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National Research University Higher School of EconomicsLaboratory of Stochastic Analysis and its ApplicationsMini-course "Topics in Applied Probability:Tempered Stable Distributions, Ad Placement, and Turing's Formula"

# Mini-course "Topics in Applied Probability:Tempered Stable Distributions, Ad Placement, and Turing's Formula"

6-17 October 2014 Assistand Proffesor UNC Charllote Michael Grabchak gave a lectures on

"Tempered Stable Distributions, Ad Placement, and Turing's Formula".

Lecture 1: Do financial returns have a finite or infinite variance? A  paradox and an explanation.
Abstract: One of the major points of contention in studying and modeling financial returns is whether or not the variance of the returns is finite or infinite. The available empirical evidence can be, and has been, interpreted in more than one way. The apparent paradox, which has puzzled many a researcher, is that the tails appear to become less heavy for less frequent (e.g. monthly) returns than for more frequent (e.g. daily) returns, a phenomenon not easily explainable by the standard models. We provide an explanation of this paradox and show that, for financial returns, natural families of models are those with tempered heavy tails. These models can generate observations that appear heavy tailed for a wide range of aggregation levels before becoming clearly light tailed at even larger aggregation scales.

Lecture 2: Tempered stable distributions and processes.
Abstract: In this lecture we introduce the class of tempered stable distributions, which is a large and flexible class of models with tempered heavy tails. We discuss many properties of these distributions and their associated Levy processes. In particular, we analyze the tails of these distributions and we characterize when their associated Levy processes are close to stable Levy process and when they are close to Brownian motion.

Lecture 3, Part 1: Does value-at-risk encourage diversification when  losses follow a tempered stable or more general Levy  processes?
Abstract: Value-at-risk (VaR) is one of the most popular measures of financial risk even though it is not a coherent risk measure and may discourage diversification. In this lecture we show that portfolio selection based on VaR always encourages diversification when losses follow a certain very large class of tempered stable Levy processes. Further, we also give sufficient conditions for more general Levy processes.

Lecture 3, Part 2: How do tempered stable distributions appear in  applications?
Abstract: In this lecture we discuss limits theorems for two models of summation of iid random variables with a parameter. The parameter determines how heavy the tails of the distribution are. The first model explains how tempered stable distributions arise in applications. Our discussion is motivated by the problem of modeling mobility, which is important in applications to computer science, anthropology, and ecology.

Lecture 4: Applications of stochastic knapsack and multi-armed bandit  problems to internet ad placement.
Abstract: One of the most important questions for internet companies is how to choose which ad to display in order to maximize their revenue. In this lecture we discuss two situations. In the first we must guarantee a certain number of clicks by a prespecified time before we get paid. We formulate this as a stochastic knapsack problem and give several strategies. The second is the situation where we know nothing about the ads. We must find a balance between gaining new information and exploiting our knowledge. This is formulated as a multiarmed bandit problem and we give several strategies for doing this.

Lecture 5: Turing's formula and its statistical implications.
Abstract: Turing's formula is a nonparametric estimator for seeing an observation of a previously unobserved type. It can be thought of as an estimator of the tail. We will discuss this formula and show how the ideas behind it can be used for the estimation of entropy and related quantities. Applications to ecology and linguistics will be stressed.

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