Aspects of Mathematical Finance
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Probability Seminar
PROBABILITY SEMINAR The Probability Seminar takes place Fridays from 11:00 to 12:00 noon . Unless indicated otherwise, talks in Spring 2008 are held in Room 903, 9 th Floor in the School of Social Work (Department of Statistics), 1255 Amsterdam Avenue between 121 st - 122 nd streets. Half an hour before each talk, we usually meet for coffee and tea in the Statistics Departmental Lounge on the 10 th floor of that building. For information on how to get to Columbia University and find the Department of Statistics, please check our campus map at http://www.columbia.edu/about_columbia/map/ ----------------------------------------------------------------------------------- SPECIAL MATHEMATICAL FINANCE SEMINAR Tuesday, January 22 nd , 1:10-2:25 in Rm. 307 Math Professor Alexander CHERNY, Moscow State University & Bloomberg Combining Factor Risks in Risk Measurement Schemes Abstract: This paper is related to the measurement of the risk of a portfolio, the risk being driven by multiple factors. The problem under consideration is proper combining of the risks brought by individual factors, with a view of dependence between them. The practical importance of the problem stems from the fact that one can effectively estimate the joint law of a portfolio and each single factor but, due to the lack of data, cannot estimate the joint law of portfolio and all factors. We propose a methodology based on: the notion of factor risks independently introduced in the industry by the RiskData and in the literature by Cherny and Madan; selecting a minimal multifactor risk profile matching single-factor risk profiles. Within this approach we provide a solution in the model, where the distribution of risk factors is a Gaussian copula. -
Topics in Stochastic Analysis Informal Seminar
Topics in Stochastic Analysis Informal Seminar Spring 2008, Tuesdays and Thursdays, 1:10-2:25pm, Room 307 Math (unless in- dicated otherwise), Columbia University, 2990 Broadway, New York, NY 22 Jan (Tue) Professor Alexander CHERNY (Moscow State University & Bloomberg) ”Combinging Factor Risks in Risk Measurement Schemes” Joint work with Raphael Douady (RiskData) and Stanislav Molchanov (UNCC Charlotte) Abstract: This paper is related to the measurement of the risk of a portfolio, the risk being driven by multiple factors. The problem under consideration is proper combining of the risks brought by individual factors, with a view of dependence between them. The practical importance of the problem stems from the fact that one can effectively estimate the joint law of a portfolio and each single factor but, due to the lack of data, cannot estimate the joint law of portfolio and all factors. We propose a methodology based on: the notion of factor risks independently intro- duced in the industry by the RiskData and in the literature by Cherny and Madan; selecting a minimal multifactor risk profile matching single-factor risk profiles. Within this approach we provide a solution in the model, where the distribution of risk factors is a Gaussian copula. The solution is computationally feasible as it is reduced to inverting a matrix of a reasonable dimension. 5 Feb (Tue) 1:10-2:25, 903 SSW Dr. Emmanuel SCHERTZER (Courant, NYU) 1 ”The Dynamical Web and the Brownian Net” Abstract: The discrete web is an infinite collection of coalescing random paths. This model arises in various contexts, such as the voter model in statistical me- chanics and drainage networks in hydrology. -
Combinatorial Stochastic Processes
University of California, Berkeley From the SelectedWorks of Jim Pitman January 2006 Combinatorial Stochastic Processes Contact Start Your Own Notify Me Author SelectedWorks of New Work Available at: http://works.bepress.com/jim_pitman/1 Combinatorial Stochastic Processes Jim Pitman 1 2 Jim Pitman Combinatorial stochastic processes Preliminaries 0.0. Preface This is a collection of expository articles about various topics at the interface between enumerative combinatorics and stochastic processes. These articles ex- pand on a course of lectures given at the Ecole´ d'Et´ ´e de Probabilit´es de St. Flour in July 2002. The articles are called 'chapters' and numbered according to the order of these chapters in a printed volume to appear in Springer Lecture Notes in Mathematics. Each chapter is fairly self-contained, so readers with ad- equate background can start reading any chapter, with occasional consultation of earlier chapters as necessary. Following this Chapter 0, there are 10 chapters, each divided into sections. Most sections conclude with some Exercises. Those for which I don't know solutions are called Problems. Acknowledgments Much of the research reviewed here was done jointly with David Aldous. Much credit is due to him, especially for the big picture of contin- uum approximations to large combinatorial structures. Thanks also to my other collaborators in this work, especially Jean Bertoin, Michael Camarri, Steven Evans, Sasha Gnedin, Ben Hansen, Jacques Neveu, Mihael Perman, Ravi Sheth, Marc Yor and Jim Young. A preliminary version of these notes was developed in Spring 2002 with the help of a dedicated class of ten graduate students in Berkeley: Noam Berger, David Blei, Rahul Jain, Serban Nacu, Gabor Pete, Lea Popovic, Alan Hammond, Antar Bandyopadhyay, Manjunath Krishnapur and Gregory Miermont. -
A Guide to Brownian Motion and Related Stochastic Processes
Vol. 0 (0000) A guide to Brownian motion and related stochastic processes Jim Pitman and Marc Yor Dept. Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA e-mail: [email protected] Abstract: This is a guide to the mathematical theory of Brownian mo- tion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical the- ory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley [43] and Durrett [106], and who wants a broader perspective on the theory of Brow- nian motion and related stochastic processes than can be found in these texts. Keywords and phrases: Markov process, random walk, martingale, Gaus- sian process, L´evyprocess, diffusion. AMS 2000 subject classifications: Primary 60J65. Contents 1 Introduction . .3 1.1 History . .3 1.2 Definitions . .4 2 BM as a limit of random walks . .5 3 BM as a Gaussian process . .7 3.1 Elementary transformations . .8 3.2 Quadratic variation . .8 3.3 Paley-Wiener integrals . .8 3.4 Brownian bridges . 10 3.5 Fine structure of Brownian paths . 10 3.6 Generalizations . 10 3.6.1 Fractional BM . 10 3.6.2 L´evy's BM . 11 3.6.3 Brownian sheets . 11 3.7 References . 11 4 BM as a Markov process . 12 4.1 Markov processes and their semigroups .