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A stochastic maximum principle for processes driven by fractional Brownian motion. (English) Zbl 1064.93048

A stochastic maximum principle for a controlled process X(t) governed by the stochastic differential equation of the form

dX(t)=b(t,X(t),u(t))dt+σ(t,X(t),u(t))dB H (t)

is proved where B H (t) is an m-dimensional fractional Brownian motion with the Hurst parameter H(1/2,1) m , b:[0,T]×R n ×UR n and σ:[0,T]×R n ×UR n×n are given C 1 functions and the control process u:[0,T]×ΩUR 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.

93E20Optimal stochastic control (systems)
60H05Stochastic integrals
60H10Stochastic ordinary differential equations