Portfolio optimization with consumption in a fractional Black-Scholes market

Abstract

We consider the classical Merton problem of flnding the optimal consumption rate and the optimal portfolio in a Black-Scholes market driven by fractional Brownian motion B H with Hurst parameter H > 1=2. The integrals with respect to B H are in the Skorohod sense, not pathwise which is known to lead to arbitrage. We explicitly flnd the optimal consumption rate and the optimal portfolio in such a market for an agent with logarithmic utility functions. A true self-flnancing portfolio is found to lead to a con- sumption term that is always favorable to the investor. We also present a numerical implementation by Monte Carlo simulations.