Stochastic processes with proportional increments and the last-arrival problem

The notion of stochastic processes with proportional increments is introduced. This notion is of general interest as indicated by its relationship with several stochastic processes, as counting processes, Lévy processes, and others, as well as martingales related with these processes. The focus of this article is on the motivation to introduce processes with proportional increments, as instigated by certain characteristics of stopping problems under weak information. We also study some general properties of such processes. These lead to new insights into the mechanism and characterization of Pascal processes. This again will motivate the introduction of more general f-increment processes as well as the analysis of their link with martingales. As a major application we solve the no-information version of the last-arrival problem which was an open problem. Further applications deal with the impact of proportional increments on modelling investment problems, with a new proof of the 1/e-law of best choice, and with other optimal stopping problems.