For constant coefficients, the density process $Y_tB_t/B_0$ is a true exponential martingale and defines the <risk-neutral measure> $Q$. Under $Q$,
$$
dS_t=rS_tdt+\sigma S_t dW_t^Q.
$$
The minimal value process is therefore
$$
V(t,s)=e^{-r(T-t)}
\mathbb E^Q[g(S_T)\mid S_t=s].
$$
The <Markov property> and the lognormal transition law make this a deterministic function of $(t,s)$, and
$$
\boxed{V(t,S_t)=\phi_tB_t+\pi_tS_t.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-211/3/c/solution