A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales

By the local time method we prove comparison theorems for systems of stochastic differential inequalities with respect to semimartingales. Furthermore, we construct the 'maximal/minimal solution' of a system of stochastic differential inequalities by the monotone iterative technique. In one-dimensional case, using the comparison results, we give a stochastic Bihari-type inequality and its application to multi-dimensional stochastic differential equations.