Fractionalization of the complex-valued Brownian motion of order \(n\) using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics.

*(English)*Zbl 1099.60025The paper deals with the procedure of fractionalization of the complex-valued Brownian motion. Some results on the fractional Taylor’s series of analytic functions expressed in terms of Mittag-Leffler function are presented. Then these results are used to solve some linear partial differential equations involving both \(dz\) and \((dz)^\alpha\). The procedure of \(\alpha\)-fractionalization is applied to complex-valued Brownian motion of order \(n\), by using fractional derivative.

Then applications of this procedure are considered. At first, an exponential process of which the increase rate is driven by an \(\alpha\)-fractional Brownian motion of order \(n\), is studied. And it is shown that its expression involves explicitly the Mittag-Leffler function. Then in the framework of mathematical finance, the optimal management of a portfolio in the presence of the stock market driven by fractional noises is investigated. This stochastic problem is converted into a non-random problem the solution of which can be obtained by variational techniques. At last, applications to variational mechanics and to stochastic mechanics are examined.

Then applications of this procedure are considered. At first, an exponential process of which the increase rate is driven by an \(\alpha\)-fractional Brownian motion of order \(n\), is studied. And it is shown that its expression involves explicitly the Mittag-Leffler function. Then in the framework of mathematical finance, the optimal management of a portfolio in the presence of the stock market driven by fractional noises is investigated. This stochastic problem is converted into a non-random problem the solution of which can be obtained by variational techniques. At last, applications to variational mechanics and to stochastic mechanics are examined.

Reviewer: Yuliya S. Mishura (Kyïv)

##### MSC:

60G15 | Gaussian processes |

91B28 | Finance etc. (MSC2000) |

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\textit{G. Jumarie}, Chaos Solitons Fractals 28, No. 5, 1285--1305 (2006; Zbl 1099.60025)

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