Log-optimal currency portfolios and control Lyapunov exponents

P. Algoet and T. Cover characterized log-optimal portfolios in a stationary market without friction. There is no analogous result for markets with friction, of which a currency market is a typical example. In this paper we restrict ourselves to simple static strategies. The problem is then reduced to the analysis of products of random matrices, the top-Lyapunov exponent giving the growth rate. New insights to products of random matrices will be given and an algorithm for optimizing top-Lyapunov exponents will be presented together with some key steps of its analysis. Simulation results will also be given. Let X = (Xn) be a stationary process of k x k real-valued ess of k × k real-valued matrices, depending on some vector-valued parameter θ∈R^p, satisfying Elog⁺||X0(θ)||<∞ for all θ. The top-Lyapunov exponent of X is defined as λ(θ)=limn1/nElog||Xn·Xn-1...·X0||. We develop an iterative procedure for the optimization of λ(θ). In the case when X is a Markov-process, the proposed procedure is formally within the class defined in [3]. However the analysis of the general case requires different techniques: an ODE method defined in terms of asymptotically stationary random fields. The verification of some standard technical conditions, such as a uniform law of large numbers for the error process is hard. For this we need some auxiliary results which are interes ting in their own right.