zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

91B28Finance etc. (MSC2000)
91B62Growth models in economics
Full Text: DOI
[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 · doi:10.1023/A:1008634027843
[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, No. 1, 3-29 (1968)
[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 · doi:10.1137/S036301299834171X
[6] Hu, Y.; Ksendal, B. ø: 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) · Zbl 0054.05803
[8] Jumarie, G.: Stochastic differential equations with fractional Brownian motion input, International journal of systems and sciences 24, No. 6, 1113-1132 (1993) · Zbl 0771.60043 · doi:10.1080/00207729308949547
[9] Jumarie, G.: On the representation of fractional Brownian motion as an integral with respect to (dt)${\alpha}$, Applied mathematics letters 18, 739-748 (2005) · Zbl 1082.60029 · doi:10.1016/j.aml.2004.05.014
[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 · doi:10.1016/j.aml.2004.09.012
[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 · doi:10.1016/j.insmatheco.2005.06.003
[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 · doi:10.1016/j.camwa.2006.02.001
[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 · 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, Applied Mathematics Letters (2006c), in press (doi:10.1016/j.aml.2006.08.015) (Available online) · Zbl 1142.82013
[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 · doi:10.1016/j.chaos.2006.07.053
[16] Kober, H.: On fractional integrals and derivatives, The quarterly journal of mathematics, Oxford 11, 193-215 (1940) · Zbl 0025.18502
[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 · doi:10.1103/PhysRevLett.80.214
[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)
[21] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications, SIAM review 10, 422-437 (1968) · Zbl 0179.47801 · doi:10.1137/1010093
[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 · doi:10.1016/0304-4149(95)00046-1
[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 · doi:10.1016/S0304-4149(96)00089-0
[24] Osler, T. J.: Taylor’s series generalized for fractional derivatives and applications, SIAM journal of mathematical analysis 2, No. 1, 37-47 (1971) · Zbl 0215.12101 · doi:10.1137/0502004
[25] Stratonovich, R. L.: A new form of representing stochastic integrals and equations, Journal of SIAM control 4, 362-371 (1966)