On the Time of the First Achievement of a Level by an Ascending–Descending Process

Abstract

We consider a stochastic process whose trajectories are characterized by alternate linear growth and linear decrease over time intervals of random length, while the process can also maintain its value unchanged for random periods of time between growth and decrease. This process can be considered as a mathematical model of accumulation and consumption of materials, where random periods of time for accumulation, consumption, and interruptions in operation are combined. We study the mean value \( \mathbf {E} N \) of the time of first achievement of a fixed level by trajectories of this process, including finding exact formulas for \( \mathbf {E} N \), producing an upper bound in the form of an inequality, and obtaining the asymptotics of \( \mathbf {E} N \) under the conditions of an infinitely receding level.