On Equilibrium of an Adaptive Single Component Market

A mathematical model of an adaptive Samuel-Marshall type single component market described by quasi-linear functional differential equations with dependent on phase coordinates and frequently switched an ergodic Markov process is presented. The proposed method is based on an averaging procedure with respect to time along the critical solutions of the generative average linear equation and with respect to the invariant measure of the Markov process. It is proved that exponential stability of the resulting deterministic equation is sufficient for exponential p stability of the initial random system for all positive numbers P and for sufficiently fast switching.