Application of the fractional problem of the calculus of variations and the fractional path integral approach to stochastic modeling

Fizika A

Volumn 19, Issue 1, 2010, Pages 11-30

  El Nabulsi, R A ^a  

^a Jeju National University   (South Korea)

Author keywords

Financial modeling; Fractional path integral; Options pricing; Stochastic models

Indexed keywords

EID: 79957498614     PISSN: 13300008     EISSN: 13339125     Source Type: Journal    
DOI: None     Document Type: Article

References (79)

1. K. B. Oldham, J. Spanier, The Fractional Falculus, Acad. Press, New York, London (1974).
2. S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (1993).
3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons Inc., New York (1993).
4. I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equa-tions, to Methods of their Solution and some of their Applications, Academic Press, New York-London (1999).
5. R. Hilfer, Editor, Applications of Fractional Calculus in Physics, Word Scientific, New Jersey, London, Hong Kong (2000).
6. T. E. Duncan, Y. Hu and B. Pasic-Duncan, Stochastic Calculus for Fractional Brownian Motion I: Theory, SIAM Journal on Control and Optimization 38 (2000) 582.
7. Y. Yu and B. Oksendal, Fractional White Noise Calculus and Applications to Finance, Preprint, University of Oslo (2003).
8. B. B. Mandelbrot and J. W. Ness, SIAM Review 10 (1968) 422.
9. V. Kobolev and E. Romanov, Fractional Langevin Equation to Describe Anomalous Diffusion, chaodyn/9908001.
10. F. Mainardi and P. Pironi, Extracta Mathematicae 10, 1 (1996) 140.
11. E. Lutz, Phys. Rev. E 64 (2001) 051106.
12. S. C. Kou and X. Sunney Xie, Phys. Rev. Lett. 93, 18 (2004) 180603-1.
13. W. R. Parke, Rev. Econ. Stat. 81, 4 (1999) 632.
14. F. Lillo and J. D. Farmer, Studies in Nonlin. Dynamics and Econometrics 8, 3 (2004) Article 1.
15. J. T. Barkoulas, W. C. Labys and J. I. Onochie, Financial Review 34, 1 (2005) 91.
16. K. H. Liow, Long-Term Memory in Volatility: Some Evidence from International Se-curitized Real Estate Markets, J. Real Estate Finance and Economics (in press).
17. R. T. Baillie, econometrics, J. Econometrics 73, 1 (1996) 1.
18. L. Bachelier, Louis Bachelier's Theory of Speculation: The Origins of Modern Finance, translated and with a commentary by: Mark Davis and Alison Etheridge with a fore-word by Paul A Samuelson, Princeton University Press (2006).
19. B. B. Mandelbrot and H. W. Taylor, Operations Research 15 (1967) 1057.
20. S. I. Boyarchenko and S. Z. Levendorskii, Int. J. Theor. Appl. Fin. 3 (2000) 549.
21. A. Matacz, Int. J. Theor. Appl. Fin. 3, 1 (2000) 143.
22. R. C. Merton, Continuous Time Finance, Blackwell, Cambridge, MA (1990).
23. E. Scalas, R. Goronflo and F. Mainardi, Fractional Calculus and Continuous-Time Finance, cond-mat/0001120.
24. J. Masoliver et al., J. Econ. Behavior and Organization 61, 4 (2006) 577.
25. F. Riewe, Phys. Rev. E 53 (1996) 1890.
26. M. Klimek, Czech. J. Phys. 51, 12 (2001) 1348.
27. M. Klimek, Czech. J. Phys. 55, 11 (2005) 1447.
28. O. P. Agrawal, J. Math. Anal. Appl. 272 (2002) 368.
29. D. Baleanu and T. Avkar, Nuovo Cimento 119 (2004) 73.
30. R. A. El-Nabulsi, Fizika A 14, 4 (2005) 289.
31. R. A. El-Nabulsi, Int. J. Mod. Geom. Meth. Mod. Phys. 5, 6 (2008), 863, and references therein.
32. R. A. El-Nabulsi, Int. J. Geom. Meth. Mod. Phys. 6, 1 (2009) 25.
33. R. A. El-Nabulsi, Chaos, Solitons and Fractals 42, 1 (2009) 52.
34. R. A. El-Nabulsi, Chaos, Solitons and Fractals 42, 1 (2009) 84.
35. R. A. El-Nabulsi, Chaos, Solitons and Fractals 41, 5 (2009) 2262.
36. R. A. El-Nabulsi, Chaos, Solitons and Fractals 42, 5 (2009) 2614.
37. R. A. El-Nabulsi, Chaos, Solitons and Fractals 42, 5 (2009) 2924.
38. R. A. El-Nabulsi, Chaos, Solitons and Fractals 42, 4 (2009) 2384.
39. R. A. El-Nabulsi, Complexified Dynamical Systems from Real Fractional Actionlike with Time-Dependent Fractional Dimension on Multifractal Sets, Invited contribution to The 3rd Int. Conf. Complex Systems and Applications, University of Le Havre, Le Havre, Normandy, France, June 29-July 02, 2009.
40. R. A. El-Nabulsi, Fractional Quantum Field Theory on Multifractal Sets, Symposium Fractional Signals and Systems, Lisbon09, American J. Eng. Appl. Sci. (in press).
41. R. A. El-Nabulsi, Mod. Phys. Lett. B 23, 28 (2009) 3369.
42. R. A. El-Nabulsi, Int. J. Mod. Phys. B 23, 6 (2009) 3349.
43. R. A. El-Nabulsi, Oscillating Flat FRW Dark Energy Dominated Cosmology from Pe-riodic Functional Approach, Comm. Theor. Phys. (accepted for publication, in press).
44. R. A. El-Nabulsi, Modifications at Large Distances from Fractional and Fractal Argu-ments, FRACTALS, (accepted for publication, in press).
45. R. A. El-Nabulsi, Fizika A 17, 2 (2008) 71.
46. R. A. El-Nabulsi, Int. J. Appl. Math. & Statistics 5, S06 (2006) 50 (Special Issue Dedicated to Prof. Jagannath Mazumdar).
47. R. A. El-Nabulsi and D. F. M. Torres, Math. Meth. Appl. Sci. 30, 15 (2007) 1931.
48. R. A. El-Nabulsi and D. F. M. Torres, J. Math. Phys. 49 (2008) 053521.
49. R. A. El-Nabulsi, Fizika B 17, 3 (2008) 369.
50. R. A. El-Nabulsi, EJTP4 15 (2007) 157.
51. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, 2nd Edition, World Scientific (1995).
52. I. Calvo and R. Sanchez, J. Phys. A: Math. Theor. 41 (2008) 282002.
53. D. P. Ellerman, SIAM Review 26, 2 (1983) 241.
54. B. E. Baaquie, C. Coriano and M. Skirant, Quantum Mechanics, Path Integrals and Option Pricing: Reducing the Complexity of Finance, cond-mat/0208191.
55. B. E. Baaquie, C. Coriano and M. Skirant, Physica A 334 (2004) 531.
56. V. Linetsky, Comp. Economics 11 (1998) 129.
57. E. J. Sullivan and T. M. Weithers, J. Economic Education 22 (1991) 165.
58. I. Avramidi, Analytic and Geometric Aspects of Heat Kernel Applications in Finance, NATIXIS Corporate and Investment Bank, Paris, France, (2007) pp. 249.
59. G. S. F. Frederico and D. F. M. Torres, WSEAS Transact. Math. 7, 1 (2008) 6.
60. B. E. Baaquie, Quantum Finance, Cambridge University Press (2004).
61. G. Sachs, Option Pricing Using Feynman Path Integral, Talk given Loomis Laboratory of Physics, University of Illinois at Urbana-Champaign, October 13 (2000).
62. J. L. McCauley, G. H. Gunaratne and K. E. Bassler, Martingale Option Pricing, physics/0606011.
63. B. Dumas, J. Fleming and R. E. Whaley, J. Finance 53 (1998) 2059.
64. M. Vlad and F. Spineanu, Plasma Phys. Controlled Fusion 47, 2 (2005) 281.
65. L. Mascarenhas, Transact. Am. Math. Soc. 281, 1 (1984) 179.
66. Y. Weitsman, J. Composite Materials 10, 3 (1976) 193.
67. L. L. Latour et al., Proc. Natl. Acad. Sci. USA 91 (1994) 1229.
68. J. Wu and K. M Berland, Biophys. J. 95 (2008) 2049.
69. E. L. Pankratov, Techn. Phys. 49, 1 (2004) 114.
70. C. H. Hui, J. Futures Markets 17 (1997) 667.
71. G. O. Roberts and C. F. Shortland, Math. Finance 7 (1997) 83.
72. C. F. Lo, H. C. Lee and C. H. Hui, Quantitative Finance 3 (2003) 98.
73. C. H. Hui and C. F. Lo, J. Future Markets 26, 10 (2006) 939.
74. M. R. Rodrigo and R. S. Mamon, Appl. Math. Letters 19 (2006) 398.
75. R. Vilela Mendes, Physica A 387, 15 (2008) 3987.
76. R. Vilela Mendes, A Fractional Calculus Interpretation of the Fractional Volatility Problem, Nonl. Dynamics (in press).
77. D. O. Cajueiro and J. F. Barbachan, Volatility Estimation and Option Pricing with Fractional Brownian Motion, FLWP-06 (2003) (IBMEC-Sao Paulo).
78. V. V. Piterbarg, Appl. Math. Finance 12, 2 (2005) 147.
79. A. Kogure, Asia-Pacific Financial Market 3, 1 (1996) 1.