The paper studies the transient and asymptotic behavior of the solution of the sign-indefinite differential Riccati equation (DRE) arising in finite-horizon H-infinity-filtering and control problems. Differently from the sign-definite H-2-DRE, the solution of the H-infinity-DRE can have finite escape times even for nonnegative initial conditions. Sufficient and necessary conditions for boundedness and convergence are derived in correspondence to a fixed value of the H-infinity-norm attenuation level gamma. Finally, a stepwise gamma-switching strategy is devised to guarantee boundedness as well as asymptotic performance.