Lower semicontinuous risk measure

Question

I am looking for some risk measures that hold the lower semi-continuous property.

I am not sure whether Expected Shortfall is a such a measure or not. Can anyone give me some help？

Thanks.

Answer 1

The literature is replete with relevant examples. The expected shortfall is, indeed, a lower semi-continuous risk measure on $\mathcal{L}_p(\Omega, \mathcal{F}, \mathrm{P})$, $p\in[1,+\infty]$.

Recall that $\operatorname{AV@R}_\alpha[Z]$ is defined as

$$ \operatorname{AV@R}_\alpha[Z] = \inf_{t\in\mathbb{R}}\{t + \mathbb{E}[Z-t]_+ \}, $$

that is, it is the inf-projection of the function

$$ \phi(t)=t + \mathbb{E}[Z-t]_+, $$

which is proper, convex and lower-semicontinuous. Average value-at-risk (aka expected shortfall) is a coherent risk measure.

There are many such risk measures you can find in: A. Shapiro, D. Dentcheva and A.Ruszczynski, Lectures on stochastic programming: modeling and theory, MPS-SIAM series on optimization, 2009.

Another such risk measure is the very simple mean-variance risk measure

$$ mv[Z] = \mathbb{E}[Z] + c\mathrm{Var}[Z], $$

which, however, fails to be coherent (it is not positively homogeneous).

Yet another example is the Mean-Upper-Semideviation of order $p$ which is well-defined and finite for $Z\in\mathcal{L}_p(\Omega, \mathcal{F}, \mathrm{P})$. This is defined as

$$ \mathrm{MUS}_p[Z] = \mathbb{E}[Z] + c \left( \mathbb{E}[Z-\mathbb{E}[Z]]_+^p \right)^{1/p}. $$

This risk measure is coherent.

All well-established risk measures I know of are lower-semicontinuous.