The proposal continuous stochastic differential equation for conditional variance is constructed as a diffusion approximation of discrete ARCH process. In contrast to classical auto regressive models with independent random perturbations our paper deals with uncertainty given as a stationary ergodic Markov chain. The method is based on stochastic analysis approach to finite dimensional difference equations with proportional to small parameter ε increments. Writing a point-form solution of this difference equation as vertexes of a time-dependent continuous broken line given on the segment [0,1] with ε-dependent scaling of intervals between vertexes and tending ε to zero we apply probabilistic limit theorems for dynamical systems with rapid Markov switching. The distribution of stationary solution of resulting stochastic equation may be successfully used for analysis of initial discrete model. This method permits to discuss a correlation effect on log of cumulative excess return with stochastic volatility. model-based analysis shows that it is important to take into account possible serial residual correlation in conditional variance process. The proposed method is applied to investigate the GARCH(1,1) and GARCH(2,1) processes under assumption that random variables are serially correlated. As a result it is possible to find continuous stochastic differential equations these processes converge to in distribution.