On Boundedness and Stability of Solutions of Nonlinear Difference Equation with Nonmartingale Type Noise

Consider a stochastic difference equation xi+1 = Gi [xi, xi−1, . . . , x0] + fi [xi, xi−1, . . . , x0] + k0 ∑ k=0 σi−k [xi−k, xi−k−1, . . . , x0] ξi+1−k, (1) with the Volterra type nonlinear main term G and the Volterra type noise f + ∑k0 k=0 σi−kξi+1−k. Functions Gi, fi, σi are random while ξi is a martingale-difference. In general, k0 ≥ 1 so Eqn (1) cannot be considered as a stochastic equation with respect to the discrete semimartingale because the term ∑k0 k=0 σi−kξi+1−k is not a martingale-difference. The main aim of this paper is to establish the sufficient conditions on the almost sure boundedness, asymptotic and exponential stability of the solutions. Eqn (1) can be interpreted as a generalization of the equation describing the gain incurred by the insurance company in year i+1. We therefore explore the possible application of our theory in this area.