# Stochastic constraints

Where in this post we discuss various types of constraints on stochastic optimal control problems starting from the classic uniform constraints to chance/probabilistic constraints and more elaborate risk constraints. We point out some caveats and propose a few remedies. Continue reading →

# Conditional risk mappings: robust representations

Coherent risk measures can be written as the support function of a set of random variables or as a worst-case expectation over a set of probability measures. This is the so-called dual or robust representation of risk measures. These representations extend to the conditional variants of risk measures. In this post we revisit the construction and properties of conditional risk mappings and derive the dual representation of compositions of conditional risk mappings. Continue reading →

# Interchangeability of infimum in risk measures

In this post we discuss the interchangeability of the infimum with (monotone) risk measures in finite probability spaces. In particular, we show that under the common monotonicity assumption (which is satisfied by all well-behaving risk measures), for a risk measure $\rho:\mathbb{R}^n\to\mathbb{R}$ and a mapping $f:\mathbb{R}^m\to\mathbb{R}^n$, we have

\begin{aligned} \rho\left(\inf_x f(x)\right) = \inf_x \rho(f(x)) \end{aligned}

and $\mathbf{argmin}_x f(x) \subseteq \mathbf{argmin}_x \rho(f(x))$, while, under additional conditions (which are typically met in finite-dimensional spaces), we have $\mathbf{argmin}_x f(x) = \mathbf{argmin}_x \rho(f(x))$ Continue reading →

# What is a non-law-invariant risk measure?

Someone asked me recently, what is a risk-measure which is not law-invariant? Admittedly, the law invariance property seems so natural that any meaningful risk measure should have this property. Of course we can struggle to construct risk measures which are not law invariant, but are there any naturally occurring risk measures which do not have this property? The answer is yes, but let us take things from the start. Continue reading →

# Zero risk

What does it mean for a random variable to exhibit zero risk? It of course depends on the risk measure we are using to quantify it. Continue reading →

# Average value-at-risk of simple quadratic form

The average value-at-risk of a quadratic form $y'y$, where $y\sim \mathcal{N}(0_n,I_n)$ is given by a particularly complex closed-loop formula which I’ll describe below.

# Strict monotonicity of expected shortfall

The expected shortfall, also known as average value-at-risk or conditional value-at-risk, is a coherent risk measure defined as

$\mathrm{AV@R}_{\alpha}[Z]=\inf_{t\in\mathbb{R}} \{t+\alpha^{-1}\mathbb{E}[Z-t]_+\}$

for $Z\in\mathcal{Z}:=\mathcal{L}_p(\Omega,\mathcal{F},\mathrm{P})$ for some $p\in[1,+\infty]$.

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