Comparison theorems for some backward stochastic Volterra integral equations

For some backward stochastic Volterra integral equations (BSVIEs) in multi-dimensional Euclidean spaces, comparison theorems are established in a systematic way for the adapted solutions and adapted M-solutions. For completeness, comparison theorems for (forward) stochastic differential equations, backward stochastic differential equations, and (forward) stochastic Volterra integral equations (FSVIEs) are also presented. Duality principles are used in some relevant proofs. Also, it is found that certain kinds of monotonicity conditions play crucial roles to guarantee the comparison theorems for FSVIEs and BSVIEs to be true. Various counterexamples show that the assumed conditions are almost necessary in some sense.