Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 6 / c

The positive density process is the stochastic exponentialSince $a^{\prime }$ is bounded, the Novikov condition holds; $Z$ is a true martingale with $\mathbb{E}{Z}_T=1$ and $Z_T>0$. Thus it defines the stated equivalent probability measure. The positive sign in the exponent means that the Girsanov theorem givesUnder this change of measure, applying the Itô formula to ${\pi }_t=U(t,S_t)$ produces precisely the drift in its backward equation, which vanishes:Bounded $a$ and $U_S$ make $\pi$ a true martingale under $\hat{\mathbb{P}}$. Its terminal value is $g^{\prime }(S_T)$. ConsequentlyThe sign can also be checked before changing measure: under the original measure $d\pi =-a{a}^{\prime }{U}_{S}dt+a{U}_{S}dW$, while the quadratic covariation term in $d(Z\pi )$ is $Za{a}^{\prime }{U}_{S}dt$, exactly cancelling its drift.