Problems of construction and stochastical stability of recurrent Monte Carlo algorithms for approximation of the functions satisfying a recurrent procedure φⁿ = A₀φⁿ + A₁φ^n-1 + fⁿ, n≥1, where A₀, A₁ are linear integral operators are considered. A recurrent algorithm is introduced, where a nonparametric estimation procedure for approximation of functions on each iteration levels is used. The sufficient conditions of stochastic stability in the sense of boundedness of a covariation function of the estimators are obtained.