The behavior of solutions of stochastic differential inequalities

Summary LetX andZ be ^d -valued solutions of the stochastic differential inequalities dX _t a(t,X _t )dt+(t,X _t )dW _t andb(t, Z _t )dt+(t, Z _t )dW _t dZ _t , respectively, with a fixed ^m -valued Wiener processW. In this paper we give conditions ona, b and under which the relationX ₀Z ₀ of the initial values leads to the same relation between the solutions with probability one. Further we discuss whether in general our conditions can be weakened or not. Then we deal with notions like maximal/minimal solution of a stochastic differential inequality. Using the comparison result we derive a sufficient condition for the existence of such solutions as well as some Gronwall-type estimates.