Mertonʹs model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence

One considers the model of optimal portfolio first proposed by Merton, the simplest one, but here one assumes that the noises involved in the dynamics of the wealth are fractional Brownian motions (in the sense of fractional derivative of Gaussian white noises) with short-range dependence, that is to say with a Hurst parameter lower than 1/2. Instead of using the dynamic programming approach, the stochastic optimal control problem is converted into a non-random optimization involving the state moments as state variables, and then Taylor expansion of fractional order provides a way to circumvent some of the difficulties due to the presence of the fractal terms. The mathematical framework is essentially engineering mathematics, and mainly one will work formally by using the Maruyama notation of fractional order.