Riccati moment-explosion horizon (${\tau }_{\ast }=\sup \{\tau :R\text{ is finite on }[0,\tau ]\}$)

The maximal time on which the coefficient solution of the exponential-quadratic transform stays finite. For $R^{\prime }=2R^2-2R+1$, $R(0)=0$, the solution is $R=(1+\tan (\tau -\pi /4))/2$ and explodes at $3\pi /4$. Thus local solvability of the coefficient ordinary differential equations does not imply an unrestricted global moment formula. For payoff exponent $0\le \theta \le 1$, the Riccati forcing is nonpositive and a negative equilibrium bounds the solution.