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Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio. (English) Zbl 1189.91230
Summary: By using the new fractional Taylor’s series of fractional order $f\left(x+h={E}_{\alpha }\left({h}^{\alpha }{D}_{x}^{\alpha }f\left(x\right)$ where ${E}_{\alpha }\left(·\right)$ denotes the Mittag-Leffler function, and ${D}_{x}^{\alpha }$ 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, one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a (usual) Gaussian white noise. One can then derive two new families of fractional Black-Scholes equations, and one shows how one can obtain their solutions. Merton’s optimal portfolio is once more considered and some new results are contributed, with respect to the modeling on one hand, and to the solution on the other hand. Finally, one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics.
##### MSC:
 91G80 Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 26A33 Fractional derivatives and integrals (real functions) 35R11 Fractional partial differential equations 45K05 Integro-partial differential equations
##### References:
 [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, 271-287 (2008) · Zbl 1141.91455 · doi:10.1016/j.insmatheco.2007.03.001 [2] Kober, H.: On fractional integrals and derivatives, Quart. J. Math. Oxford 11, 193-215 (1940) · Zbl 0025.18502 [3] Letnivov, A. V.: Theory of differentiation of fractional order, Math. sb. 3, 1-7 (1868) [4] Liouville, J.: Sur le calcul des differentielles à indices quelconques, J. ecole polytechnique 13, 71 (1832) [5] Osler, T. J.: Taylor’s series generalized for fractional derivatives and applications, SIAM J. Math. anal. 2, No. 1, 37-47 (1971) · Zbl 0215.12101 · doi:10.1137/0502004 [6] Jumarie, G.: Stochastic differential equations with fractional Brownian motion input, Internat. J. Systems sci. 24, No. 6, 1113-1132 (1993) · Zbl 0771.60043 · doi:10.1080/00207729308949547 [7] Jumarie, G.: On the representation of fractional Brownian motion as an integral with respect to (dt)$\alpha$, Appl. math. Lett. 18, 739-748 (2005) · Zbl 1082.60029 · doi:10.1016/j.aml.2004.05.014 [8] Jumarie, G.: On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. math. Lett. 18, 817-826 (2005) · Zbl 1075.60068 · doi:10.1016/j.aml.2004.09.012 [9] Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of non-differentiable functions. Further results, Comput. math. Appl. 51, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001 [10] Djrbashian, M. M.; Nersesian, A. B.: Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv. acad. Nauk armjanskoi SSR 3, No. 1, 3-29 (1968) [11] Duncan, T. E.; Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion, I. Theory, SIAM J. Control optim. 38, 582-612 (2000) · Zbl 0947.60061 · doi:10.1137/S036301299834171X [12] Caputo, M.: Linear model of dissipation whose Q is almost frequency dependent II, Geophys. J. R. astron. Soc. 13, 529-539 (1967) [13] Jumarie, G.: New stochastic fractional models for malthusian growth, the Poissonian birth prodess and optimal management of populations, Math. comput. Modelling 44, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003 [14] Jumarie, G.: Path integral for the probability of the trajectories generated by fractional dynamics subject to Gaussian white noise, Appl. math. Lett. 20, 846-852 (2007) · Zbl 1142.82013 · doi:10.1016/j.aml.2006.08.015 [15] Kolwankar, K. M.; Gangal, A. D.: Holder exponents of irregular signals and local fractional derivatives, Pramana J. Phys 48, 49-68 (1997) [16] Kolwankar, K. M.; Gangal, A. D.: Local fractional Fokker–Planck equation, Phys. rev. Lett. 80, 214-217 (1998) · Zbl 0945.82005 · doi:10.1103/PhysRevLett.80.214 [17] Jumarie, G.: Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of non differentiable functions, Chaos solitons fractals 32, 969-987 (2006) · Zbl 1154.70011 · doi:10.1016/j.chaos.2006.07.053 [18] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications, SIAM rev. 10, 422-437 (1968) · Zbl 0179.47801 · doi:10.1137/1010093 [19] Mandelbrot, B. B.; Cioczek-Georges, R.: A class of micropulses and antipersistent fractional Brownian motions, Stochastic process. Appl. 60, 1-18 (1995) · Zbl 0846.60055 · doi:10.1016/0304-4149(95)00046-1 [20] Mandelbrot, B. B.; Cioczek-Georges, R.: Alternative micropulses and fractional Brownian motion, Stochastic process. Appl. 64, 143-152 (1996) · Zbl 0879.60076 · doi:10.1016/S0304-4149(96)00089-0 [21] Black, F.; Scholes, H.: The pricing of options and corporate liabilities, J. political economy 81, 81-98 (1973) [22] Wyss, W.: The fractional black–Scholes equation, Fract. calc. Appl. anal. 3, No. 1, 51-61 (2000) · Zbl 1058.91045 [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, 585-598 (2005) · Zbl 1104.91034 · doi:10.1016/j.insmatheco.2005.06.003 [24] Hu, Y.; øksendal, B.: Fractional white noise calculus and applications to finance, Infin. dimens. Anal. quantum probab. Relat. top. 6, No. 6, 1-32 (2003) · Zbl 1045.60072 · doi:10.1142/S0219025703001110 [25] Itô, K.: On stochastic differential equations, Mem. amer. Soc. 4 (1951) · Zbl 0054.05803 [26] Stratonovich, R. L.: A new form of representing stochastic integrals and equations, J. SIAM control 4, 362-371 (1966) [27] Decreusefond, L.; Ustunel, A. S.: Stochastic analysis of the fractional Brownian motion, Potential anal. 10, 177-214 (1999) · Zbl 0924.60034 · doi:10.1023/A:1008634027843