On Random Sums and Compound Process Models in Financial Mathematics

We study the process composed from random increments occurring at random moments. Resulting compound process is therefore characterized by the intensity of random time points and by the distribution of increments. We propose a model considering the compound process as a two-dimensional random point process and expressing the mutual dependence of both components via the multiplicative hazard regression (Cox) model. The method of estimation of model components is presented and the prediction of process behaviour is studied. The application deals with the process of financial transactions and with the problem of detection of atypical trajectories.