Strange facts about the marginal distributions of processes based on the Ornstein-Uhlenbeck process

ABSTRACT The Ornstein-Uhlenbeck process is particularly useful for modeling stochastic processes in financial applications. Furthermore, functions of such a process can be used to model random volatility of other processes, resulting in more flexible models for financial risk variables. The distribution of such a financial risk variable is of particular interest in value-at-risk analysis. As we know, the far quantiles of the distribution function provide information on the level of capital reserves required to accommodate extreme stress situations. This paper presents an approximation for the distribution function that works surprisingly well, even for the far tails of the distribution of levels as small as 10.7. While theoretically unjustified and strange, it may still be very useful in practice.