Of concern is the degenerate Riccati equation of the form TR + RT = TA + TBR + RTC + RTDR (*). This models a certain transport equation in the half-space. Our first result will concern a unique, positive solution of the operator equation SR + RT = K in a Banach space equipped with a cone structure. We then proceed to prove existence of positive solutions for (*) on an LI space of sa-finite measure and a Banach space X, respectively, under different assumptions. Some estimates of the solution with respect to certain norms are obtained.