Fractional calculus and continuous-time finance

By Enrico Scalas, Rudolf Gorenflo, Francesco Mainardi. 2000

869 citations recorded by the source corpus.

In this paper we present a rather general phenomenological theory of tick-by-tick dynamics in financial markets. Many well-known aspects, such as the Levy scaling form, follow as particular cases of the theory. The theory fully takes into account the non-Markovian and non-local character of financial time series. Predictions on the long-time behaviour of the waiting-time probability density are presented. Finally, a general scaling form is given, based on the solution of the fractional diffusion equation.

Subfields: Economics and Econometrics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/42611bb713f50a3f1f05b547c665d064ca8fc6d2

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Citations in the corpus

TEMPERED FRACTIONAL CALCULUS (2015, background): “They are now used to model a wide variety of problems in physics [24, 35,38,50,51,55,65], finance [23,27,37,42,60,61], biology [2,4,22,26,36], and hydrology [1,7,8,15,17,62].”
Lyapunov functions for fractional-order systems in biology: Methods and applications (2020, neutral): “, [15, 17, 18, 25, 30, 53, 65].”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, methodology): “Fractional diffusion equations are used to model problems in physics (see a comprehensive review by Metzler and Klafter [25]), finance [10,16,28,31,30], and hydrology [2,4,5,32,33].”
On the sliding‐mode control of fractional‐order nonlinear uncertain dynamics (2016, background): “Known examples are found in the areas of bioengineering [2], transport phenomena [3, 4], economy [5], mechanics [6], and others [1, 7].”
Fractional order models of leaves (2014, background): “, 2011), ranging from the dynamics of financial markets (Scalas et al., 2000; Duarte et al., 2010), biological systems (Ionescu et al.”
Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives (2005, background): “Important physical applications of CTRW models include diffusion of carriers in amorphous photo-conductors, diffusion in turbulent flow, a percolation model in porous media (α = 1/2) [21], fractal media [26], various biological phenomena [5] and finance [28].”
The application of continuous-time random walks in finance and economics (2006, methodology): “[72] and the rigorous proof was submitted to the proceedings of FDA 04 [73].”
SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations (2009, background): “In the last few decades, fractional Kolmogorov or equivalently fractional Fokker-Planck equations have appeared as an essential tool for the study of dynamics of various complex stochastic processes arising in anomalous diffusion in physics [19,31], finance [8], hydrology [2], and cell biology [22].”
The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics (2004, methodology): “For the waiting time between two transactions power-law statistics were observed (Kim and Yoon 2003, Raberto et al 2002, Scalas et al 2000, Mainardi et al 2000).”
A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions (2018, background): “hydrology [5], chemistry and biochemistry [6], and finance [7].”
A fractional generalization of the Poisson processes (2007, background): “Raberto [20, 14, 9, 18], and, later, by Weiss and co-workers [15].”
Finite difference methods for two-dimensional fractional dispersion equation (2006, neutral): “Fractional diffusion equations have recently been applied to many problems in physics (see an excellent review article by Metzler and Klafter [29]), finance [16,21,33,36,35], and hydrology [2,5,6,37,38].”
VECTOR GRÜNWALD FORMULA FOR FRACTIONAL DERIVATIVES (2004, background): “Introduction Fractional derivatives have been around for centuries [22, 26] but recently they have found new applications in physics [2, 6, 7, 9, 15, 18, 19, 29], hydrology [1, 4, 5, 10, 14, 28], and finance [24, 25, 27].”
Finite difference approximations for two-sided space-fractional partial differential equations (2006, background): “For some applications to physics and hydrology, see [1, 2, 4, 13].”