Vasicek beyond the normal
Professor Ragnar Norberg, Department of Statistics, London School of Economics


A general Ornstein-Uhlenbeck (OU) process is obtained upon replacing the Brownian motion appearing in the defining stochastic differential equation with a general Levy process. Certain properties of the Brownian ancestor are distribution-free and carry over to the general OU process. Explicit expressions are obtainable for expected values of a number of functionals of interest also in the general case. Special attention is paid here to gamma and Poisson driven OU processes. The Brownian, Poisson, and gamma versions of the OU process are compared in various respects and, in particular, their aptitude to describe stochastic interest rates is discussed in view of some standard issues in financial and actuarial mathematics; prices of zero-coupon bonds, moments of present values, and probability distributions of present values of perpetuities. The problem of possible negative interest rates finds its resolution in the general set-up by taking the driving Levy process to be non-decreasing (a subordinator).