Coupled nonlinear stochastic differential equations generating arbitrary distributed observable with 1/f noise

Abstract

Abstract. Nonlinear stochastic differential equations provide one of the mathematical models yielding 1/f noise. However, the drawback of a single equation as a source of 1/f noise is the necessity of power-law steady-state probability density of the signal. In this paper we generalize this model and propose a system of two coupled nonlinear stochastic differential equations. The equations are derived from the scaling properties necessary for the achievement of 1/f β noise. The first equation describes the changes of the signal, whereas the second equation represents a fluctuating rate of change. The proposed coupled stochastic differential equations allows us to obtain 1/f β spectrum in a wide range of frequencies together with the almost arbitrary steady-state density of the signal.

Order book model with herd behavior exhibiting long-range memory

“…Though fully integrating the both parts could prove to be a challenging task. There could be a couple of possible approaches: a superstatistical or a coupled SDE approach as used in [62,63] or a coarse-grained approximation of the model as discussed in [64].…”

Understanding the Nature of the Long-Range Memory Phenomenon in Socioeconomic Systems

“…Proposed SDEs ( 8 ), ( 16 ), and ( 21 ) have served as a basis to study heterogeneous diffusion in a non-homogeneous medium [ 90 , 96 , 97 ] and time subordinated processes [ 98 , 99 ] as well as the effects of non-linear variable transformations [ 100 , 101 ].…”

“…In later sections of this paper, we show that the superstatistical approach is not the only approach that allows us to change the observed signal PDF. The coupled SDE approach, proposed in [ 99 ], allows for more flexibility and easier interpretation of how the statistical properties become independent of each other. The general form of the set of coupled SDEs was derived from the scaling properties needed for the realization of noise [ 99 ] …”

“…The coupled SDE approach, proposed in [ 99 ], allows for more flexibility and easier interpretation of how the statistical properties become independent of each other. The general form of the set of coupled SDEs was derived from the scaling properties needed for the realization of noise [ 99 ] …”

“…Deviations from this law could indicate the presence of true long-range memory. Though the method may fail when the stochastic process is not one-dimensional, the study of what happens in the multidimensional case, e.g., as in [ 99 ], is pending. Other challenges may also arise, as discussed in Section 4 .…”