Approximations for the Distribution of Perpetuities with Small Discount RatesJ. Blanchet and P. W. Glynn Technical Report. General perpetuities (i.e. random variables of the form D = \int_0^\infty \exp{\Gamma(t-)}d\Lambda(t), also known as infinite horizon discounted rewards) play an important role in several application settings (e.g. insurance, finance and time series analysis). Our focus is on developing approximations for the distribution of D that are asymptotically valid when the “accumulated short rate process” or “accumulated force of interest” (represented by \Gamma) is small. In this paper, we emphasize approximations that are good around the “center” of the distribution of D. We provide: 1) characterizations, in terms of solutions to certain linear equations, for the distribution of D when \Gamma and \Lambda are driven by Markov processes; 2) General sufficient conditions under which weak convergence results can be derived for D; 3) Edgeworth expansions for the distribution of D in the iid case and the case in which \Lambda is a Levy process and \dot{\Gamma}(t) is a function of a Markov process, this last setting is of particular interest in applications to life and non-life insurance problems. |