Let $\tau=T-t$ and set
$$
V(t,z)=\exp\{P(\tau)+Q(\tau)z+R(\tau)z^2\}.
$$
Then
$$
\frac{V_z}{V}=Q+2Rz,
\qquad
\frac{V_{zz}}V=2R+(Q+2Rz)^2,
\qquad
\frac{V_t}V=-(\dot P+\dot Qz+\dot Rz^2).
$$
For $B(z)=a-bz$ and $C(z)=c$, matching constant, linear, and quadratic coefficients gives
$$
\dot R
=2c^2R^2+2(\theta\rho c-b)R+\frac12\theta(\theta-1),
$$
$$
\dot Q
=2aR+(\theta\rho c-b)Q+2c^2QR,
$$
and
$$
\dot P=aQ+c^2R+\frac12c^2Q^2.
$$
The terminal condition becomes $P(0)=Q(0)=R(0)=0$. The first equation is a <Riccati equation>; once it is solved, the second is linear in $Q$, followed by direct integration for $P$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 6 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-211/6/c/solution