Stochastic representation of subdiffusion processes with time-dependent drift

Stochastic Processes and their Applications

Volumn 119, Issue 10, 2009, Pages 3238-3252

  Magdziarz, Marcin ^a  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

stable distribution; First passage time; Fractional Fokker Planck equation; Inverse subordinator; Subdiffusion

Indexed keywords

APPROXIMATE SOLUTION; APPROXIMATION SCHEME; FIRST-PASSAGE TIME; FRACTIONAL FOKKER-PLANCK EQUATION; INVERSE SUBORDINATOR; LINEAR TIME; MATHEMATICAL DESCRIPTIONS; MEAN SQUARE DISPLACEMENT; POWER-LAW; STATISTICAL PHYSICS; STOCHASTIC INTEGRAL; STOCHASTIC PROCESS; STOCHASTIC REPRESENTATIONS; SUBDIFFUSION; SUBDIFFUSION PROCESS; SUBORDINATORS; TIME-DEPENDENT; UNIFORMLY CONVERGENT;

MONTE CARLO METHODS; PROBABILITY DENSITY FUNCTION; RECONNAISSANCE AIRCRAFT; TIMING JITTER;

FOKKER PLANCK EQUATION;

EID: 69749117030     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2009.05.006     Document Type: Article

References (28)

1. Bertoin J. Lévy Processes (1996), Cambridge University Press, Cambridge
2. Bertoin J., van Harn K., and Steutel F.W. Renewal theory and level passage by subordinators. Statist. Probab. Lett. 45 (1999) 65-69
3. Daley D.J., and Vere-Jones D. An Introduction to the Theory of Point Processes. 2nd edn. vol. 1 (2003), Springer, New York
4. Feller W. An Introduction to Probability Theory and Its Applications. 2nd edn. vol. 2 (1971), Wiley, New York
5. Heinsalu E., Patriarca M., Goychuk I., and Hänggi P. Use and abuse of a fractional Fokker-Planck dynamics for time-dependent driving. Phys. Rev. Lett. 99 (2007) 120602
6. Janicki A., and Weron A. Simulation and Chaotic Behaviour of α-Stable Stochastic Processes (1994), Marcel Dekker, New York
7. Jurlewicz A. Limit theorems for randomly coarse grained continuous-time random walks. Dissertationes Math. 431 (2005) 1-45
8. Kingman J.F.C. On doubly stochastic Poisson processes. Math. Proc. Cambridge Philos. Soc. 60 (1964) 923-930
9. Kloeden P.E., and Platen E. Numerical Solution of Stochastic Differential Equations (1995), Springer, Berlin
10. Lagerås A.N. A renewal-process-type expression for the moments of inverse subordinators. J. Appl. Probab. 42 (2005) 1134-1144
11. M. Magdziarz, Black-Scholes formula in subdiffusive regime, 2009 (submitted for publication)
12. Magdziarz M., and Weron A. Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation. Phys. Rev. E 75 (2007) 056702
13. Magdziarz M., Weron A., and Weron K. Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation. Phys. Rev. E 75 (2007) 016708
14. Meerschaert M.M., and Scheffler H.P. Limit Theorems for Sums of Independent Random Vectors (2001), Wiley, New York
15. Meerschaert M.M., Benson D.A., Scheffler H.P., and Baeumer B. Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65 (2002) 041103
16. Meerschaert M.M., and Scheffler H.P. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004) 623-638
17. Meerschaert M.M., and Scheffler H.P. Stochastic model for ultraslow diffusion. Stochastic Process. Appl. 116 (2006) 1215-1235
18. Metzler R., Barkai E., and Klafter J. Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82 (1999) 3563-3567
19. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
20. Mikosch T., and Resnick S. Activity rates with very heavy tails. Stochastic Process. Appl. 116 (2006) 131-1556
21. Montroll E.W., and Weiss G.H. Random walks on lattices II. J. Math. Phys. (1965) 167-181
22. Piryatinska A., Saichev A.I., and Woyczynski W.A. Models of anomalous diffusion: the subdiffusive case. Physica A 349 (2005) 375-420
23. Samko S.G., Kilbas A.A., and Maritchev D.I. Integrals and Derivatives of the Fractional Order and Some of Their Applications (1993), Gordon and Breach, Amsterdam
24. Sato K.-I. Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press, Cambridge
25. Sokolov I.M., and Klafter J. Field-induced dispersion in subdiffusion. Phys. Rev. Lett. 97 (2006) 140602
26. van Harn K., and Steutel F.W. Stationarity of delayed subordinators. Stoch. Models 17 (2001) 369-374
27. Weron A., Magdziarz M., and Weron K. Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation. Phys. Rev. E 77 (2008) 036704
28. Winkel M. Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Probab. 42 (2005) 138-152