Optimal portfolio in a fractional Black \& Scholes market (Q2712771)

Suppose that an investor chooses a postfolio \({\theta}(t)=\left(\alpha(t),\beta(t)\right)\), giving the number of units \({\alpha}(t), {\beta}(t)\) held at time \(t\) into the bonds and stocks, respectively. Let NEWLINE\[NEWLINEZ^{\theta}(t,\omega)= {\alpha}(t)A(t)+{\beta(t)S(t)}NEWLINE\]NEWLINE be the value process corresponding to this portfolio. The strategy \({\theta}\) is supposed to be self-financing and admissible, with a given initial value \(z>0\). The authors solve the problem to find \(V(z)\) and \({{\theta}^{\ast}}{\in}{\mathcal A}\) such that NEWLINE\[NEWLINEV(z)= \sup_{\theta\in\mathcal A}E^{z}\left[u \left(z^\theta(T) \right) \right]= E^{x}\left[u\left(z^{\theta^\ast}(T) \right) \right]NEWLINE\]NEWLINE with utility function \(u(x)={\gamma}^{-1} x^{\gamma}\). It is explained in detail how the martingale approach of Cox and Huang can be used to solve this problem for an Itô type Black \& Scholes market with Hurst parameter \(H>0,5\). The explicit values of \(V(z)\) and \({\theta}^{\ast}\) are presented. The results are compared to the corresponding well-known results for the standard Black \& Scholes market.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00063].