There exists a solution of the SP for all if and only if R is completely-S
(Mandelbaum-Van d. Heyden, Bernard-ElKharroubi ’91)
Uniqueness holds if R=I+Q where
|Q| has spectral radius less than one
(Harrison-Reiman ’81, Dupuis-Ishii ’91)
There is non-uniqueness (and potentially non-adaptedness) for some R that are completely-S.