Robust Backward Linear-Quadratic Differential Game and Team: A soft-constraint analysis

The robust design with model uncertainty is an extensively investigated research issue but traditionally focuses on forward stochastic system for which the initial condition is given as a priori. This paper makes a systematic continuation of this issue but in backward linear-quadratic context. That is, the state dynamics is evolved by a linear backward stochastic differential equation with terminal condition being specified, and cost functionals to be optimized are of quadratic forms. Three types of dynamic backward decisions, are introduced in both game and team setting in presence of drift uncertainty (disturbance). Specifically, in (Stackelberg) game setting, both cases for the follower or leader with unknown disturbance are discussed respectively; whereas in the team setting, the case with disturbance unknown to the leader is highlighted. For each of these three types, corresponding robust backward strategies are designed and associated optimal values for either leader or follower are given.