Triangular array limits for continuous time random walks

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Levy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Levy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space-time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed.

[1]  M. Shlesinger,et al.  Stochastic pathway to anomalous diffusion. , 1987, Physical review. A, General physics.

[2]  D. Benson,et al.  Multidimensional advection and fractional dispersion. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  V. Zolotarev,et al.  Chance and Stability, Stable Distributions and Their Applications , 1999 .

[4]  Rina Schumer,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[5]  Enrico Scalas,et al.  Coupled continuous time random walks in finance , 2006 .

[6]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[7]  R. Nagel,et al.  One-parameter Semigroups of Positive Operators , 1986 .

[8]  M. Shlesinger,et al.  Random walks with infinite spatial and temporal moments , 1982 .

[9]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[10]  Mark M. Meerschaert,et al.  Inhomogeneous Fractional Diffusion Equations , 2005 .

[11]  Francesco Mainardi,et al.  Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk , 2007, 0709.3990.

[12]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[13]  Mark M. Meerschaert,et al.  Limit theorems for continuous-time random walks with infinite mean waiting times , 2004, Journal of Applied Probability.

[14]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[15]  Peter Richmond,et al.  Waiting time distributions in financial markets , 2002 .

[16]  P. Spreij.  Probability and Measure , 1996 .

[17]  J. Mason,et al.  Operator-limit distributions in probability theory , 1993 .

[18]  M. Meerschaert,et al.  Ultrafast subordinators and their hitting times , 2006 .

[19]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[20]  M. Meerschaert,et al.  Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice , 2001 .

[21]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[22]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

[23]  Tosio Kato,et al.  ON SUBORDINATED HOLOMORPHIC SEMIGROUPS , 1991 .

[24]  M. Meerschaert,et al.  Space-time fractional derivative operators , 2005 .

[25]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

[26]  Harry Kesten,et al.  Hitting probabilities of single points for processes with stationary independent increments , 1969 .

[27]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[28]  P. Embrechts,et al.  High Risk Scenarios and Extremes , 2007 .

[29]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[30]  D. Benson,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[31]  R. Wolpert.  Lévy Processes , 2000 .

[32]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[33]  M. Meerschaert,et al.  Portfolio Modeling with Heavy Tailed Random Vectors , 2003 .

[34]  David A. Benson,et al.  Advection and dispersion in time and space , 2002 .

[35]  Alexander I. Saichev,et al.  Fractional kinetic equations: solutions and applications. , 1997, Chaos.

[36]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[37]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[38]  A. Chaves,et al.  A fractional diffusion equation to describe Lévy flights , 1998 .

[39]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[40]  Yuji Kasahara,et al.  Limit Theorems of Occupation Times for Markov Processes , 1976 .

[41]  L. Bondesson,et al.  Infinite divisibility of random variables and their integer parts , 1996 .

[42]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[43]  Enrico Scalas.  Five Years of Continuous-time Random Walks in Econophysics , 2005 .

[44]  M. Meerschaert,et al.  Stochastic model for ultraslow diffusion , 2006 .

[45]  D. Benson,et al.  Operator Lévy motion and multiscaling anomalous diffusion. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[47]  J. Williamson.  Harmonic Analysis on Semigroups , 1967 .

[48]  Mark M. Meerschaert,et al.  Limit theorem for continuous-time random walks with two time scales , 2004, Journal of Applied Probability.

[49]  Takashi Komatsu,et al.  Pseudo-differential operators and Markov processes , 1984 .

[50]  P. Phillips,et al.  Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets , 1994 .

[51]  Rina Schumer,et al.  Multiscaling fractional advection‐dispersion equations and their solutions , 2003 .

[52]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[53]  Dennis W. Jansen,et al.  On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective , 1989 .

[54]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[55]  Mark M. Meerschaert,et al.  STOCHASTIC SOLUTIONS FOR FRACTIONAL CAUCHY PROBLEMS , 2003 .

[56]  Enrico Scalas,et al.  Speculative option valuation and the fractional diffusion equation , 2005 .

[57]  George M. Zaslavsky,et al.  Fractional kinetic equation for Hamiltonian chaos , 1994 .

[58]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[59]  I. Podlubny.  Fractional differential equations , 1998 .

[60]  S. Rachev.  Handbook of heavy tailed distributions in finance , 2003 .

[61]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[62]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .