Category Archives: Probability theory

Probability Theory Cookbook

I have been working on a probability theory cookbook which is intended to serve as a collection of important results in general probability theory. It can be used for a quick brush up, as a quick reference or cheat sheet, but not as primary tutorial material. This is a first draft version, but I will keep adding material

Download it here: probability cookbook v0.1

Probability Cookbook

Feedback is more than welcome!

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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 haveĀ \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 →

What is (not) the Markov property

Let (\Omega, \mathcal{F}, \{\mathcal{F}_k\}_k, \mathrm{P}) be a filtered probability space with a discrete filtration (although the results we are going to discuss hold for continuous random processes as well). We say that a random process \{X_k\}_k on (\Omega, \mathcal{F}, \{\mathcal{F}_k\}_k, \mathrm{P}) possesses the Markov property if

\mathrm{P}[X_{k+1} \in A\mid \mathcal{F}_{s}] = \mathrm{P}[X_{k+1} \in A\mid \mathcal{F}_k]

for all s\leq k. We see this often in the following form

\mathrm{P}[X_{k+1} \in A\mid X_s, X_{s-1},\ldots, X_0] = \mathrm{P}[X_{k+1} \in A\mid X_s]

or simply with with s=k. This is the Markov property.

The following is just wrong: For a sequence of sets \{B_k\}_{k} where B_k\in\mathcal{F}_k

\mathrm{P}[X_{k+1} \in A\mid X_s\in B_{s}, X_{s-1}\in B_{s-1},\ldots, X_0\in B_0] = \mathrm{P}[X_{k+1} \in A\mid X_s\in B_{s}]

A very easy and straightforward way to verify that this is false is to set B_{k}=\Omega, B_{k-2}=\Omega, \ldots, B_{0}=\Omega, that is, provide no information about X_{k}, X_{k-2}, \ldots, X_{0} and provide the information X_{k-1}=x, i.e., B_{k-1}=\{x\} which actually offers some information. Then, according to the wrong statement above, it would be

\mathrm{P}[X_{k+1} \in A\mid X_s\in \Omega, X_{k-1}=x, X_{k-2}\in \Omega, \ldots] = \mathrm{P}[X_{k+1} \in A].

Although, it should naturally be \mathrm{P}[X_{k+1} \in A\mid X_{k-2}=x]. That is, the information that X_{k-2}=x is completely expunged.

Reference: K.L. Chung, Green, Brown and Probability, World Scientific, 1995 [Chap. 5].

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.

Continue reading →

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].

Continue reading →

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