Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations

In this paper, we present a new approach to obtain the comparison theorem of two 1-dimensional SDEs with diffusion and jumps. The two equations is treated as one two-dimensional SDE and the comparison requirement is regarded as to keep the solution (Xt1,Xt2) within the constraint K={(x1,x2);x1⩽x2}. We then apply a new criteria of “viability condition” which is a necessary and sufficient condition to keep the solution to be inside the constraint K.