Continuous-time averaged models of discrete-time stochastic systems: Survey and open problems

A survey of a number of papers devoted to continuous-time modeling of discrete-time stochastic systems is given. It is concluded that, although different approaches to averaged (approximate) models justifying are in use, the procedures of building the averaged (approximate) models are similar in different papers. In addition to the deterministic (ODE) model some stochastic continuous-time models described by SDE are introduced. A new result concerning evaluation of the ODE model accuracy over the infinite time interval under partial stability condition is presented. Applications in adaptation, optimization and control are discussed. ¹