Entrance times of random walks: With applications to pension fund modeling

The purpose of the paper is twofold. First, we consider entrance times of random walks, i.e. the time of first entry to the negative axis. Partition sum formulas are given for entrance time probabilities, the nth derivative of the generating function, and the nth falling factorial entrance time moment. Similar formulas for the characteristic function of the position of the random walk both conditioned on entry and conditioned on no entry are also established. Second, we consider a model for a with-profits collective pension fund. The model has previously been studied by approximate methods, but we give here an essentially complete theoretical description of the model based on the entrance time results. We also conduct a mean-variance analysis for a fund in stationarity. To facilitate the analysis we devise a simple and effective exact simulation algorithm for sampling from the stationary distribution of a regenerative Markov chain.