Controlled Lyapunov-exponents in optimization and finance

Let X = (Xn) 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) = limn 1/n E log || Xn · Xn−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 a convergent 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 [1], however the general case requires fundamentally different techniques.