The behavior of solutions of stochastic differential inequalities (Q1902867)

Let \(X\) and \(Z\) be \(R^d\)-valued solutions of the stochastic differential inequalities \[ dX_t \leq a(t,X_t)dt + \sigma(t,X_t)dW_t \quad \text{and} \quad b(t,Z_t)dt + \sigma(t,Z_t)dW_t \leq dZ_t, \] respectively, with a fixed \(R^m\)-valued Wiener process \(W\). The authors give conditions on \(a\), \(b\) and \(\sigma\) under which the relation \(X_0 \leq Z_0\) of the initial values leads to the same relation between the solutions with probability one. A detailed discussion whether in general their conditions can be weakened or not is given. Inspired by the deterministic theory they introduce the notions of maximal/mimimal solution of a stochastic differential inequality. Using the comparison result a sufficient condition for existence of such solutions as well as some Gronwall-type estimates are derived.