Merton's Optimal Portfolio: An Approach via Fractional Taylor's Series

One way to take account of large volatility in stock exchange market is to use a modelingvia stochastic processes of fractional order. The volatility of stock exchange variations canbe suitably represented by a time variation of order (dt)h where h, referred to as the Hurstexponent, is a real-valued parameter which ful ls the condition 0 < h < 1. In this paper,we use the Mittag-Le er function and modi ed Riemann-Liouville fractional derivative tointroduce fractional Taylor's series that this expansion provides a way to circumvent some ofthe di culties due to the presence of the fractal terms. Then we can meaningfully considera modeling of fractional stochastic di erential equations as a fractional dynamics driven bya Gaussian white noise. Furthermore, we preset the adaptation and solutions of two classesof fractional Black-Scholes equations. Finally, Merton's optimal portfolio is more consideredas a practical model.

Keywords:

Merton's portfolio , Fractional Taylor's series , Fractional stochastic dierential equations , Mittag-Leer function , Riemann-Liouville fractional derivative , Gaussian white noise