Preferential growth: solution and application to modeling stock market

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a probability proportional to the size thereof. We calculate exactly the probability that the size of the ith cluster at time t is k. We applied our model as a background for a microscopic economic model.