A stochastic maximum principle for processes driven by fractional Brownian motion. (Q1766033)

A stochastic maximum principle for a controlled process \(X(t)\) governed by the stochastic differential equation of the form \[ \text{d}X(t)=b(t,X(t),u(t))\,\text{d}t +\sigma (t,X(t),u(t))\,\text{d}B^H(t) \] is proved where \(B^H(t)\) is an \(m\)-dimensional fractional Brownian motion with the Hurst parameter \(H\in (1/2,1)^m\), \(b:[0,T]\times R^n\times U\to R^n\) and \(\sigma : [0,T]\times R^n\times U\to R^{n\times n}\) are given \(C^1\) functions and the control process \(u:[0,T]\times \Omega \to U\subset R^k\) is adapted. The stochastic fractional backward equation for the adjoint process is discussed in the linear case. As an application a problem of minimal variance hedging in an incomplete market driven by fractional Brownian motion is solved.