Thou shalt not say "at random" in vain: Bertrand's paradox exposed

We review the well known Bertrand paradoxes, and we first maintain that they do not point to any probabilistic inconsistency, but rather to the risks incurred with a careless use of the locution "at random". We claim then that these paradoxes spring up also in the discussion of the celebrated Buffon's needle problem, and that they are essentially related to the definition of (geometrical) probabilities on "uncountably" infinite sets. A few empirical remarks are finally added to underline the difference between "passive" and "active" randomness, and the prospects of any experimental decisionComment: 17 pages, 4 figures. Added: Appendix A; References 7, 8, 10; Modified: Abstract; Section 4; a few sentences elsewher

A semi-Markov model with memory for price changes

We study the high frequency price dynamics of traded stocks by a model of returns using a semi-Markov approach. More precisely we assume that the intraday returns are described by a discrete time homogeneous semi-Markov which depends also on a memory index. The index is introduced to take into account periods of high and low volatility in the market. First of all we derive the equations governing the process and then theoretical results have been compared with empirical findings from real data. In particular we analyzed high frequency data from the Italian stock market from first of January 2007 until end of December 2010

Selfdecomposability and selfsimilarity: a concise primer

We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Levy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.Comment: 24 pages, 3 figures; corrected misprint in the title; redactional modifications required by the referee; added references from [16] to [28];. Accepted and in press on Physica

Controlled quantum evolutions and transitions

We study the nonstationary solutions of Fokker-Planck equations associated to either stationary or nonstationary quantum states. In particular we discuss the stationary states of quantum systems with singular velocity fields. We introduce a technique that allows to realize arbitrary evolutions ruled by these equations, to account for controlled quantum transitions. The method is illustrated by presenting the detailed treatment of the transition probabilities and of the controlling time-dependent potentials associated to the transitions between the stationary, the coherent, and the squeezed states of the harmonic oscillator. Possible extensions to anharmonic systems and mixed states are briefly discussed and assessed.Comment: 24 pages, 4 figure

Measures of lexical distance between languages

The idea of measuring distance between languages seems to have its roots in the work of the French explorer Dumont D'Urville \cite{Urv}. He collected comparative words lists of various languages during his voyages aboard the Astrolabe from 1826 to 1829 and, in his work about the geographical division of the Pacific, he proposed a method to measure the degree of relation among languages. The method used by modern glottochronology, developed by Morris Swadesh in the 1950s, measures distances from the percentage of shared cognates, which are words with a common historical origin. Recently, we proposed a new automated method which uses normalized Levenshtein distance among words with the same meaning and averages on the words contained in a list. Recently another group of scholars \cite{Bak, Hol} proposed a refined of our definition including a second normalization. In this paper we compare the information content of our definition with the refined version in order to decide which of the two can be applied with greater success to resolve relationships among languages

A semi-Markov model for price returns

We study the high frequency price dynamics of traded stocks by a model of returns using a semi-Markov approach. More precisely we assume that the intraday return are described by a discrete time homogeneous semi-Markov process and the overnight returns are modeled by a Markov chain. Based on this assumptions we derived the equations for the first passage time distribution and the volatility autocorreletion function. Theoretical results have been compared with empirical findings from real data. In particular we analyzed high frequency data from the Italian stock market from first of January 2007 until end of December 2010. The semi-Markov hypothesis is also tested through a nonparametric test of hypothesis

Observability of Market Daily Volatility

We study the price dynamics of 65 stocks from the Dow Jones Composite Average from 1973 until 2014. We show that it is possible to define a Daily Market Volatility $\sigma(t)$ which is directly observable from data. This quantity is usually indirectly defined by $r(t)=\sigma(t) \omega(t)$ where the $r(t)$ are the daily returns of the market index and the $\omega(t)$ are i.i.d. random variables with vanishing average and unitary variance. The relation $r(t)=\sigma(t) \omega(t)$ alone is unable to give an operative definition of the index volatility, which remains unobservable. On the contrary, we show that using the whole information available in the market, the index volatility can be operatively defined and detected

Spot foreign exchange market and time series

We investigate high frequency price dynamics in foreign exchange market using data from Reuters information system (the dataset has been provided to us by Ols en & Associates). In our analysis we show that a na\"ive approach to the definition of price (for example using the spot midprice) may lead to wrong conclusions on price behavior as for example the presence of short term covariances for returns. For this purpose we introduce an algorithm which only uses the non arbitrage principle to estimate real prices from the spot ones. The new definition leads to returns which are i.i.d. variables and therefore are not affected by spurious correlations. Furthermore, any apparent information (defined by using Shannon entropy) contained in the data disappears