For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and generalized Bessel meanders. We discuss the limits of the application of this method for other diffusion processes.

This article is a survey of results concerning an inequality, which may be seen as a versatile tool to solve problems in the domain of Applied Probability. The inequality, which we call BRS-inequality, gives a convenient upper bound for the expected maximum number of non-negative random variables one can sum up without exceeding a given upper bound $s>0$. One valuable property of the BRS-inequality is that it is valid without any hypothesis about independence of the random variables. Another welcome feature is that, once one sees that one can use it in a given problem, its application is often straightforward or not very involved.

This survey is focussed, and we hope that it is pleasant and inspiring to read. Focus is easy to achieve, given that the BRS-inequality and its most useful versions can be displayed in five Theorems and their proofs. We try to present these in an appealing way. The objective to be inspiring is harder, and the best we can think of is offering a variety of applications. Our examples include comparisons between sums of i.i.d. versus non-identically distributed and/or dependent random variables, problems of condensing point processes, awkward processes, monotone subsequence problems, knapsack problems, online algorithms, tiling policies, Borel-Cantelli type problems, up to applications in the theory of resource dependent branching processes.

Apart from our wish to present the inequality in an organised way, the motivation for this survey is the hope that interested readers may see potential of the inequality for their own problems.

This is a survey of results on universal algorithms for classification and prediction of stationary processes. The classification problems include discovering the order of a k-step Markov chain, determining memory words in finitarily Markovian processes and estimating the entropy of an unknown process. The prediction problems cover both discrete and real valued processes in a variety of situations. Both the forward and the backward prediction problems are discussed with the emphasis being on pointwise results. This survey is just a teaser. The purpose is merely to call attention to results on classification and prediction. We will refer the interested reader to the sources. Throughout the paper we will give illuminating examples.

We present an overview of the broad class of financial models in which the prices of assets are Lévy-Ito processes driven by an n-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is associated with an n-dimensional Lévy process. Each model consists of a pricing kernel, a money market account, and one or more risky assets. We show how the excess rate of return above the interest rate can be calculated for risky assets in such models, thus showing the relationship between risk and return when asset prices have jumps. The framework is applied to a variety of asset classes, allowing one to construct new models as well as interesting generalizations of familiar models.

We are going to discuss recent progress on many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, Lévy processes, infinitely divisible processes and fields and Lévy-driven stochastic differential equations. Essential notions and results towards practical implementation are outlined, and summaries of simulation recipes are provided throughout along with numerical illustrations. Some future research directions are highlighted.

An important feature of a Poisson limit theorem in [4] is the absence of the traditional assumption (A.2). The purpose of this addendum is to explain why assumption (A.2) is not required, and compare the assumptions of the Poisson limit theorem in [4] with traditional ones. We present details of the argument behind Theorem 3 in [4].