Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 30K (Stochastic Financial Models)

For $X={\theta }^{T}Z$,The objective has gradient $b-V\theta$ and Hessian $-V$, which is negative definite. Its unique maximizer is therefore

Put $D=b^T{V}^{-1}b$ and decompose any $\theta$ in the $V$-inner product asThenFor fixed $\lambda$, a nonzero $\eta$ leaves the mean unchanged and strictly raises variance, so strict monotonicity of $F$ excludes it from a maximizer. If $D>0$ and $\lambda <0$, the zero portfolio has a larger mean and smaller variance, so this is also impossible. Hence every maximizer isIf $b=0$, the unique variance-minimizing maximizer is ${\theta }^{\ast }=0$, which is the same conclusion with $\lambda =0$.

Assume first that $D=b^T{V}^{-1}b>0$. Since $X_M={\theta }_M^{T}Z$,and for $Y={\varphi }^{T}Z$,The least-squares affine-regression coefficients are thereforeThe minimum iswhereThis matrix is symmetric and positive semidefinite. If $b=0$, then $X_M=0$, one may take $\alpha =0$ and arbitrary $\beta$, and the minimum is ${\varphi }^{T}V\varphi$, corresponding to $Q=V$.