Linear quadratic stochastic integral games and related topics

Abstract

This paper studies linear quadratic games problem for stochastic Volterra integral equations (SVIEs in short) where necessary and sufficient conditions for the existence of saddle points are derived in two different ways. As a consequence, the open problems raised by Chen and Yong (2007) are solved. To characterize the saddle points more clearly, coupled forward-backward stochastic Volterra integral equations and stochastic Fredholm-Volterra integral equations are introduced. Compared with deterministic game problems, some new terms arising from the procedure of deriving the later equations reflect well the essential nature of stochastic systems. Moreover, our representations and arguments are even new in the classical SDEs case.