Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optimal portfolio

Computers and Mathematics with Applications

Volumn 59, Issue 3, 2010, Pages 1142-1164

  Jumarie, Guy ^a  

^a UNIVERSITÉ DU QUÉBEC À MONTRÉAL   (Canada)

Author keywords

Fractional Black Scholes equation; Fractional Brownian motion; Fractional Merton's portfolio; Fractional stochastic differential equation; Fractional Taylor's series

Indexed keywords

BASIC EQUATIONS; BLACK SCHOLES EQUATIONS; COARSE-GRAINED; FRACTIONAL BROWNIAN MOTION; FRACTIONAL DYNAMICS; FRACTIONAL ORDER; FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION; FRACTIONAL TAYLOR'S SERIES; GAUSSIAN WHITE NOISE; INITIAL VALUES; MITTAG-LEFFLER FUNCTIONS; NEW RESULTS; OPTIMAL PORTFOLIOS; RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES; SPACE AND TIME; STOCK EXCHANGE; VIRTUAL DATA;

BROWNIAN MOVEMENT; MEASUREMENT THEORY; STOCHASTIC SYSTEMS; TAYLOR SERIES; WHITE NOISE;

EQUATIONS OF MOTION;

EID: 74249118189     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2009.05.015     Document Type: Article

References (27)

1. Jumarie G. Stock exchange fractional dynamics defined as fractional growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes. Insurance, Math. Econom. 12 (2008) 271-287. http://dx.doi.10.1016/j.insmatheco.2007.03.001 Available on line doi:10.1016/j.insmatheco.2007.03.001
2. Kober H. On fractional integrals and derivatives. Quart. J. Math. Oxford 11 (1940) 193-215
3. Letnivov A.V. Theory of differentiation of fractional order. Math. Sb. 3 (1868) 1-7
4. Liouville J. Sur le calcul des differentielles à indices quelconques. J. Ecole Polytechnique 13 (1832) 71 (in French)
5. Osler T.J. Taylor's series generalized for fractional derivatives and applications. SIAM. J. Math. Anal. 2 1 (1971) 37-47
6. Jumarie G. Stochastic differential equations with fractional Brownian motion input. Internat. J. Systems Sci. 24 6 (1993) 1113-1132
7. α. Appl. Math. Lett. 18 (2005) 739-748
8. Jumarie G. On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion. Appl. Math. Lett. 18 (2005) 817-826
9. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions. Further results. Comput. Math. Appl. 51 (2006) 1367-1376
10. Djrbashian M.M., and Nersesian A.B. Fractional derivative and the Cauchy problem for differential equations of fractional order. Izv. Acad. Nauk Armjanskoi SSR 3 1 (1968) 3-29 (in Russian)
11. Duncan T.E., and Pasik-Duncan B. Stochastic calculus for fractional Brownian motion, I. Theory. SIAM J. Control Optim. 38 (2000) 582-612
12. Caputo M. Linear model of dissipation whose Q is almost frequency dependent II. Geophys. J. R. Astron. Soc. 13 (1967) 529-539
13. Jumarie G. New stochastic fractional models for Malthusian growth, the Poissonian birth prodess and optimal management of populations. Math. Comput. Modelling 44 (2006) 231-254
14. Jumarie G. Path integral for the probability of the trajectories generated by fractional dynamics subject to Gaussian white noise. Appl. Math. Lett. 20 (2007) 846-852
15. Kolwankar K.M., and Gangal A.D. Holder exponents of irregular signals and local fractional derivatives. Pramana J. Phys 48 (1997) 49-68
16. Kolwankar K.M., and Gangal A.D. Local fractional Fokker-Planck equation. Phys. Rev. Lett. 80 (1998) 214-217
17. Jumarie G. Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor's series of non differentiable functions. Chaos Solitons Fractals 32 (2006) 969-987
18. Mandelbrot B.B., and van Ness J.W. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422-437
19. Mandelbrot B.B., and Cioczek-Georges R. A class of micropulses and antipersistent fractional Brownian motions. Stochastic Process. Appl. 60 (1995) 1-18
20. Mandelbrot B.B., and Cioczek-Georges R. Alternative micropulses and fractional Brownian motion. Stochastic Process. Appl. 64 (1996) 143-152
21. Black F., and Scholes H. The pricing of options and corporate liabilities. J. Political Economy 81 (1973) 81-98
22. Wyss W. The fractional Black-Scholes equation. Fract. Calc. Appl. Anal. 3 1 (2000) 51-61
23. 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 Economice 37 (2005) 585-598
24. Hu Y., and Øksendal B. Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 6 (2003) 1-32
25. Itô K. On stochastic differential equations. Mem. Amer. Soc. 4 (1951)
26. Stratonovich R.L. A new form of representing stochastic integrals and equations. J. SIAM. Control 4 (1966) 362-371
27. Decreusefond L., and Ustunel A.S. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999) 177-214