Brown and Levy Steady-State Motions

Abstract

This paper introduces and explores a novel class of Brown and Levy steady-state motions. These motions generalize, respectively, the Ornstein-Uhlenbeck process (OUP) and the Levy-driven OUP. As the OUP and the Levy-driven OUP: the motions are Markov; their dynamics are Langevin; and their steady-state distributions are, respectively, Gauss and Levy. As the Levy-driven OUP: the motions can display the Noah effect (heavy-tailed amplitudal fluctuations); and their memory structure is tunable. And, as Gaussian-stationary processes: the motions can display the Joseph effect (long-ranged temporal dependencies); and their correlation structure is tunable. The motions have two parameters: a critical exponent which determines the Noah effect and the memory structure; and a clock function which determines the Joseph effect and the correlation structure. The novel class is a compelling stochastic model due to the following combination of facts: on the one hand the motions are tractable and amenable to analysis and use; on the other hand the model is versatile and the motions display a host of both regular and anomalous features.

Keywords: joseph effect and long-range dependence; levy-driven processes; markov processes and Langevin dynamics; memory and correlation; noah effect and heavy tails; ornstein-uhlenbeck processes.

Figure 1

The functions $\varphi \left(r\right)$ specified in rows #1 to #5 of Table 2. Left panel: red curve—row #1; blue curve—rows #3 and #4. Right panel: green curve—row #2; purple curve—row #5.

Figure 2

The function ${\varphi }_{\lambda }\left(r\right)$ specified in row #6 of Table 3. Left panel: red curve $\lambda$ = 0.2; blue curve $\lambda$ = 0.4; green curve $\lambda$ = 0.6; purple curve $\lambda$ = 0.8. Right panel: red curve $\lambda$ = 1.2; blue curve $\lambda$ = 1.4; green curve $\lambda$ = 1.6; purple curve $\lambda$ = 1.8.

Figure 3

The function ${\varphi }_{\lambda }\left(r\right)$ specified in row #8 of Table 3. Left panel: red curve $\lambda$ = 0.2; blue curve $\lambda$ = 0.4; green curve $\lambda$ = 0.6; purple curve $\lambda$ = 0.8. Right panel: red curve $\lambda$ = 1.2; blue curve $\lambda$ = 1.4; green curve $\lambda$ = 1.6; purple curve $\lambda$ = 1.8.

Figure 4

The functions ${\varphi }_{\lambda }\left(r\right)$ specified in row #6 (left panel) and in row #8 (right panel) of Table 3, depicted with respect to their parameter $\lambda$. In both panels: red curve r = 0.2; blue curve r = 0.4; green curve r = 0.6; purple curve r = 0.8.