THE BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL-INEQUALITIES

Let X and Z be R(d)-valued solutions of the stochastic differential inequalities dX(t) less than or equal to a(t,X(t))dt + sigma(t,X(t))dW(t) and b(t,Z(t))dt + sigma(t,Z(t))dW(t) less than or equal to dZ(t), respectively, with a fixed R(m)-valued Wiener process W. In this paper we give conditions on a,b and sigma under which the relation X(0) less than or equal to Z(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.