Simulation of Brown–Resnick processes

Abstract

Brown–Resnick processes form a flexible class of stationary max-stable processes based on Gaussian random fields. With regard to applications, fast and accurate simulation of these processes is an important issue. In fact, Brown–Resnick processes that are generated by a dissipative flow do not allow for good finite approximations using the definition of the processes. On large intervals we get either huge approximation errors or very long operating times. Looking for solutions of this problem, we give different representations of the Brown–Resnick processes—including random shifting and a mixed moving maxima representation—and derive various kinds of finite approximations that can be used for simulation purposes. Furthermore, error bounds are calculated in the case of the original process by Brown and Resnick (J Appl Probab 14(4):732–739, 1977). For a one-parametric class of Brown– Resnick processes based on the fractional Brownian motion we perform a simulation study and compare the results of the different methods concerning their approximation quality. The presented simulation techniques turn out to provide remarkable improvements.