Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 5

A predictable strategy $H$ is self-financing when its wealth $X=H\cdot P$ satisfiesIt is admissible when its wealth obeys the stipulated lower bound, here taken to be nonnegative. A strictly positive Itô process $Y$ is a martingale deflator when every deflated asset price $Y{P}^i$ is a local martingale.

The product rule and self-financing identity show that $XY$ is the stochastic integral of $H$ against the vector of deflated prices $YP$, hence is a local martingale. It is nonnegative by admissibility and positivity of $Y$. Every nonnegative local martingale is a supermartingale, so

The definition of the concave conjugate gives the pointwise Fenchel–Young inequalityApply it to $(X_T,Y_T)$, take expectations, and use part b:If $U^{\prime }(X_T)=Y_T$, the first inequality is equality; if $XY$ is a true martingale, the second is equality.

Applying the Itô product rule to $YB$ makes its drift vanish automatically. For $YS$, the drift isThus both deflated prices are local martingales whenSince self-financing gives $dX=\varphi dB+\pi dS$, another application of the product rule, including $d[X,Y]$, cancels the drift and yields

Normalize $Y_0=1/X_0$. Solving its stochastic differential equation givesFor logarithmic utility, $\hat{U}(y)=-\log y-1$. Part c and $\mathbb{E}{W}_T=0$ therefore giveIf ${\pi }_t=X_t\lambda /(S_t\sigma )$, part d gives $d(XY)=0$. Hence $X_t{Y}_t=X_0{Y}_0=1$, so $U^{\prime }(X_T)=1/X_T=Y_T$ and both inequalities in part c are equalities.