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. (C) 2005 Elsevier B.V. All rights reserved.