Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 30K (Stochastic Financial Models)

If $y_1\le y_2$, then the utility function is increasing, so for every $X\in \mathcal{X}$,Taking suprema gives $F(y_1)\le F(y_2)$.

For $0\le t\le 1$, let $X_1$ and $X_2$ attain the suprema at $y_1$ and $y_2$. Because $\mathcal{X}$ is a vector space,The concavity of $U$ now givesThus $F$ is increasing and concave, as stated by optimized affine shift of concave utility.

For $0\le t\le 1$, concavity of $U$ impliesso $G$ is concave.

Since $\mathbb{E}Z=0$, Jensen inequality givesIf $0\le s<t$, concavity and $s=(1-s/t)0+(s/t)t$ yieldHence $G$ is decreasing on $[0,\infty )$; this is scaled centered risk under concave utility.

Let the value function at time $n$ be the greatest conditional expected terminal utility attainable from wealth $x$. Independence of the innovations gives the Bellman equationfor $0\le n<N$. A policy obtained from the maximizing choices is admissible because each time-$n$ holding is a predictable process, hence is known before ${\xi }_n$ is observed.

We prove the claimed properties by backward induction. They hold at $n=N$ because $V(N,\cdot )=U$ is increasing and concave. Suppose they hold at time $n+1$. For fixed $\theta$, the mapis increasing because $1+r>0$, and taking a supremum preserves this inequality.

For concavity, let ${\theta }_x$ and ${\theta }_y$ be optimal at wealths $x$ and $y$. For $0\le t\le 1$, use the admissible portfolioThe next wealth is the same convex combination of the next wealths generated from $(x,{\theta }_x)$ and $(y,{\theta }_y)$. ThereforeThis completes the induction and proves monotonicity and concavity of a portfolio value function.

Condition on the information available before time $n$. The current wealth is then fixed, while ${\xi }_n\sim N(b,\Sigma )$ is independent of that information. PutFor any portfolio $\theta$, use the inner product induced by the positive-definite matrix $\Sigma$ to writeThe linear image of a multivariate normal vector shows that the two portfolio returns are jointly normal, andThus independence of uncorrelated jointly normal variables makes the residual return ${\eta }^T{\xi }_n$ independent of the aligned return; it also has expected value zero.

The continuation value $V(n,\cdot )$ is concave by part (c). Conditional Jensen inequality therefore shows that adding the independent centered residual cannot improve the objective:where $m=(1+r)X_{n-1}$. Since the optimal portfolio is unique, its residual must be zero.

It remains to determine the sign. If $\lambda <0$, the portfolios $\lambda {\Sigma }^{-1}b$ and $-\lambda {\Sigma }^{-1}b$ have the same return variance, namely ${\lambda }^{2}B$, while their means are $\lambda B$ and $-\lambda B$. The latter return has the distribution of the former plus the positive constant $-2\lambda B$. Since $V(n,\cdot )$ is increasing, replacing $\lambda$ by $-\lambda$ cannot reduce the objective, contradicting uniqueness. Hence $\lambda \ge 0$.

Applying this conditional argument at every time gives nonnegative, past-measurable random variables ${\lambda }_n$ such thatThis is the Gaussian one-fund theorem.