Stochastic approximation procedure in semi-Markov environment applied to alcohol consumption model

In this paper, we consider a stochastic approximation procedure with semi-Markov switchings in an averaging scheme with a small parameter. Introduction.Due to the wide use of stochastic diffusion problems, conditions for stability and control of such systems seem to be important.In [6] sufficient conditions for stability of stochastic systems via Lyapunov function properties are given and estimates of large deviations of linear diffusion systems are obtained.Problems of optimal control of diffusion processes described by stochastic differential equations with acceptable control are described in [20].On the other hand, asymptotic behaviour is considered in [25] and [26].For conditions of weak convergence of random processes in [11,16,12] the method of small parameter and a singular perturbation problem solution was used for the construction of the generator limiting process.This method is applied to schemes of averaging of diffusion approximation and to schemes of averaging of asymptotically small diffusion.In particular in [16] the cases of random evolution of Markov and semi-Markov switchings were examined.Construction of semi-Markov processes and investigation of asymptotic properties of random processes with semi-Markov switchings are considered in [1,2,3,4].