2007/03/23 by D. Sornette, Didier Sornette, В. Ф. Писаренко +1
Economics, Econometrics and Finance · Physics and Astronomy · #Complex Systems and Time Series Analysis #Financial Risk and Volatility Modeling #Stochastic processes and financial applications #physics.data-an #physics.soc-ph #q-fin.ST
paper · pdf · doi:10.1016/j.physd.2007.08.020
35 pages, 5 figures, 8 tables
arxiv created 2007/03/23 · openalex publication_date 2007/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it has no linear predictability (zero two-point correlation) but a certain nonlinear predictability (non-zero three-point correlation). It can thus be considered as a paradigm for testing the existence of a possible nonlinear predictbility in a given time series. We present a rather exhaustive study of the process, including its ability to produce fat-tailed distribution from Gaussian innovations, the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super-exponential sensitivity of the inversion of the innovations in the presence of a large impluse. Our study emphasizes the conditions under which a degree of predictability can be achieved and describes a number of different attempts, which overall illuminates the properties of the process. In conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic process exhibits remarkably rich and complex behavior, which makes it a serious candidate for the modeling of financial times series and of other complex systems.