Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit

By Rudolf Gorenflo, Francesco Mainardi, Enrico Scalas, Marco Raberto. 2001

284 citations recorded by the source corpus.

A proper transition to the so-called diffusion or hydrodynamic limit is discussed for continuous time random walks. It turns out that the probability density function for the limit process obeys a fractional diffusion equation. The relevance of these results for financial applications is briefly discussed.

Subfields: Economics and Econometrics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/b06dc6ba7d0cafcb02b0d61334cde8c6e37bb7a7

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Citations in the corpus

Finite di$erence approximations for fractional advection-dispersion &ow equations (2004, methodology): “In this paper we develop practical numerical methods to solve one dimensional fractional advection–dispersion equations with variable coe4cients on a 5nite domain.”
A fractional generalization of the Poisson processes (2007, background): “Raberto [20, 14, 9, 18], and, later, by Weiss and co-workers [15].”
Waiting-times and returns in high-frequency financial data: an empirical study (2002, background): “Wp, 89.90.+n Stochastic processes; Continuous-time random walk; Statistical finance; Econophysics; Autocorrelation function”
Continuous-time random walk and parametric subordination in fractional diffusion (2007, unknown): “…to the variable x) of the particle being in point x at instant t , see e.g. [14, 16, 30, 46, 47, 48], p(x, t) = δ(x)Ψ(t) + ∫ t 0 φ(t− t′) [∫ +∞ −∞ w(x− x′) p(x′, t′) dx′ ] dt′ , (3.1) in which the survival function Ψ(t) = ∫ ∞ t φ(t′) dt′ (3.2) denotes the probability that at instant t the…”
Economic interpretation of fractional derivatives (2017, methodology): “Recently, non-integer order derivatives have been used to describe the financial processes [6, 7, 8, 9, 10] and the economic processes with memory and [14, 15, 16, 17, 18].”
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].”
TEMPERED FRACTIONAL CALCULUS (2015, background): “…✩ Farzad Sabzikar a, Mark M. Meerschaert a,∗, Jinghua Chen b a Department of Statistics and Probability, Michigan State University, East Lansing, MI 48823, United States b School of Sciences, Jimei University, Xiamen, Fujian, 361021, China a r t i c l e i n f o a b s t r a c t Article history:…”
Finite difference methods for two-dimensional fractional dispersion equation (2006, methodology): “In this paper we discuss a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain.”