ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH RANDOM PERTURBATIONS

Let there be giwQn a sequence of differential equations (1) X = =-À„X + S^{t, X, CD^), a sequence of initial conditions х, ̂ (0, a)„), a sequence of numbers v,,, a sequence of intervals <0, T„> and a sequence of (ß„, ^ „ , P„). The S „ is a stochastic process defined for [t, x, ш„] e <0, oo) x (-00,00) x ß„, and describes random perturba-tions. Finally X is a scalar variable, Я „ are positive numbers. We shall investigate two different kinds of assumptions concerning S„. Firstly, we may assume that the perturbations act only at discrete instants tll' \ causing discontinuities of the solution; secondly, they may act on the entire intervals {t^^ \ t^^'^^^) being bounded there however a mixed case may also be considered. For the sake of simplicity we shall assume that these instants tH"" ^ are regularly distributed, i.e. there exist numbers d ^> 0 such that rf ^ = kd „ (k are nonnegative integers). The assumptions on S „ are given more precisely in § 2: in the discrete case in (16) to (18), and in the continuous case in (19) or (20) together with a further assumption. However, these assumptions do not ensure the uniqueness of the process S „ nor of the solutions x