Linear Backward Stochastic Differential Equations of Descriptor Type: Regular Systems

In this article, a class of linear backward stochastic differential equations of descriptor type with time-invariant coefficients are introduced. Necessary and sufficient conditions for their solvability are obtained. It turns out that such equations may not always have a solution, and even when they do, some components of the solution could have a jump at terminal time. Exact controllability of linear descriptor stochastic control systems is also considered.