Identification of rank one rational spectral densities from noisy observations: A stochastic realization approach

Let S be a q × q spectral density matrix which is the sum of a rational rank one spectral density S₁ and of a constant positive definite matrix Q. The identification problem of S₁ and Q from S is addressed. The conditions under which S₁ and Q are identifiable are first derived. Then, an identification method is proposed. It is based on a parametrization of the external stochastic realizations of S whose innovation sequence has a prescribed dimension.     

Numerical simulation of stochastic replicator models in catalyzed RNA-like polymers

A stochastic model for replicators in catalyzed RNA-like polymers is presented and numerically solved. The model consists of a system of reaction–diffusion equations describing the evolution of a population formed by RNA-like molecules with catalytic capabilities in a prebiotic process. The diffusion effects and the catalytic reactions are deterministic. A stochastic excitation with additive noise is introduced as a force term. To numerically solve the governing equations we apply the stochastic method of lines. A finite-difference reaction–diffusion system is constructed by discretizing the space and the associated stochastic differential system is numerically solved using a class of stochastic Runge–Kutta methods. Numerical experiments are carried out on a prototype of four catalyzed selfreplicator species along with an activated and an inactivated residues. Results are given in two space dimensions.