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
Feedback is more than welcome!
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 and a mapping , we have
and , while, under additional conditions (which are typically met in finite-dimensional spaces), we have Continue reading →
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 →
Let 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 on possesses the Markov property if
for all . We see this often in the following form
or simply with with . This is the Markov property.
The following is just wrong: For a sequence of sets where
A very easy and straightforward way to verify that this is false is to set , that is, provide no information about and provide the information , i.e., which actually offers some information. Then, according to the wrong statement above, it would be
Although, it should naturally be . That is, the information that is completely expunged.
Reference: K.L. Chung, Green, Brown and Probability, World Scientific, 1995 [Chap. 5].
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 →