Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black–Scholes equations

Abstract

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 α ( h α D x α ) f ( x ) where E α ( . ) denotes the Mittag–Leffler function, and D x α 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.