SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.

[1]  D. V. Widder,et al.  The Laplace Transform , 1943 .

[2]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[3]  E. Montroll.  Random walks on lattices , 1969 .

[4]  G. Weiss,et al.  Expected Number of Distinct Sites Visited by a Random Walk with an Infinite Variance , 1970 .

[5]  J. Jacod.  Calcul stochastique et problèmes de martingales , 1979 .

[6]  L. Hörmander,et al.  Differential operators with constant coefficients , 1983 .

[7]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[8]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[9]  P. Protter.  Stochastic integration and differential equations , 1990 .

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

[11]  W. Webb,et al.  Automated detection and tracking of individual and clustered cell surface low density lipoprotein receptor molecules. , 1994, Biophysical journal.

[12]  Niels Jacob,et al.  Pseudo-Differential Operators and Markov Processes , 1996 .

[13]  Denis R. Bell.  Degenerate Stochastic Differential Equations and Hypoellipticity , 1996 .

[14]  Boris Rubin,et al.  Fractional Integrals and Potentials , 1996 .

[15]  K. Jacobson,et al.  Single-particle tracking: applications to membrane dynamics. , 1997, Annual review of biophysics and biomolecular structure.

[16]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[17]  E. Bazhlekova.  Subordination principle for fractional evolution equations , 1999 .

[18]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

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

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

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

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

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

[24]  O. Barndorff-Nielsen,et al.  Lévy processes : theory and applications , 2001 .

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

[26]  Jinqiao Duan,et al.  Fractional Fokker-Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises , 1999, math/0409486.

[27]  Francesco Mainardi,et al.  Random walk models approximating symmetric space-fractional diffusion processes , 2012, 1210.6589.

[28]  N. Jacob,et al.  Pseudo Differential Operators and Markov Processes: Volume II: Generators and Their Potential Theory , 2002 .

[29]  Raisa E. Feldman,et al.  Limit Distributions for Sums of Independent Random Vectors , 2002 .

[30]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[31]  D. Stroock.  Markov processes from K. Itô's perspective , 2003 .

[32]  M. Czubak,et al.  PSEUDODIFFERENTIAL OPERATORS , 2020, Introduction to Partial Differential Equations.

[33]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[34]  R. Situ.  Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

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

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

[37]  Rudolf Gorenflo,et al.  Cauchy and Nonlocal Multi-Point Problems for Distributed Order Pseudo-Differential Equations, Part One , 2005 .

[38]  Stanly Steinberg,et al.  Random walk models associated with distributed fractional order differential equations , 2006 .

[39]  Marcin Magdziarz,et al.  Competition between subdiffusion and Lévy flights: a Monte Carlo approach. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Francesco Mainardi,et al.  Continuous-time random walk and parametric subordination in fractional diffusion , 2007 .

[41]  Chjan C. Lim,et al.  The Monte Carlo Approach , 2007 .

[42]  Francesco Mainardi,et al.  Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects , 2007, 0705.0797.

[43]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[44]  J. Klafter,et al.  Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: the case of a time-dependent force. , 2008, Physical review letters.

[45]  Mark M. Meerschaert,et al.  Triangular array limits for continuous time random walks , 2008 .

[46]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[47]  Mark M. Meerschaert,et al.  Erratum to Triangular array limits for continuous time random walks [Stochastic Process. Appl. 118 (9) (2008) 1606-1633] , 2010 .

[48]  Kei Kobayashi.  Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations , 2009, 0906.5385.

[49]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[50]  Integral and Differential Equations of Fractional Order Integral and Differential Equations of Fractional Order , .