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Fractionalization of the complex-valued Brownian motion of order $$n$$ using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics. (English) Zbl 1099.60025
The 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.

##### MSC:
 60G15 Gaussian processes 91B28 Finance etc. (MSC2000)
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