penalizing negative skewness by linking $U(\mu)$ and $U(\Sigma)$

Question

Consider $U_1(\mu,\Sigma)$ and $U_2(\mu,\Sigma)$, where $U_1(\mu, \cdot) = U_2(\mu, \cdot)$, $U_1(\cdot, \Sigma) = U_2(\cdot, \Sigma)$ such that

\begin{equation*} arg\inf\limits_{\mu \in U_1(\mu, \cdot), \Sigma \in U_1(\cdot, \Sigma)} \left(w^T \mu – \alpha w^T \Sigma w \right) \equiv arg\inf\limits_{\mu \in U_2(\mu, \cdot), \Sigma \in U_2(\cdot, \Sigma)} \left(w^T \mu – \alpha w^T \Sigma w\right) \end{equation*}

Solutions for inner problems are the same

$(\mu, \Sigma)$ from $U_1$ are positively related ($\Sigma \uparrow$ as $\mu \uparrow \Rightarrow$ positively skewed)

$(\mu, \Sigma)$ from $U_2$ are negatively related ($\Sigma \downarrow$ as $\mu \uparrow \Rightarrow$ negatively skewed)

Solutions for inner problem: low $\mu$, high $\Sigma$

\begin{equation} arg\inf\limits_{(\mu, \Sigma) \in U_1(\mu, \Sigma)} \left(w^T \mu – \alpha w^T \Sigma w \right) \geq arg\inf\limits_{(\mu, \Sigma) \in U_2(\mu, \Sigma)} \left(w^T \mu – \alpha w^T \Sigma w \right) \quad (1) \end{equation}

$\Rightarrow$ negative skewness is penalized by linking $U(\mu)$ and $U(\Sigma)$.

$\begin{align} \max\limits_w \left(w^T \mu – \alpha w^T \Sigma w\right) &\geq \max\limits_w\inf\limits_{(\mu, \Sigma) \in U(\mu, \Sigma)} \left(w^T \mu – \alpha w^T \Sigma w\right)\\ &\geq \max\limits_w\inf\limits_{\mu \in U(\mu), \Sigma \in U(\Sigma)} \left(w^T \mu – \alpha w^T \Sigma w\right)\quad (2) \end{align}$

Can anyone help me in proving the equation 1 and 2? here the uncertain set for mu(mean) could be ellipsoid and will change the problem in second order conic problem ......and the uncertain set for the sigma(variance matrix) could be a rectangular box which will change the problem in semi definite programming problem.also we can choose some others as i mention in the comments.

Answer 1

@amber -

As a general hint: try to solve a small-scale case first. Pick a two- or better three-asset $(\mu,\Sigma)$ and construct the objectives. Construct the "skewness tensor" (it's not a matrix, you can think of it like a "cube" or something - consult this book on how you can actually represent it as a matrix, or Fabozzi et al's textbook for an accessible discussion to the whole idea of introducing skewness tensors).

In your case, actually, it will be difficult to construct the feasible sets, since you don't seem to have a clear grasp of the optimization problem - there are many calls for clarifications. Try to spell them out explicitly.

Next, note that the skewness tensor alters both the feasible sets and the objective. It is easy to establish the monotonicity, once the tensor is explicitly introduced in both. To do so, you can even use the simple Karush-Kuhn-Tucker conditions.

Both inequalities are very easy to establish.

Good luck with your thesis.