4. Simulation Algorithms and Simulation Studies

In this chapter, we present the basic ideas for the simulation of a stationary Gaussian process from which we deduce the simulation of a fBm and the simulation of processes driven by a fBm. Our approach is based on Durbin-Levinson’s algorithm. Since the process formed by the first order increments of a fBm is a stationary Gaussian one, we first simulate the increments of the process and then, by a simple “integration”, we obtain a trajectory of the fBm. For models defined by differential equations, first an observation of the fBm is generated and then, it is transformed according to differential equation. Simulating these processes, we can explore the statistical properties of the estimators defined in the previous chapter from an empirical point of view. We study the distribution of the estimators of H and of σ. Special attention is devoted to the construction of a confidence interval for H. Some simulation results concern the estimation of the parameters of a pure fBm, some others are for the parameters of models that are excited by a fBm. To generate the uniform deviates, we recommend the use of linear congruential generator given in Langlands et al. (Bull Am Math Soc (New Ser) 30(1):1–61, 1994) and for the normal deviates, and we recommend the use of Algorithm M described in Knuth (The art of computer programming. Vol. 2. Seminumerical algorithms, Addison-Wesley series in computer science and information processing, 2nd edn. Addison-Wesley, Reading, 1981). It is a very fast generator. Pascal programs are given in Chap. 8.