Level Crossings of an Oscillating Marked Random Walk

This paper deals with a class of real-valued random-walk processes, observed over random epochs of time, that forms a delayed renewal process. The present model does not restrict this class to a merely monotone random walk, which is easier to analyze and find explicit form functional. The objective is to find the first passage of the process exiting a rectangular set and registering the value of the process at this time, thus generalizing past models where either the observed process was monotone or the first passage time reduced to the moment of the first drop. The joint transformation of the named random characteristics of the process are derived in a closed form. The paper concludes with examples, including numerical examples, demonstrating the use of the results as well as practical applications to finance.