Waiting time distributions in financial markets

Sabatelli, Lorenzo; Keating, Shane R.; Dudley, Joe; Richmond, Peter

doi:10.1140/epjb/e20020151

Cited by 162 publications

(96 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Fractional differential equations have recently made a renaissance, mainly driven by scientists in Physics [5,10,13,31,33,35,37,45,51], Finance [22,42,44,46], and Hydrology [3,[6][7][8]47,48], as they can be derived via stochastic limit theorems and hence provide robust and parsimonious models predicting power-law tails. This is because fractional derivatives derive from sums of random movements with power law probability tails [39,47], for which the usual central limit theorem is replaced by its heavy tail analogue [20,34].…”

Section: Application To Fractional Advection-dispersion Equationsmentioning

confidence: 99%

Unbounded functional calculus for bounded groups with applications

2009

View full text Add to dashboard Cite

Abstract. In this paper, we develop the unbounded extension of the Hille-Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy-Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines.

show abstract

Section: Application To Fractional Advection-dispersion Equationsmentioning

confidence: 99%

Unbounded functional calculus for bounded groups with applications

2009

View full text Add to dashboard Cite

show abstract

“…Smethurst and Williams (2001) find = 1.4 in the waiting times for a doctor appointment. Sabatelli et al (2002) report = 0.4 for 19th century Irish stock prices, and = 1.87 for the modern Japanese Yen currency market. Benson et al (2007) discuss the important implications of random waiting times between observations, for extreme values in geophysics.…”

Section: Introductionmentioning

confidence: 98%

Extremal limit theorems for observations separated by random power law waiting times

Meerschaert

Stoev

2009

Journal of Statistical Planning and Inference

View full text Add to dashboard Cite

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.

show abstract

“…Fractional derivatives are used in physics to model anomalous diffusion, where a cloud of particles spreads farther and faster than the classical diffusion model predicts (Barkai et al, 2000;Blumen et al, 1989;Bouchaud and Georges, 1990;Klafter et al, 1987;Meerschaert et al, 1999Meerschaert et al, , 2002aMeerschaert et al, , 2002bKlafter, 2000, 2004;Saichev and Zaslavsky, 1997;Zaslavsky, 1994). Fractional derivative models have also been proposed in finance (Gorenflo et al, 2001;Raberto et al, 2002;Sabatelli et al, 2002;Scalas et al, 2000Scalas et al, , 2005Scalas, 2006) to model price volatility, and in hydrology , Benson et al, 2000a, 2000bSchumer et al, 2001Schumer et al, , 2003 to model fast spreading of pollutants. In each case, the fractional derivative term substitutes for the classical second derivative term, resulting in a wider and faster spread.…”

Section: Introductionmentioning

confidence: 99%

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

2007

View full text Add to dashboard Cite

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. IntegroDifference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1 < α ≤ 2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.

show abstract

Waiting time distributions in financial markets

Cited by 162 publications

References 1 publication

Unbounded functional calculus for bounded groups with applications

Unbounded functional calculus for bounded groups with applications

Extremal limit theorems for observations separated by random power law waiting times

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Contact Info

Product

Resources

About