Aureli Alabert

Some Remarks on the Conditional Independence and the Markov Property

Recently, there has been some work on stochastic differential equations with boundary conditions (cf., for instance, [1, 2, 7, 8, 9]). This has been possible thanks to the development of the extended stochastic calculus for anticipating processes (see, for example, Nualart-Pardoux [6]), since the solutions to stochastic boundary problems are not in general adapted to the driving Brownian motion. In these papers, two types of problems have been considered. First to prove the existence and uniqueness of a solution for different kinds of equations, and secondly, to study the Markov properties of the solution.

No-Free-Lunch theorems in the continuum

No-Free-Lunch Theorems state, roughly speaking, that the performance of all search algorithms is the same when averaged over all possible objective functions. This fact was precisely formulated for the first time in a now famous paper by Wolpert and Macready, and then subsequently refined and extended by several authors, usually in the context of a set of functions with finite domain and codomain. Recently, Auger and Teytaud have studied the situation for continuum domains. In this paper we provide another approach, which is simpler, requires less assumptions, relates the discrete and continuum cases, and we believe that clarifies the role of the cardinality and structure of the domain.

A second-order Stratonovich differential equation with boundary conditions

In this paper we show that the solution of a second-order stochastic differential equation with diffusion coefficient and boundary conditions X0 = 0 and X1 = 1 is a 2-Markov field if and only if the drift is a linear function. The proof is based on the method of change of probability and makes use of the techniques of Malliavin calculus.

Classifying the typefaces of the Gutenberg 42-line bible

We have measured the dissimilarities among several printed characters of a single page in the Gutenberg 42-line bible, and we prove statistically the existence of several different matrices from which the metal types were constructed. This is in contrast with the prevailing theory, which states that only one matrix per character was used in the printing process of Gutenberg’s greatest work. The main mathematical tool for this purpose is cluster analysis, combined with a statistical test for outliers. We carry out the research with two letters,

A Conditional Independence Property for the Solution of a Linear Stochastic Differential Equation with Lateral Conditions

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Markov Field Property of Stochastic Differential Equations

and

Exit Times from Equilateral Triangles

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