A stochastic process cumulating random increments at random moments is studied. We model it as a two-dimensional random point process and study advantages of such an approach. First, a rather general model allowing for the dependence of both components mutually as well as on covariates is formulated, then the case where the increments depend on time is analyzed with the aid of the multiplicative hazard regression model. Special attention is devoted to the problem of prediction of process behaviour. To this end, certain results on risk processes and crossing probabilities are recalled and utilized. The application deals with the process of financial transactions and the problem of detection of outlied trajectories.