Monte Carlo computation of optimal portfolios in complete markets

We introduce a method that relies exclusively on Monte Carlo simulation in order to compute numerically optimal portfolio values for utility maximization problems. Our method is quite general and only requires complete markets and knowledge of the dynamics of the security processes. It can be applied regardless of the number of factors and of whether the agent derives utility from intertemporal consumption, terminal wealth or both. We also perform some comparative statics analysis. Our comparative statics show that risk aversion has by far the greatest influence on the value of the optimal portfolio

Chebyshev type inequalities for Hilbert space operators

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$, $\alpha: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\left(\int_{T}\alpha(t) A_t d\mu(t)\right)\circ\left(\int_{T}\alpha(s) B_s d\mu(s)\right). \end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\times n$ matrices. Several applications are given as well.Comment: 18 pages, to appear in J. Math. Anal. Appl. (JMAA