Controlled Lyapunov-exponents with applications

Let X = (X_n) be a stationary process of k × k real-valued matrices, depending on some vector-valued parameter θεR^p, satisfying E log⁺ ||X0(θ)|| < ∞ for all θ. The top-Lyapunov exponent of X is defined as λ(θ) = _n^lim 1/n E log ||X_n·X_n-1...·X₀||. Top-Lyapunov exponents play a prominent role in randomization procedures for optimization, such as SPSA, and in finance, giving the growth-rate of a self-financing currency-portfolio with a fixed strategy. 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 (Beneviste, 1990). 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 interesting in their own right. These are given in the Appendix. Simulation results are also presented.