Gaussian volatility exponential-quadratic transform ($V(t,\sigma )=e^{P(T-t)+Q(T-t)\sigma +R(T-t){\sigma }^2}$)

For Ornstein-Uhlenbeck process volatility, matching powers of $\sigma$ reduces the transform PDE to a scalar Riccati equation for $R$, a linear equation for $Q$ and an integral for $P$. All coefficients start at zero. The representation exists up to the Riccati moment-explosion horizon; it need not remain finite on every horizon for arbitrary exponential powers.

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.