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 ( X t 1 , X t 2 ) within the constraint K = { ( x 1 , x 2 ) ; x 1 ⩽ x 2 } . 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.