Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 28J (Stochastic Financial Models)

The product $z_n=z_0{\prod }_{j=0}^{n-1}{X}_j$ uses only past multipliers, so all $X_n,\dots ,X_{N-1}$ are independent of it. With $m=\mathbb{E}{X}_0$, independence givesThe formula is finite when $m<\infty$; since the variables are positive, it also has the extended value $+\infty$ for $m=\infty$ and $n<N$. At $n=N$, it simply reads $z_N$.

Put $b=1-R>0$ and $m_b=\mathbb{E}{X}_0^b$. Homogeneity of the constant relative risk aversion utility gives $U(z_N)=U(z_n){\prod }_{j=n}^{N-1}{X}_j^b$. ThusAs before this is an extended-valued conditional expectation if the required moment is infinite and days remain.

Assume the riskless gross return $q=1+r>0$, as required by the positive-wealth model, and a finite moment $m_b=\mathbb{E}{X}_0^b$. DefineThis maximum exists: the control interval is compact and dominated convergence gives continuity, using the integrable bound $(X_0+q{)}^b\le X_0^b+q^b$ for $0<b\le 1$.

The terminal value is $V_N(w)=U(w)$. If $V_{n+1}(w)=a_{n+1}U(w)$, the Bellman equation and independence of the next return giveBackward induction therefore provesAny maximizing proportion can be used each day; allowing adapted choices cannot exceed the Bellman bound. For $\alpha =1$, $C_{\ast }=m_b$, exactly agreeing with part (b). If $m_b=\infty$ and at least one day remains, the allowed choice $\theta =1$ already gives infinite expected utility, so the value is infinite rather than a finite coefficient. This is the necessary moment qualification to the finite optimization formula.