Comparison Theorems for Backward Stochastic Volterra Integral Equations

摘要

For backward stochastic Volterra integral equations (BSVIEs) inmulti-dimensional Euclidean spaces, comparison theorems are established in asystematic way for the adapted solutions and adapted M-solutions. Forcompleteness, comparison theorems for (forward) stochastic differentialequations, backward stochastic differential equations, and (forward) stochasticVolterra integral equations (FSVIEs) are also presented. Duality principles areused in some relevant proofs. Also, it is found that certain kind ofmonotonicity conditions play crucial roles to guarantee the comparison theoremsfor FSVIEs and BSVIEs to be true. Various counterexamples show that the assumedconditions are almost necessary in some sense.