# Inner and Outer Semicontinuity of Multifunctions

Where in this post we introduce the notions of inner and outer semicontinuity for multi-valued functions using the Painlevé-Kuratowski notion of set-convergence which we introduced previously. We then provide a few useful characterisations of (semi)continuity for multi-valued mappings. We discuss the similarities and differences between these definitions and there counterparts for single-valued functions. Continue reading →

# On Set Convergence III

Where in this, third and final, post on the topic of set convergence we introduce the notion of Painleve-Kuratowski convergence of sequences of sets in topological spaces and we discuss about the unique properties of this notion of converges. We see that the K-limit of a constant sequence may not be equal to the elements of that sequence and many other interesting properties. Continue reading →

# On Set Convergence II

This post comes as a sequel of On Set Convergence I where we introduced some necessary notions that are useful for studying how sets converge in a topological space. Alongside, we provided two non-topological definitions of convergence, namely the limits inferior and superior. In this post we will delve a bit further into the topological properties of the inner and outer limits of sequences of sets. Continue reading →

# On Set Convergence I

We give the definitions of inner and outer limits for sequences of sets in topological and normed spaces and we provide some important facts on set convergence on topological and normed spaces. We juxtapose the notions of the limit superior and limit inferior for sequences of sets and we outline some facts regarding the Painlevé-Kuratowski convergence of set sequences. Continue reading →

# Linearisation bounds for smooth mutlivalued functions

How far is a function $f$ from its linearisation? Typically, one would assume that $f$ is twice continuously differentiable and use the following second-order version of the mean value theorem:

$f(y) = f(x) + \langle J f(x),y-x\rangle + \int_0^1 (1-\tau) (y-x)'\nabla^2 f(x+\tau (y-x))(y-x)\mathrm{d}\tau$

This is typically used in the context of linearisation of nonlinear dynamical systems as in [Sec. 2.5.1.3, 1]. The requirement that $f$ is twice continuously differentiable, can, however, be reduced to $f$ being continuously differentiable with Lipschitz gradient. 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].

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