 Solution

Set $\tau =T-t$, $\kappa =\lambda -\rho b\theta$ and $k=\theta (\theta -1)/2$. For the Ornstein-Uhlenbeck process volatility, the transformed partial differential equation isUse the Gaussian volatility exponential-quadratic transform ansatz $V=\exp (P+Q\sigma +R{\sigma }^2)$, where the coefficients depend on $\tau$. Its derivatives satisfyMatching the constant, linear and quadratic powers of $\sigma$ givesThe terminal condition requiresThese polynomial ordinary differential equations have a unique local solution by the Picard's theorem for ordinary differential equations. Substitution then proves the desired partial differential equation solution on every horizon for which the coefficient solution remains finite. The Riccati equation for $R$ is solved first; $Q$ subsequently solves a linear equation and $P$ is an integral of known coefficients.

Unrestricted global existence needs a qualification. Take $a=0$, $\lambda =b=1$, $\rho =0$ and $\theta =2$. Then $Q=0$ and the Riccati equation becomesThis solves the initial condition but explodes at $\tau =3\pi /4$. Therefore no finite real exponential-quadratic solution with the required terminal condition exists on an entire horizon $T\ge 3\pi /4$ for these allowed parameters. The correct general claim is local existence, or existence before the Riccati moment-explosion horizon.

A useful sufficient global condition is $0\le \theta \le 1$, so $k\le 0$. If $b\ne 0$, the lower equilibriumtraps the solution in $r_{-}\le R(\tau )\le 0$: the polynomial vector field points inward at the upper endpoint and vanishes at the lower endpoint. If $b=0$, the equation for $R$ is linear. In either case there is no finite-time explosion; $Q$ is then a linear equation with coefficients bounded on compact time intervals, and $P$ is finite on those intervals. This proves the intended ansatz globally under that sufficient parameter restriction, without asserting it for every real $\theta$.