Controlled Lyapunov-exponents with applications

Abstract

Let X = (X/sub n/) be a stationary process of k /spl times/ k real-valued matrices, depending on some vector-valued parameter /spl theta//spl epsiv//spl Ropf//sup p/, satisfying E log/sup +/ /spl par/X0(/spl theta/)/spl par/ < /spl infin/ for all /spl theta/. The top-Lyapunov exponent of X is defined as /spl lambda/(/spl theta/) = /sub n//sup lim/ 1/n E log /spl par/X/sub n//spl middot/X/sub n-1/.../spl middot/X/sub 0//spl par/. 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 /spl lambda/(/spl theta/). 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.