  We give a formula for the Lipschitz constant in Thompson's part metric of any
order-preserving flow on the interior of a (possibly infinite dimensional)
closed convex pointed cone. This provides an explicit form of a
characterization of Nussbaum concerning non order-preserving flows. As an
application of this formula, we show that the flow of the generalized Riccati
equation arising in stochastic linear quadratic control is a local contraction
on the cone of positive definite matrices and characterize its Lipschitz
constant by a matrix inequality. We also show that the same flow is no longer a
contraction in other natural Finsler metrics on this cone, including the
standard invariant Riemannian metric. This is motivated by a series of
contraction properties concerning the standard Riccati equation, established by
Bougerol, Liverani, Wojtowski, Lawson, Lee and Lim: we show that some of these
properties do, and that some other do not, carry over to the generalized
Riccati equation.
