Rodkina, Alexandra E.; Mao, Xuerong
Author Affiliation, Ana.
Department of Mathematics and Compuiter Science
On boundedness and stability of solutions of nonlinear difference equation with nonmartingale type noise
Journal of Difference Equations and Applications
Date of Publication
Consider a stochastic difference equation xi+1 = Gi [xi, xi-1, ...,xo] + fi[xi, xi-1,..., xo]+ k0 åsi-k[xi-k, xi-k-1,...,xo]xi+1-k, k=0 with the volterra type nonlinear main term G and Volterra type noise f+ åsi-k xi+1-k. Functions G, f, s supposed to be random, xi is a martingale-difference. In general, ko³ 1 so Eq. (1) cannot be considered as a stochastic equation with respect to the discrete semimartingale because the term åsi-k xi+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. Equation (1) can be interpreted as a generalization of the equation describing the gain incurred by the insurance company in a year i+1. we therefore explore the possible application of our theory in this area.....