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

The contraction rate in Thompson metric of order-preserving flows on a cone - application to generalized Riccati equations