##
**Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations.**
*(English)*
Zbl 1141.91455

Summary: Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order \(f(x+h) = E_\alpha (h^\alpha D^\alpha _x)f(x)\) where \(E_\alpha (.)\) denotes the Mittag-Leffler function, and \(D^\alpha _x\) is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô’s lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton’s optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.

### MSC:

91B28 | Finance etc. (MSC2000) |

91B62 | Economic growth models |

### Keywords:

fractional Gaussian noises; fractional stochastic differential equation; fractional exponential growth; fractional Brownian motion; path probability density; fractional Black-Scholes equation
PDF
BibTeX
XML
Cite

\textit{G. Jumarie}, Insur. Math. Econ. 42, No. 1, 271--287 (2008; Zbl 1141.91455)

Full Text:
DOI

### References:

[1] | Black, F.; Scholes, H., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 81-98 (1973) · Zbl 1092.91524 |

[2] | Caputo, M., Linear model of dissipation whose Q is almost frequency dependent II, Geophysics Journal of Royal Astronomical Society, 13, 529-539 (1967) |

[3] | Decreusefond, L.; Ustunel, A. S., Stochastic analysis of the fractional Brownian motion, Potential Analysis, 10, 177-214 (1999) · Zbl 0924.60034 |

[4] | Djrbashian, M. M.; Nersesian, A. B., Fractional derivative and the Cauchy problem for differential equations of fractional order, Izvestiya Academic Nauk Armjanskoi SSR, 3, 1, 3-29 (1968), (in Russian) |

[5] | Duncan, T. E.; Hu, Y.; Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion, I. Theory, SIAM Journal of Control and Optimization, 38, 582-612 (2000) · Zbl 0947.60061 |

[6] | Hu, Y.; Ø ksendal, B., Fractional white noise calculus and applications to finance, Infinite Dimensional Analysis, and Quantum Probability and Related Topics 6, 6, 1-32 (2003) · Zbl 1045.60072 |

[7] | Itô, K., On stochastic differential equations, Memoirs of the American Society, 4 (1951) |

[8] | Jumarie, G., Stochastic differential equations with fractional Brownian motion input, International Journal of Systems and Sciences, 24, 6, 1113-1132 (1993) · Zbl 0771.60043 |

[9] | Jumarie, G., On the representation of fractional Brownian motion as an integral with respect to \((d t)^\alpha \), Applied Mathematics Letters, 18, 739-748 (2005) · Zbl 1082.60029 |

[10] | Jumarie, G., On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Applied Mathematics Letters, 18, 817-826 (2005) · Zbl 1075.60068 |

[11] | Jumarie, G., Merton’s model of optimal portfolio in a Black-Scholes market driven by a fractional Brownian motion with short-range dependence, Insurance: Mathematics and Economics, 37, 585-598 (2005) · Zbl 1104.91034 |

[12] | Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Computers and Mathematics with Applications, 51, 1367-1376 (2006) · Zbl 1137.65001 |

[13] | Jumarie, G., New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Mathematical and Computer Modelling, 44, 231-254 (2006) · Zbl 1130.92043 |

[15] | Jumarie, G., Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor’s series of non differentiable functions, Chaos, Solitons and Fractals, 32, 969-987 (2006) · Zbl 1154.70011 |

[16] | Kober, H., On fractional integrals and derivatives, The Quarterly Journal of Mathematics, Oxford, 11, 193-215 (1940) · JFM 66.0520.02 |

[17] | Kolwankar, K. M.; Gangal, A. D., Holder exponents of irregular signals and local fractional derivatives, Pramana Journal of Physics, 48, 49-68 (1997) |

[18] | Kolwankar, K. M.; Gangal, A. D., Local fractional Fokker-Planck equation, Physics Review Letters, 80, 214-217 (1998) · Zbl 0945.82005 |

[19] | Letnivov, A. V., Theory of differentiation of fractional order, Matematicheskiĭ Sbornik, 3, 1-7 (1868) |

[20] | Liouville, J., Sur le calcul des differentielles à indices quelconques, Journal of Ecole Polytechnique, 13, 71 (1832), (in French) |

[21] | Mandelbrot, B. B.; van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Review, 10, 422-437 (1968) · Zbl 0179.47801 |

[22] | Mandelbrot, B. B.; Cioczek-Georges, R., A class of micropulses and antipersistent fractional Brownian motions, Stochastic Processes and their Applications, 60, 1-18 (1995) · Zbl 0846.60055 |

[23] | Mandelbrot, B. B.; Cioczek-Georges, R., Alternative micropulses and fractional Brownian motion, Stochastic Processes and their Applications, 64, 143-152 (1996) · Zbl 0879.60076 |

[24] | Osler, T. J., Taylor’s series generalized for fractional derivatives and applications, SIAM Journal of Mathematical Analysis, 2, 1, 37-47 (1971) · Zbl 0215.12101 |

[25] | Stratonovich, R. L., A new form of representing stochastic integrals and equations, Journal of SIAM Control, 4, 362-371 (1966) · Zbl 0143.19002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.