Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 29K (Stochastic Financial Models)

For a bounded stopping time $T\le N$, writeThe indicator is measurable at time $n-1$. Taking conditional expectations term by term and using the supermartingale inequality gives $\mathbb{E}{\hat{M}}_T\le \mathbb{E}{\hat{M}}_0$. The same calculation gives equality for the martingale $M$. Since their initial values agree, $\mathbb{E}{M}_T\ge \mathbb{E}{\hat{M}}_T$.

If ${\theta }_n$ shares are held during period $n$, the remaining wealth $X_{n-1}-{\theta }_n{S}_{n-1}$ is in the bank account. Self-financing therefore giveswhich rearranges to the required equation.

Condition on the information through time $n-1$. Independence of ${\xi }_n$ and the definition of the Bellman equation for terminal-wealth utility implybecause the actual holding is one candidate in the supremum. Thus $V(n,X_n)$ is a supermartingale for every strategy.

For the stated strategy it is a martingale, soEvery competing strategy has expected terminal value at most $V(0,X_0)$ by the supermartingale inequality from part (a), evaluated at the bounded time $N$. Hence ${\theta }^{\ast }$ is optimal.