Well-posedness and order preservation for neutral type stochastic differential equations of infinite delay with jumps

Abstract

<abstract><p>In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term.</p></abstract>