Notes provided by the speakers for their talks.

Peter Carr, February 26, 2008

• ./BUpres.pdf
  
  
• ./skorohod3.pdf
  
  
• ./Cunypres.pdf (A pdf containing the lecture)
  
  
• ./HyperOptions.pdf (A pdf containing supplemental material, but in a preliminary - not paper - format.)
  
  

Gerardo Hernandez-del-Valle, March 5, 2008

See http://www.math.csi.cuny.edu/probability/Notebook/abstract-Hernandez-del-Valle.pdf

Rama Cont March 11, 2008

My talk will be based on Chapter 5 of our book

Financial Modelling with Jump Processes http://www.cmap.polytechnique.fr/~rama/Jumps/

and the paper by

Kallsen Tankov Characterization of dependence of multidimensional Lévy processes using Lévy copulas Journal of Multivariate Analysis Volume 97, Issue 7, August 2006, Pages 1551-1572

Xia Chen, March 18, 2008

I recommend two survey papers by Yimin Xiao on local non-determinism.

You may go to his webpage

http://www.stt.msu.edu/~xiaoyimi/Allpub.html

56. Y. Xiao, Strong local nondeterminism of Gaussian random fields and its applications. Asymptotic Theory in Probability and Statistics with Applications (T.-L. Lai, Q.-M. Shao and L. Qian, eds), pp. 136-176, Higher Education Press, Beijing, 2007.

40. Y. Xiao, Properties of local nondeterminism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV (2006), 157-193.

Your students may read the part of applications to local times.

Omri Sarig, March 25, 2008

"Generalized laws of large numbers for horocycle flows"

I'll try to be a self contained as time will allow. I will only assume that people know measure theoretic prob theory, and the definition of a Riemannian surface, especially the "hyperbolic plane".

There is a very accessible account of hyperbolic geometry in John Stillwell's book "geometry of surfaces" (Universitext), chapter 4,5 for people who have never seen these things before. I am NOT going to assume that the material there is known.

Maria Cristina Mariani, April 8th

Take a look at:

• P. Willmott, J.N. Dewynne, S.D. Howison, "Option Pricing: Mathematical Models and Computation." Oxford Financial Press, Oxford 1993. or
  
  
• Investment Science, D. Luenberger. or
  
  
• Options, Futures, and Other Derivatives, by Hull and R. N. Mantegna, H. E. Stanley, "An Introduction to Econophysics: Correlations and Complexity in Finance, " Cambridge University Press, Cambridge 1999.
  
  

William Cuckler, April 15th

Here is a reference that would be valuable for a student to read before the talk:

http://www.tcs.tifr.res.in/~jaikumar/Papers/EntropyAndCounting.pdf

This is a survey paper written by Jaikumar Radhakrishnan about using entropy ideas to solve combinatorial problems. The discussion about Bregman's Theorem on page 5 and 6 will be especially important for the talk. The following terms will also be important, and can be found in any graph theory textbook: Hamiltonian cycle, perfect matching, bipartite graph. However, I will remind the audience about these defintions.

Andrew Heunis, April 29

My goal is to spend the first two-thirds of the talk on background, dealing mainly with the Rockafellar-Moreau approach to convex optimization and the Yosida-Hewitt characterization of the dual of the space of essentially bounded measurable functions (the L^¥-space) I will be doing this because neither Rockafellar-Moreau nor Yosida-Hewitt seem to be all that widely known, and in fact have hardly made it into most of the textbooks. The background needed for this part of the talk will be quite straightforward, namely:

(1) basic measure theory, in particular the Radon-Nikodym theorem;

(2) basic functional analysis (really just understanding what is meant by a continuous linear functional on a normed vector space);

(3) basic convex analysis (the idea of convex sets and functions; it would also help if people were familiar with the idea of the conjugate of a convex function, but I'll explain that).

For the last third of the talk I'll be dealing with the problem of math. finance, building on the preceding discussion of Rockafellar-Moreau and Yosida-Hewitt. In fact, I will really talk about a stripped-down mini problem which keeps the most essential features of the math. finance problem. This is where probability theory comes in, and here the following background will be useful:

(4) the ideas of filtration, martingale, Brownian motion, and Ito integral.

Some background reading will definitely help for the ideas in (4) if people are not already familiar with these. A good reference, which will be ample for the needs of the talk, is Chapters 1 - 3 of the book

"Stochastic Differential Equations" (6th edition) by Bernt Oksendal (Springer Universitext Series).

These chapters should be reasonably accessible to people with a first year graduate course in probability.